    Internet Research Task Force (IRTF)                   S. Smyshlyaev, Ed.
    Request for Comments: 8645                                     CryptoPro
    Category: Informational                                      August 2019
    ISSN: 2070-1721

                    Re-keying Mechanisms for Symmetric Keys

    Abstract

       A certain maximum amount of data can be safely encrypted when
       encryption is performed under a single key.  This amount is called
       the "key lifetime".  This specification describes a variety of
       methods for increasing the lifetime of symmetric keys.  It provides
       two types of re-keying mechanisms based on hash functions and block
       ciphers that can be used with modes of operations such as CTR, GCM,
       CBC, CFB, and OMAC.

       This document is a product of the Crypto Forum Research Group (CFRG)
       in the IRTF.

    Status of This Memo

       This document is not an Internet Standards Track specification; it is
       published for informational purposes.

       This document is a product of the Internet Research Task Force
       (IRTF).  The IRTF publishes the results of Internet-related research
       and development activities.  These results might not be suitable for
       deployment.  This RFC represents the consensus of the Crypto Forum
       Research Group of the Internet Research Task Force (IRTF).  Documents
       approved for publication by the IRSG are not candidates for any level
       of Internet Standard; see Section 2 of RFC 7841.

       Information about the current status of this document, any errata,
       and how to provide feedback on it may be obtained at
       https://www.rfc-editor.org/info/rfc8645.

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    Copyright Notice

       Copyright (c) 2019 IETF Trust and the persons identified as the
       document authors.  All rights reserved.

       This document is subject to BCP 78 and the IETF Trust's Legal
       Provisions Relating to IETF Documents
       (https://trustee.ietf.org/license-info) in effect on the date of
       publication of this document.  Please review these documents
       carefully, as they describe your rights and restrictions with respect
       to this document.

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    Table of Contents

       1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
       2.  Conventions Used in This Document . . . . . . . . . . . . . .   7
       3.  Basic Terms and Definitions . . . . . . . . . . . . . . . . .   7
       4.  Choosing Constructions and Security Parameters  . . . . . . .   9
       5.  External Re-keying Mechanisms . . . . . . . . . . . . . . . .  11
         5.1.  Methods of Key Lifetime Control . . . . . . . . . . . . .  14
         5.2.  Parallel Constructions  . . . . . . . . . . . . . . . . .  14
           5.2.1.  Parallel Construction Based on a KDF on a Block
                   Cipher  . . . . . . . . . . . . . . . . . . . . . . .  15
           5.2.2.  Parallel Construction Based on a KDF on a Hash
                   Function  . . . . . . . . . . . . . . . . . . . . . .  16
           5.2.3.  Tree-Based Construction . . . . . . . . . . . . . . .  16
         5.3.  Serial Constructions  . . . . . . . . . . . . . . . . . .  17
           5.3.1.  Serial Construction Based on a KDF on a Block Cipher   19
           5.3.2.  Serial Construction Based on a KDF on a Hash Function  19
         5.4.  Using Additional Entropy during Re-keying . . . . . . . .  19
       6.  Internal Re-keying Mechanisms . . . . . . . . . . . . . . . .  20
         6.1.  Methods of Key Lifetime Control . . . . . . . . . . . . .  22
         6.2.  Constructions that Do Not Require a Master Key  . . . . .  23
           6.2.1.  ACPKM Re-keying Mechanisms  . . . . . . . . . . . . .  23
           6.2.2.  CTR-ACPKM Encryption Mode . . . . . . . . . . . . . .  25
           6.2.3.  GCM-ACPKM Authenticated Encryption Mode . . . . . . .  26
         6.3.  Constructions that Require a Master Key . . . . . . . . .  29
           6.3.1.  ACPKM-Master Key Derivation from the Master Key . . .  29
           6.3.2.  CTR-ACPKM-Master Encryption Mode  . . . . . . . . . .  31
           6.3.3.  GCM-ACPKM-Master Authenticated Encryption Mode  . . .  33
           6.3.4.  CBC-ACPKM-Master Encryption Mode  . . . . . . . . . .  37
           6.3.5.  CFB-ACPKM-Master Encryption Mode  . . . . . . . . . .  39
           6.3.6.  OMAC-ACPKM-Master Authentication Mode . . . . . . . .  40
       7.  Joint Usage of External and Internal Re-keying  . . . . . . .  42
       8.  Security Considerations . . . . . . . . . . . . . . . . . . .  43
       9.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  43
       10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  44
         10.1.  Normative References . . . . . . . . . . . . . . . . . .  44
         10.2.  Informative References . . . . . . . . . . . . . . . . .  45
       Appendix A.  Test Examples  . . . . . . . . . . . . . . . . . . .  48
         A.1.  Test Examples for External Re-keying  . . . . . . . . . .  48
           A.1.1.  External Re-keying with a Parallel Construction . . .  48
           A.1.2.  External Re-keying with a Serial Construction . . . .  49
         A.2.  Test Examples for Internal Re-keying  . . . . . . . . . .  52
           A.2.1.  Internal Re-keying Mechanisms that Do Not
                   Require a Master Key  . . . . . . . . . . . . . . . .  52
           A.2.2.  Internal Re-keying Mechanisms with a Master Key . . .  56
       Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .  69
       Contributors  . . . . . . . . . . . . . . . . . . . . . . . . . .  69
       Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  69

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    1.  Introduction

       A certain maximum amount of data can be safely encrypted when
       encryption is performed under a single key.  Hereinafter, this amount
       will be referred to as the "key lifetime".  The need for such a
       limitation is dictated by the following methods of cryptanalysis:

       1.  Methods based on the combinatorial properties of the used block
           cipher mode of operation

              These methods do not depend on the underlying block cipher.
              Common mode restrictions derived from such methods are of
              order 2^{n/2}, where n is a block size defined in Section 3.
              [Sweet32] includes an example of an attack that is based on
              such methods.

       2.  Methods based on side-channel analysis issues

              In most cases, these methods do not depend on the used
              encryption modes and weakly depend on the used cipher
              features.  Limitations resulting from these considerations are
              usually the most restrictive ones.  [TEMPEST] is an example of
              an attack that is based on such methods.

       3.  Methods based on the properties of the used block cipher

              The most common methods of this type are linear and
              differential cryptanalysis [LDC].  In most cases, these
              methods do not depend on the used modes of operation.  In the
              case of secure block ciphers, bounds resulting from such
              methods are roughly the same as the natural bounds of 2^n and
              are dominated by the other bounds above.  Therefore, they can
              be excluded from the considerations here.

       As a result, it is important to replace a key when the total size of
       the processed plaintext under that key approaches the lifetime
       limitation.  A specific value of the key lifetime should be
       determined in accordance with some safety margin for protocol
       security and the methods outlined above.

       Suppose L is a key lifetime limitation in some protocol P.  For
       simplicity, assume that all messages have the same length m.  Hence,
       the number of messages q that can be processed with a single key K
       should be such that m * q <= L.  This can be depicted graphically as
       a rectangle with sides m and q enclosed by area L (see Figure 1).

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                          +------------------------+
                          |                      L |
                          | +--------m---------+   |
                          | |==================|   |
                          | |==================|   |
                          | q==================|   |       m * q <= L
                          | |==================|   |
                          | |==================|   |
                          | +------------------+   |
                          +------------------------+

             Figure 1: Graphic Display of the Key Lifetime Limitation

       In practice, the amount of data that corresponds to limitation L may
       not be enough.  The simplest and obvious solution in this situation
       is a regular renegotiation of an initial key after processing this
       threshold amount of data L.  However, this reduces the total
       performance, since it usually entails termination of application data
       transmission, additional service messages, the use of a random number
       generator, and many other additional calculations, including
       resource-intensive public key cryptography.

       For protocols based on block ciphers or stream ciphers, a more
       efficient way to increase the key lifetime is to use various
       re-keying mechanisms.  This specification considers re-keying
       mechanisms for block ciphers only; re-keying mechanisms typical for
       stream ciphers (e.g., [Pietrzak2009], [FPS2012]) are beyond the scope
       of this document.

       Re-keying mechanisms can be applied at the different protocol levels:
       the block cipher level (this approach is known as fresh re-keying and
       is described, for instance, in [FRESHREKEYING]; the block cipher mode
       of operation level (see Section 6); and the protocol level above the
       block cipher mode of operation (see Section 5).  The usage of the
       first approach is highly inefficient due to the key changing after
       each message block is processed.  Moreover, fresh re-keying
       mechanisms can change the block cipher internal structure and,
       consequently, can require an additional security analysis for each
       particular block cipher.  As a result, this approach depends on
       particular primitive properties and cannot be applied to any
       arbitrary block cipher without additional security analysis.
       Therefore, fresh re-keying mechanisms go beyond the scope of this
       document.

       Thus, this document contains the list of recommended re-keying
       mechanisms that can be used in the symmetric encryption schemes based
       on the block ciphers.  These mechanisms are independent from the

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       particular block cipher specification, and their security properties
       rely only on the standard block cipher security assumption.

       This specification presents two basic approaches to extending the
       lifetime of a key while avoiding renegotiation, which were introduced
       in [AAOS2017]:

       1.  External re-keying

          External re-keying is performed by a protocol, and it is
          independent of the underlying block cipher and the mode of
          operation.  External re-keying can use parallel and serial
          constructions.  In the parallel case, data processing keys K^1,
          K^2, ... are generated directly from the initial key K
          independently of each other.  In the serial case, every data-
          processing key depends on the state that is updated after the
          generation of each new data-processing key.

          As a generalization of external parallel re-keying, an external
          tree-based mechanism can be considered.  It is specified in
          Section 5.2.3 and can be viewed as the tree generalization in
          [GGM].  Similar constructions are used in the one-way tree
          mechanism ([OWT]) and [AESDUKPT] standard.

       2.  Internal re-keying

          Internal re-keying is built into the mode, and it depends heavily
          on the properties of the mode of operation and the block size.

       The re-keying approaches extend the key lifetime for a single initial
       key by allowing the leakages to be limited (via side channels) and by
       improving the combinatorial properties of the used block cipher mode
       of operation.

       In practical applications, re-keying can be useful for protocols that
       need to operate in hostile environments or under restricted resource
       conditions (e.g., those that require lightweight cryptography, where
       ciphers have a small block size that imposes strict combinatorial
       limitations).  Moreover, mechanisms that use external or internal
       re-keying may provide some protection against possible future attacks
       (by limiting the number of plaintext-ciphertext pairs that an
       adversary can collect) and some properties of forward or backward
       security (meaning that past or future data-processing keys remain
       secure even if the current key is compromised; see [AbBell] for more
       details).  External or internal re-keying can be used in network
       protocols as well as in the systems for data-at-rest encryption.

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       Depending on the concrete protocol characteristics, there might be
       situations in which both external and internal re-keying mechanisms
       (see Section 7) can be applied.  For example, a similar approach was
       used in Taha's tree construction (see [TAHA]).

       Note that there are key-updating (key regression) algorithms (e.g.,
       [FKK2005] and [KMNT2003]) that are called "re-keying" as well, but
       they pursue goals other than increasing the key lifetime.  Therefore,
       key regression algorithms are excluded from the considerations here.

       This document represents the consensus of the Crypto Forum Research
       Group (CFRG).

    2.  Conventions Used in This Document

       The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
       "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
       "OPTIONAL" in this document are to be interpreted as described in
       BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
       capitals, as shown here.

    3.  Basic Terms and Definitions

       This document uses the following terms and definitions for the sets
       and operations on the elements of these sets:

       V*      the set of all bit strings of a finite length (hereinafter
               referred to as strings), including the empty string;

       V_s     the set of all bit strings of length s, where s is a
               non-negative integer;

       |X|     the bit length of the bit string X;

       A | B   the concatenation of strings A and B both belonging to V*,
               i.e., a string in V_{|A|+|B|}, where the left substring in
               V_|A| is equal to A and the right substring in V_|B| is equal
               to B;

       (xor)   the exclusive-or of two bit strings of the same length;

       Z_{2^n} the ring of residues modulo 2^n;

       Int_s: V_s -> Z_{2^s}
               the transformation that maps the string a = (a_s, ... , a_1)
               in V_s into the integer Int_s(a) = 2^{s-1} * a_s + ... + 2 *
               a_2 + a_1 (the interpretation of the binary string as an
               integer);

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       Vec_s: Z_{2^s} -> V_s
               the transformation inverse to the mapping Int_s (the
               interpretation of an integer as a binary string);

       MSB_i: V_s -> V_i
               the transformation that maps the string a = (a_s, ... , a_1)
               in V_s into the string MSB_i(a) = (a_s, ... , a_{s-i+1}) in
               V_i (most significant bits);

       LSB_i: V_s -> V_i
               the transformation that maps the string a = (a_s, ... , a_1)
               in V_s into the string LSB_i(a) = (a_i, ... , a_1) in V_i
               (least significant bits);

       Inc_c: V_s -> V_s
               the transformation that maps the string a = (a_s, ... , a_1)
               in V_s into the string Inc_c(a) = MSB_{|a|-c}(a) |
               Vec_c(Int_c(LSB_c(a)) + 1(mod 2^c)) in V_s (incrementing the
               least significant c bits of the bit string, regarded as the
               binary representation of an integer);

       a^s     the string in V_s that consists of s 'a' bits;

       E_{K}: V_n -> V_n
               the block cipher permutation under the key K in V_k;

       ceil(x) the smallest integer that is greater than or equal to x;

       floor(x)
               the biggest integer that is less than or equal to x;

       k       the bit length of the K; k is assumed to be divisible by 8;

       n       the block size of the block cipher (in bits); n is assumed to
               be divisible by 8;

       b       the number of data blocks in the plaintext P (b =
               ceil(|P|/n));

       N       the section size (the number of bits that are processed with
               one section key before this key is transformed).

       A plaintext message P and the corresponding ciphertext C are divided
       into b = ceil(|P|/n) blocks, denoted as P = P_1 | P_2 | ... | P_b and
       C = C_1 | C_2 | ... | C_b, respectively.  The first b-1 blocks P_i
       and C_i are in V_n for i = 1, 2, ... , b-1.  The b-th blocks P_b and
       C_b may be incomplete blocks, i.e., in V_r, where r <= n if not
       otherwise specified.

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    4.  Choosing Constructions and Security Parameters

       External re-keying is an approach assuming that a key is transformed
       after encrypting a limited number of entire messages.  The external
       re-keying method is chosen at the protocol level, regardless of the
       underlying block cipher or the encryption mode.  External re-keying
       is recommended for protocols that process relatively short messages
       or protocols that have a way to divide a long message into manageable
       pieces.  Through external re-keying, the number of messages that can
       be securely processed with a single initial key K is substantially
       increased without a loss of message length.

       External re-keying has the following advantages

       1.  It increases the lifetime of an initial key by increasing the
           number of messages processed with this key.

       2.  It has minimal impact on performance when the number of messages
           processed under one initial key is sufficiently large.

       3.  It provides forward and backward security of data-processing
           keys.

       However, the use of external re-keying has the following
       disadvantage: in cases with restrictive key lifetime limitations, the
       message sizes can become obstructive due to the impossibility of
       processing sufficiently large messages, so it may be necessary to
       perform additional fragmentation at the protocol level.  For example,
       if the key lifetime L is 1 GB and the message length m = 3 GB, then
       this message cannot be processed as a whole, and it should be divided
       into three fragments that will be processed separately.

       Internal re-keying is an approach assuming that a key is transformed
       during each separate message processing.  Such procedures are
       integrated into the base modes of operations, so every internal
       re-keying mechanism is defined for the particular operation mode and
       the block size of the used cipher.  Internal re-keying is recommended
       for protocols that process long messages: the size of each single
       message can be substantially increased without loss in the number of
       messages that can be securely processed with a single initial key.

       Internal re-keying has the following advantages:

       1.  It increases the lifetime of an initial key by increasing the
           size of the messages processed with one initial key.

       2.  It has minimal impact on performance.

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       3.  Internal re-keying mechanisms without a master key do not affect
           short-message transformation at all.

       4.  It is transparent (works like any mode of operation): it does not
           require changes of initialization vectors (IVs) and a restart of
           MACing.

       However, the use of internal re-keying has the following
       disadvantages:

       1.  a specific method must not be chosen independently of a mode of
           operation.

       2.  internal re-keying mechanisms without a master key do not provide
           backward security of data-processing keys.

       Any block cipher modes of operations with internal re-keying can be
       jointly used with any external re-keying mechanisms.  Such joint
       usage increases both the number of messages processed with one
       initial key and their maximum possible size.

       If the adversary has access to the data-processing interface, the use
       of the same cryptographic primitives both for data-processing and
       re-keying transformation decreases the code size but can lead to some
       possible vulnerabilities (the possibility of mounting a chosen-
       plaintext attack may lead to the compromise of the following keys).
       This vulnerability can be eliminated by using different primitives
       for data processing and re-keying, e.g., block cipher for data
       processing and hash for re-keying (see Section 5.2.2 and
       Section 5.3.2).  However, in this case, the security of the whole
       scheme cannot be reduced to standard notions like a pseudorandom
       function (PRF) or pseudorandom permutation (PRP), so security
       estimations become more difficult and unclear.

       Summing up the abovementioned issues briefly:

       1.  If a protocol assumes processing of long records (e.g., [CMS]),
           internal re-keying should be used.  If a protocol assumes
           processing of a significant number of ordered records, which can
           be considered as a single data stream (e.g., [TLS], [SSH]),
           internal re-keying may also be used.

       2.  For protocols that allow out-of-order delivery and lost records
           (e.g., [DTLS], [ESP]), external re-keying should be used as, in
           this case, records cannot be considered as a single data stream.
           If the records are also long enough, internal re-keying should
           also be used during each separate message processing.

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       For external re-keying:

       1.  If it is desirable to separate transformations used for data
           processing and key updates, hash function-based re-keying should
           be used.

       2.  If parallel data processing is required, then parallel external
           re-keying should be used.

       3.  If restrictive key lifetime limitations are present, external
           tree-based re-keying should be used.

       For internal re-keying:

       1.  If the property of forward and backward security is desirable for
           data-processing keys and if additional key material can be easily
           obtained for the data-processing stage, internal re-keying with a
           master key should be used.

    5.  External Re-keying Mechanisms

       This section presents an approach to increasing the initial key
       lifetime by using a transformation of a data-processing key (frame
       key) after processing a limited number of entire messages (frame).
       The approach provides external parallel and serial re-keying
       mechanisms (see [AbBell]).  These mechanisms use initial key K only
       for frame key generation and never use it directly for data
       processing.  Such mechanisms operate outside of the base modes of
       operations and do not change them at all; therefore, they are called
       "external re-keying" mechanisms in this document.

       External re-keying mechanisms are recommended for usage in protocols
       that process quite small messages, since the maximum gain in
       increasing the initial key lifetime is achieved by increasing the
       number of messages.

       External re-keying increases the initial key lifetime through the
       following approach.  Suppose there is a protocol P with some mode of
       operation (base encryption or authentication mode).  Let L1 be a key
       lifetime limitation induced by side-channel analysis methods (side-
       channel limitation), let L2 be a key lifetime limitation induced by
       methods based on the combinatorial properties of a used mode of
       operation (combinatorial limitation), and let q1, q2 be the total
       numbers of messages of length m that can be safely processed with an
       initial key K according to these limitations.

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       Let L = min(L1, L2), q = min(q1, q2), and q * m <= L.  As the L1
       limitation is usually much stronger than the L2 limitation (L1 < L2),
       the final key lifetime restriction is equal to the most restrictive
       limitation L1.  Thus, as displayed in Figure 2, without re-keying,
       only q1 (q1 * m <= L1) messages can be safely processed.

                             <--------m------->
                             +----------------+ ^ ^
                             |================| | |
                             |================| | |
                         K-->|================| q1|
                             |================| | |
                             |==============L1| | |
                             +----------------+ v |
                             |                |   |
                             |                |   |
                             |                |   q2
                             |                |   |
                             |                |   |
                             |                |   |
                             |                |   |
                             |                |   |
                             |                |   |
                             |                |   |
                             |                |   |
                             |              L2|   |
                             +----------------+   v

                 Figure 2: Basic Principles of Message Processing
                            without External Re-keying

       Suppose that the safety margin for the protocol P is fixed and the
       external re-keying approach is applied to the initial key K to
       generate the sequence of frame keys.  The frame keys are generated in
       such a way that the leakage of a previous frame key does not have any
       impact on the following one, so the side-channel limitation L1 is
       switched off.  Thus, the resulting key lifetime limitation of the
       initial key K can be calculated on the basis of a new combinatorial
       limitation L2'.  It is proven (see [AbBell]) that the security of the
       mode of operation that uses external re-keying leads to an increase
       when compared to base mode without re-keying (thus, L2 < L2').
       Hence, as displayed in Figure 3, the resulting key lifetime
       limitation if using external re-keying can be increased up to L2'.

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                             <--------m------->
                       K     +----------------+
                       |     |================|
                       v     |================|
                      K^1--> |================|
                       |     |================|
                       |     |==============L1|
                       |     +----------------+
                       |     |================|
                       v     |================|
                      K^2--> |================|
                       |     |================|
                       |     |==============L1|
                       |     +----------------+
                       |     |================|
                       v     |================|
                      ...    |      . . .     |
                             |                |
                             |                |
                             |              L2|
                             +----------------+
                             |                |
                            ...              ...
                             |             L2'|
                             +----------------+

                 Figure 3: Basic Principles of Message Processing
                              with External Re-keying

       Note: The key transformation process is depicted in a simplified
       form.  A specific approach (parallel and serial) is described below.

       Consider an example.  Let the message size in a protocol P be equal
       to 1 KB.  Suppose L1 = 128 MB and L2 = 1 TB.  Thus, if an external
       re-keying mechanism is not used, the initial key K must be
       renegotiated after processing 128 MB / 1 KB = 131072 messages.

       If an external re-keying mechanism is used, the key lifetime
       limitation L1 goes off.  Hence, the resulting key lifetime limitation
       L2' can be set to more than 1 TB.  Thus, if an external re-keying
       mechanism is used, more than 1 TB / 1 KB = 2^30 messages can be
       processed before the initial key K is renegotiated.  This is 8192
       times greater than the number of messages that can be processed when
       an external re-keying mechanism is not used.

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    5.1.  Methods of Key Lifetime Control

       Suppose L is an amount of data that can be safely processed with one
       frame key.  For i in {1, 2, ... , t}, the frame key K^i (see Figures
       4 and 6) should be transformed after processing q_i messages, where
       q_i can be calculated in accordance with one of the following
       approaches:

       Explicit approach:

          q_i is such that |M^{i,1}| + ... + |M^{i,q_i}| <= L, |M^{i,1}| +
          ... + |M^{i,q_i+1}| > L.
          This approach allows use of the frame key K^i in an almost optimal
          way, but it can be applied only when messages cannot be lost or
          reordered (e.g., TLS records).

       Implicit approach:

          q_i = L / m_max, i = 1, ... , t.
          The amount of data processed with one frame key K^i is calculated
          under the assumption that every message has the maximum length
          m_max.  Hence, this amount can be considerably less than the key
          lifetime limitation L.  On the other hand, this approach can be
          applied when messages may be lost or reordered (e.g., DTLS
          records).

       Dynamic key changes:

          We can organize the key change using the Protected Point to Point
          ([P3]) solution by building a protected tunnel between the
          endpoints in which the information about frame key updating can be
          safely passed across.  This can be useful, for example, when we
          want the adversary to not detect the key change during the
          protocol evaluation.

    5.2.  Parallel Constructions

       External parallel re-keying mechanisms generate frame keys K^1, K^2,
       ... directly from the initial key K independently of each other.

       The main idea behind external re-keying with a parallel construction
       is presented in Figure 4:

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       Maximum message size = m_max.
       _____________________________________________________________

                                       m_max
                                 <---------------->
                       M^{1,1}   |===             |
                       M^{1,2}   |=============== |
             +->K^1-->   ...            ...
             |         M^{1,q_1} |========        |
             |
             |
             |         M^{2,1}   |================|
             |         M^{2,2}   |=====           |
       K-----|->K^2-->   ...            ...
             |         M^{2,q_2} |==========      |
             |
            ...
             |         M^{t,1}   |============    |
             |         M^{t,2}   |=============   |
             +->K^t-->   ...            ...
                       M^{t,q_t} |==========      |

       _____________________________________________________________

                 Figure 4: External Parallel Re-keying Mechanisms

       The frame key K^i, i = 1, ... , t - 1 is updated after processing a
       certain number of messages (see Section 5.1).

    5.2.1.  Parallel Construction Based on a KDF on a Block Cipher

       The ExtParallelC re-keying mechanism is based on the key derivation
       function on a block cipher and is used to generate t frame keys as
       follows:

          K^1 | K^2 | ... | K^t = ExtParallelC(K, t * k) = MSB_{t *
          k}(E_{K}(Vec_n(0)) |
          E_{K}(Vec_n(1)) | ... | E_{K}(Vec_n(R - 1))),

       where R = ceil(t * k/n).

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    5.2.2.  Parallel Construction Based on a KDF on a Hash Function

       The ExtParallelH re-keying mechanism is based on the key derivation
       function HKDF-Expand, described in [RFC5869], and is used to generate
       t frame keys as follows:

          K^1 | K^2 | ... | K^t = ExtParallelH(K, t * k) = HKDF-Expand(K,
          label, t * k),

       where label is a string (may be a zero-length string) that is defined
       by a specific protocol.

    5.2.3.  Tree-Based Construction

       The application of an external tree-based mechanism leads to the
       construction of the key tree with the initial key K (root key) at the
       0 level and the frame keys K^1, K^2, ... at the last level, as
       described in Figure 5.

                                K_root = K
                          ___________|___________
                         |          ...          |
                         V                       V
                        K{1,1}                K{1,W1}
                   ______|______           ______|______
                  |     ...     |         |     ...     |
                  V             V         V             V
               K{2,1}       K{2,W2}  K{2,(W1-1)*W2+1} K{2,W1*W2}
                __|__         __|__     __|__         __|__
               | ... |       | ... |   | ... |       | ... |
               V     V       V     V   V     V       V     V
            K{3,1}  ...     ...   ... ...   ...     ...   K{3,W1*W2*W3}

             ...                                           ...
            __|__                   ...                   __|__
           | ... |                                       | ... |
           V     V                                       V     V
       K{h,1}   K{h,Wh}         K{h,(W1*...*W{h-1}-1)*Wh+1}  K{h,W1*...*Wh}
         //       \\                                  //       \\
       K^1       K^{Wh}        K^{(W1*...*W{h-1}-1)*Wh+1}     K^{W1*...*Wh}
       ____________________________________________________________________

                      Figure 5: External Tree-Based Mechanism

       The tree height h and the number of keys Wj, j in {1, ... , h}, which
       can be partitioned from the "parent" key, are defined in accordance
       with a specific protocol and key lifetime limitations for the used
       derivation functions.

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       Each j-level key K{j,w}, where j in {1, ... , h}, w in {1, ... , W1 *
       ... * Wj}, is derived from the (j-1)-level "parent" key K{j-1,
       ceil(w/Wi)} (and other appropriate input data) using the j-th level
       derivation function.  This function can be based on the block cipher
       function or on the hash function and is defined in accordance with a
       specific protocol.

       The i-th frame K^i, i in {1, 2, ... , W1*...*Wh}, can be calculated
       as follows:

          K^i = ExtKeyTree(K, i) = KDF_h(KDF_{h-1}(... KDF_1(K, ceil(i / (W2
          * ... * Wh)) ... , ceil(i / Wh)), i),

       where KDF_j is the j-th level derivation function that takes two
       arguments (the parent key value and the integer in a range from 1 to
       W1 * ... * Wj) and outputs the j-th level key value.

       The frame key K^i is updated after processing a certain number of
       messages (see Section 5.1).

       In order to create an efficient implementation, during frame key K^i
       generation, the derivation functions KDF_j, j in {1, ... , h-1}
       should be used only when ceil(i / (W{j+1} * ... * Wh)) != ceil((i -
       1) / (W{j+1} * ... * Wh)); otherwise, it is necessary to use a
       previously generated value.  This approach also makes it possible to
       take countermeasures against side-channel attacks.

       Consider an example.  Suppose h = 3, W1 = W2 = W3 = W, and KDF_1,
       KDF_2, KDF_3 are key derivation functions based on the
       KDF_GOSTR3411_2012_256 (hereafter simply KDF) function described in
       [RFC7836].  The resulting ExtKeyTree function can be defined as
       follows:

          ExtKeyTree(K, i) = KDF(KDF(KDF(K, "level1", ceil(i / W^2)),
          "level2", ceil(i / W)), "level3", i).

       where i in {1, 2, ... , W^3}.

       A structure similar to the external tree-based mechanism can be found
       in Section 6 of [NISTSP800-108].

    5.3.  Serial Constructions

       External serial re-keying mechanisms generate frame keys, each of
       which depends on the secret state (K*_1, K*_2, ...) that is updated
       after the generation of each new frame key; see Figure 6.  Similar
       approaches are used in the [SIGNAL] protocol and the [TLS] updating

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       traffic key mechanism and were proposed for use in the [U2F]
       protocol.

       External serial re-keying mechanisms have the obvious disadvantage of
       being impossible to implement in parallel, but they may be the
       preferred option if additional forward secrecy is desirable.  If all
       keys are securely deleted after usage, the compromise of a current
       secret state at some point does not lead to a compromise of all
       previous secret states and frame keys.  In terms of [TLS], compromise
       of application_traffic_secret_N does not compromise all previous
       application_traffic_secret_i, i < N.

       The main idea behind external re-keying with a serial construction is
       presented in Figure 6:

       Maximum message size = m_max.
       _____________________________________________________________
                                            m_max
                                      <---------------->
                            M^{1,1}   |===             |
                            M^{1,2}   |=============== |
       K*_1 = K --->K^1-->    ...            ...
         |                  M^{1,q_1} |========        |
         |
         |
         |                  M^{2,1}   |================|
         v                  M^{2,2}   |=====           |
       K*_2 ------->K^2-->    ...            ...
         |                  M^{2,q_2} |==========      |
         |
        ...
         |                  M^{t,1}   |============    |
         v                  M^{t,2}   |=============   |
       K*_t ------->K^t-->    ...            ...
                            M^{t,q_t} |==========      |

       _____________________________________________________________

                  Figure 6: External Serial Re-keying Mechanisms

       The frame key K^i, i = 1, ... , t - 1, is updated after processing a
       certain number of messages (see Section 5.1).

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    5.3.1.  Serial Construction Based on a KDF on a Block Cipher

       The frame key K^i is calculated using the ExtSerialC transformation
       as follows:

          K^i = ExtSerialC(K, i) =
          MSB_k(E_{K*_i}(Vec_n(0)) |E_{K*_i}(Vec_n(1)) | ... |
          E_{K*_i}(Vec_n(J - 1))),

       where J = ceil(k / n), i = 1, ... , t, K*_i is calculated as follows:

          K*_1 = K,

          K*_{j+1} = MSB_k(E_{K*_j}(Vec_n(J)) | E_{K*_j}(Vec_n(J + 1)) |
          ... |
          E_{K*_j}(Vec_n(2 * J - 1))),

       where j = 1, ... , t - 1.

    5.3.2.  Serial Construction Based on a KDF on a Hash Function

       The frame key K^i is calculated using the ExtSerialH transformation
       as follows:

          K^i = ExtSerialH(K, i) = HKDF-Expand(K*_i, label1, k),

       where i = 1, ... , t; HKDF-Expand is the HMAC-based key derivation
       function, as described in [RFC5869]; and K*_i is calculated as
       follows:

          K*_1 = K,

          K*_{j+1} = HKDF-Expand(K*_j, label2, k), where j = 1, ... , t - 1,

       where label1 and label2 are different strings from V* that are
       defined by a specific protocol (see, for example, the algorithm for
       updating traffic keys in TLS 1.3 [TLS]).

    5.4.  Using Additional Entropy during Re-keying

       In many cases, using additional entropy during re-keying won't
       increase security but may give a false sense of that.  Therefore, one
       can rely on additional entropy only after conducting a deep security
       analysis.  For example, good PRF constructions do not require
       additional entropy for the quality of keys, so, in most cases, there
       is no need to use additional entropy with external re-keying
       mechanisms based on secure KDFs.  However, in some situations, mixed-
       in entropy can still increase security in the case of a time-limited

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       but complete breach of the system when an adversary can access the
       frame-key generation interface but cannot reveal the master keys
       (e.g., when the master keys are stored in a Hardware Security Module
       (HSM)).

       For example, an external parallel construction based on a KDF on a
       hash function with a mixed-in entropy can be described as follows:

          K^i = HKDF-Expand(K, label_i, k),

       where label_i is additional entropy that must be sent to the
       recipient (e.g., sent jointly with an encrypted message).  The
       entropy label_i and the corresponding key K^i must be generated
       directly before message processing.

    6.  Internal Re-keying Mechanisms

       This section presents an approach to increasing the key lifetime by
       using a transformation of a data-processing key (section key) during
       each separate message processing.  Each message is processed starting
       with the same key (the first section key), and each section key is
       updated after processing N bits of the message (section).

       This section provides internal re-keying mechanisms called ACPKM
       (Advanced Cryptographic Prolongation of Key Material) and ACPKM-
       Master that do not use a master key and use a master key,
       respectively.  Such mechanisms are integrated into the base modes of
       operation and actually form new modes of operation.  Therefore, they
       are called "internal re-keying" mechanisms in this document.

       Internal re-keying mechanisms are recommended to be used in protocols
       that process large single messages (e.g., CMS messages), since the
       maximum gain in increasing the key lifetime is achieved by increasing
       the length of a message, while it provides almost no increase in the
       number of messages that can be processed with one initial key.

       Internal re-keying increases the key lifetime through the following
       approach.  Suppose protocol P uses some base mode of operation.  Let
       L1 and L2 be a side channel and combinatorial limitations,
       respectively, and for some fixed number of messages q, let m1, m2 be
       the lengths of messages that can be safely processed with a single
       initial key K according to these limitations.

       Thus, the approach without re-keying (analogous to Section 5) yields
       a final key lifetime restriction equal to L1, and only q messages of
       the length m1 can be safely processed; see Figure 7.

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                    K
                    |
                    v
          ^ +----------------+------------------------------------+
          | |==============L1|                                  L2|
          | |================|                                    |
          q |================|                                    |
          | |================|                                    |
          | |================|                                    |
          v +----------------+------------------------------------+
            <-------m1------->
            <----------------------------m2----------------------->

                 Figure 7: Basic Principles of Message Processing
                            without Internal Re-keying

       Suppose that the safety margin for the protocol P is fixed and the
       internal re-keying approach is applied to the base mode of operation.
       Suppose further that every message is processed with a section key,
       which is transformed after processing N bits of data, where N is a
       parameter.  If q * N does not exceed L1, then the side-channel
       limitation L1 goes off, and the resulting key lifetime limitation of
       the initial key K can be calculated on the basis of a new
       combinatorial limitation L2'.  The security of the mode of operation
       that uses internal re-keying increases when compared to the base mode
       of operation without re-keying (thus, L2 < L2').  Hence, as displayed
       in Figure 8, the resulting key lifetime limitation if using internal
       re-keying can be increased up to L2'.

         K-----> K^1-------------> K^2 -----------> . . .
                 |                 |
                 v                 v
       ^ +---------------+---------------+------------------+--...--+
       | |=============L1|=============L1|======          L2|    L2'|
       | |===============|===============|======            |       |
       q |===============|===============|====== . . .      |       |
       | |===============|===============|======            |       |
       | |===============|===============|======            |       |
       v +---------------+---------------+------------------+--...--+
         <-------N------->

                 Figure 8: Basic Principles of Message Processing
                              with Internal Re-keying

       Note: The key transformation process is depicted in a simplified
       form.  A specific approach (ACPKM and ACPKM-Master re-keying
       mechanisms) is described below.

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       Since the performance of encryption can slightly decrease for rather
       small values of N, the maximum possible value should be selected for
       parameter N for a particular protocol in order to provide the
       necessary key lifetime for the considered security models.

       Consider an example.  Suppose L1 = 128 MB and L2 = 10 TB.  Let the
       message size in the protocol be large/unlimited (which may exhaust
       the whole key lifetime L2).  The most restrictive resulting key
       lifetime limitation is equal to 128 MB.

       Thus, there is a need to put a limit on the maximum message size
       m_max.  For example, if m_max = 32 MB, it may happen that the
       renegotiation of initial key K would be required after processing
       only four messages.

       If an internal re-keying mechanism with section size N = 1 MB is
       used, more than L1 / N = 128 MB / 1 MB = 128 messages can be
       processed before the renegotiation of initial key K (instead of four
       messages when an internal re-keying mechanism is not used).  Note
       that only one section of each message is processed with the section
       key K^i, and, consequently, the key lifetime limitation L1 goes off.
       Hence, the resulting key lifetime limitation L2' can be set to more
       than 10 TB (in cases when a single large message is processed using
       the initial key K).

    6.1.  Methods of Key Lifetime Control

       Suppose L is an amount of data that can be safely processed with one
       section key and N is a section size (fixed parameter).  Suppose
       M^{i}_1 is the first section of message M^{i}, i = 1, ... , q (see
       Figures 9 and 10); the parameter q can then be calculated in
       accordance with one of the following two approaches:

       o  Explicit approach:
          q_i is such that |M^{1}_1| + ... + |M^{q}_1| <= L, |M^{1}_1| + ...
          + |M^{q+1}_1| > L
          This approach allows use of the section key K^i in an almost
          optimal way, but it can be applied only when messages cannot be
          lost or reordered (e.g., TLS records).

       o  Implicit approach:
          q = L / N.
          The amount of data processed with one section key K^i is
          calculated under the assumption that the length of every message
          is equal to or greater than section size N and thus can be
          considerably less than the key lifetime limitation L.  On the
          other hand, this approach can be applied when messages may be lost
          or reordered (e.g., DTLS records).

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    6.2.  Constructions that Do Not Require a Master Key

       This section describes the block cipher modes that use the ACPKM
       re-keying mechanism, which does not use a master key; an initial key
       is used directly for the data encryption.

    6.2.1.  ACPKM Re-keying Mechanisms

       This section defines a periodical key transformation without a master
       key, which is called the ACPKM re-keying mechanism.  This mechanism
       can be applied to one of the base encryption modes (CTR and GCM block
       cipher modes) to get an extension of this encryption mode that uses
       periodical key transformation without a master key.  This extension
       can be considered as a new encryption mode.

       An additional parameter that defines the functioning of base
       encryption modes with the ACPKM re-keying mechanism is the section
       size N.  The value of N is measured in bits and is fixed within a
       specific protocol based on the requirements of the system capacity
       and the key lifetime.  The section size N MUST be divisible by the
       block size n.

       The main idea behind internal re-keying without a master key is
       presented in Figure 9:

       Section size = const = N,
       maximum message size = m_max.
       ____________________________________________________________________

                     ACPKM       ACPKM              ACPKM
              K^1 = K ---> K^2 ---...-> K^{l_max-1} ----> K^{l_max}
                  |          |                |           |
                  |          |                |           |
                  v          v                v           v
       M^{1} |==========|==========| ... |==========|=======:  |
       M^{2} |==========|==========| ... |===       |       :  |
         .        .          .        .       .          .  :
         :        :          :        :       :          :  :
       M^{q} |==========|==========| ... |==========|=====  :  |
                          section                           :
                        <---------->                      m_max
                           N bit
       ___________________________________________________________________
       l_max = ceil(m_max/N).

                 Figure 9: Internal Re-keying without a Master Key

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       During the processing of the input message M with the length m in
       some encryption mode that uses the ACPKM key transformation of the
       initial key K, the message is divided into l = ceil(m / N) sections
       (denoted as M = M_1 | M_2 | ... | M_l, where M_i is in V_N for i in
       {1, 2, ... , l - 1} and M_l is in V_r, r <= N).  The first section of
       each message is processed with the section key K^1 = K.  To process
       the (i + 1)-th section of each message, the section key K^{i+1} is
       calculated using the ACPKM transformation as follows:

          K^{i+1} = ACPKM(K^i) = MSB_k(E_{K^i}(D_1) | ... | E_{K^i}(D_J)),

       where J = ceil(k/n) and D_1, D_2, ... , D_J are in V_n and are
       calculated as follows:

          D_1 | D_2 | ... | D_J = MSB_{J * n}(D),

       where D is the following constant in V_{1024}:

                 D = ( 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87
                     | 88 | 89 | 8a | 8b | 8c | 8d | 8e | 8f
                     | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97
                     | 98 | 99 | 9a | 9b | 9c | 9d | 9e | 9f
                     | a0 | a1 | a2 | a3 | a4 | a5 | a6 | a7
                     | a8 | a9 | aa | ab | ac | ad | ae | af
                     | b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7
                     | b8 | b9 | ba | bb | bc | bd | be | bf
                     | c0 | c1 | c2 | c3 | c4 | c5 | c6 | c7
                     | c8 | c9 | ca | cb | cc | cd | ce | cf
                     | d0 | d1 | d2 | d3 | d4 | d5 | d6 | d7
                     | d8 | d9 | da | db | dc | dd | de | df
                     | e0 | e1 | e2 | e3 | e4 | e5 | e6 | e7
                     | e8 | e9 | ea | eb | ec | ed | ee | ef
                     | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7
                     | f8 | f9 | fa | fb | fc | fd | fe | ff)

       Note: The constant D is such that D_1, ... , D_J are pairwise
       different for any allowed n and k values.

       Note: The highest bit of each octet of the constant D is equal to 1.
       This condition is important as, in conjunction with a certain mode
       message length limitation, it allows prevention of collisions of
       block cipher permutation inputs in cases with key transformation and
       message processing (for more details, see Section 4.4 of [AAOS2017]).

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    6.2.2.  CTR-ACPKM Encryption Mode

       This section defines a CTR-ACPKM encryption mode that uses the ACPKM
       internal re-keying mechanism for the periodical key transformation.

       The CTR-ACPKM mode can be considered as the base encryption mode CTR
       (see [MODES]) extended by the ACPKM re-keying mechanism.

       The CTR-ACPKM encryption mode can be used with the following
       parameters:

       o  64 <= n <= 512.

       o  128 <= k <= 512.

       o  The number c of bits in a specific part of the block to be
          incremented is such that 32 <= c <= 3 / 4 n, where c is a multiple
          of 8.

       o  The maximum message size m_max = n * 2^{c-1}.

       The CTR-ACPKM mode encryption and decryption procedures are defined
       as follows:

       +----------------------------------------------------------------+
       |  CTR-ACPKM-Encrypt(N, K, ICN, P)                               |
       |----------------------------------------------------------------|
       |  Input:                                                        |
       |  - section size N,                                             |
       |  - initial key K,                                              |
       |  - initial counter nonce ICN in V_{n-c},                       |
       |  - plaintext P = P_1 | ... | P_b, |P| <= m_max.                |
       |  Output:                                                       |
       |  - ciphertext C.                                               |
       |----------------------------------------------------------------|
       |  1. CTR_1 = ICN | 0^c                                          |
       |  2. For j = 2, 3, ... , b do                                   |
       |         CTR_{j} = Inc_c(CTR_{j-1})                             |
       |  3. K^1 = K                                                    |
       |  4. For i = 2, 3, ... , ceil(|P| / N)                          |
       |         K^i = ACPKM(K^{i-1})                                   |
       |  5. For j = 1, 2, ... , b do                                   |
       |         i = ceil(j * n / N),                                   |
       |         G_j = E_{K^i}(CTR_j)                                   |
       |  6. C = P (xor) MSB_{|P|}(G_1 | ... | G_b)                     |
       |  7. Return C                                                   |
       +----------------------------------------------------------------+

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       +----------------------------------------------------------------+
       |  CTR-ACPKM-Decrypt(N, K, ICN, C)                               |
       |----------------------------------------------------------------|
       |  Input:                                                        |
       |  - section size N,                                             |
       |  - initial key K,                                              |
       |  - initial counter nonce ICN in V_{n-c},                       |
       |  - ciphertext C = C_1 | ... | C_b, |C| <= m_max.               |
       |  Output:                                                       |
       |  - plaintext P.                                                |
       |----------------------------------------------------------------|
       |  1. P = CTR-ACPKM-Encrypt(N, K, ICN, C)                        |
       |  2. Return P                                                   |
       +----------------------------------------------------------------+

       The initial counter nonce (ICN) value for each message that is
       encrypted under the given initial key K must be chosen in a unique
       manner.

    6.2.3.  GCM-ACPKM Authenticated Encryption Mode

       This section defines the GCM-ACPKM authenticated encryption mode that
       uses the ACPKM internal re-keying mechanism for the periodical key
       transformation.

       The GCM-ACPKM mode can be considered as the base authenticated
       encryption mode GCM (see [GCM]) extended by the ACPKM re-keying
       mechanism.

       The GCM-ACPKM authenticated encryption mode can be used with the
       following parameters:

       o  n in {128, 256}.

       o  128 <= k <= 512.

       o  The number c of bits in a specific part of the block to be
          incremented is such that 1 / 4 n <= c <= 1 / 2 n, c is a multiple
          of 8.

       o  Authentication tag length t.

       o  The maximum message size m_max = min{n * (2^{c-1} - 2), 2^{n/2} -
          1}.

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       The GCM-ACPKM mode encryption and decryption procedures are defined
       as follows:

       +-------------------------------------------------------------------+
       |  GHASH(X, H)                                                      |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - bit string X = X_1 | ... | X_m, X_1, ... , X_m in V_n.         |
       |  Output:                                                          |
       |  - block GHASH(X, H) in V_n.                                      |
       |-------------------------------------------------------------------|
       |  1. Y_0 = 0^n                                                     |
       |  2. For i = 1, ... , m do                                         |
       |         Y_i = (Y_{i-1} (xor) X_i) * H                             |
       |  3. Return Y_m                                                    |
       +-------------------------------------------------------------------+

       +-------------------------------------------------------------------+
       |  GCTR(N, K, ICB, X)                                               |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - section size N,                                                |
       |  - initial key K,                                                 |
       |  - initial counter block ICB,                                     |
       |  - X = X_1 | ... | X_b.                                           |
       |  Output:                                                          |
       |  - Y in V_{|X|}.                                                  |
       |-------------------------------------------------------------------|
       |  1. If X in V_0, then return Y, where Y in V_0                    |
       |  2. GCTR_1 = ICB                                                  |
       |  3. For i = 2, ... , b do                                         |
       |         GCTR_i = Inc_c(GCTR_{i-1})                                |
       |  4. K^1 = K                                                       |
       |  5. For j = 2, ... , ceil(|X| / N)                                |
       |         K^j = ACPKM(K^{j-1})                                      |
       |  6. For i = 1, ... , b do                                         |
       |         j = ceil(i * n / N),                                      |
       |         G_i = E_{K_j}(GCTR_i)                                     |
       |  7. Y = X (xor) MSB_{|X|}(G_1 | ... | G_b)                        |
       |  8. Return Y                                                      |
       +-------------------------------------------------------------------+

    Smyshlyaev                    Informational                    [Page 27]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       +-------------------------------------------------------------------+
       |  GCM-ACPKM-Encrypt(N, K, ICN, P, A)                               |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - section size N,                                                |
       |  - initial key K,                                                 |
       |  - initial counter nonce ICN in V_{n-c},                          |
       |  - plaintext P = P_1 | ... | P_b, |P| <= m_max,                   |
       |  - additional authenticated data A.                               |
       |  Output:                                                          |
       |  - ciphertext C,                                                  |
       |  - authentication tag T.                                          |
       |-------------------------------------------------------------------|
       |  1. H = E_{K}(0^n)                                                |
       |  2. ICB_0 = ICN | 0^{c-1} | 1                                     |
       |  3. C = GCTR(N, K, Inc_c(ICB_0), P)                               |
       |  4. u = n * ceil(|C| / n) - |C|                                   |
       |     v = n * ceil(|A| / n) - |A|                                   |
       |  5. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                |
       |               | Vec_{n/2}(|C|), H)                                |
       |  6. T = MSB_t(E_{K}(ICB_0) (xor) S)                               |
       |  7. Return C | T                                                  |
       +-------------------------------------------------------------------+

       +-------------------------------------------------------------------+
       |  GCM-ACPKM-Decrypt(N, K, ICN, A, C, T)                            |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - section size N,                                                |
       |  - initial key K,                                                 |
       |  - initial counter block ICN,                                     |
       |  - additional authenticated data A,                               |
       |  - ciphertext C = C_1 | ... | C_b, |C| <= m_max,                  |
       |  - authentication tag T.                                          |
       |  Output:                                                          |
       |  - plaintext P or FAIL.                                           |
       |-------------------------------------------------------------------|
       |  1. H = E_{K}(0^n)                                                |
       |  2. ICB_0 = ICN | 0^{c-1} | 1                                     |
       |  3. P = GCTR(N, K, Inc_c(ICB_0), C)                               |
       |  4. u = n * ceil(|C| / n) - |C|                                   |
       |     v = n * ceil(|A| / n) - |A|                                   |
       |  5. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                |
       |               | Vec_{n/2}(|C|), H)                                |
       |  6. T' = MSB_t(E_{K}(ICB_0) (xor) S)                              |
       |  7. If T = T', then return P; else return FAIL                    |
       +-------------------------------------------------------------------+

    Smyshlyaev                    Informational                    [Page 28]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       The * operation on (pairs of) the 2^n possible blocks corresponds to
       the multiplication operation for the binary Galois (finite) field of
       2^n elements defined by the polynomial f as follows (analogous to
       [GCM]):

       n = 128:  f = a^128 + a^7 + a^2 + a^1 + 1,

       n = 256:  f = a^256 + a^10 + a^5 + a^2 + 1.

       The initial counter nonce ICN value for each message that is
       encrypted under the given initial key K must be chosen in a unique
       manner.

       The key for computing values E_{K}(ICB_0) and H is not updated and is
       equal to the initial key K.

    6.3.  Constructions that Require a Master Key

       This section describes the block cipher modes that use the ACPKM-
       Master re-keying mechanism, which use the initial key K as a master
       key, so K is never used directly for data processing but is used for
       key derivation.

    6.3.1.  ACPKM-Master Key Derivation from the Master Key

       This section defines periodical key transformation with a master key,
       which is called the ACPKM-Master re-keying mechanism.  This mechanism
       can be applied to one of the base modes of operation (CTR, GCM, CBC,
       CFB, OMAC modes) for getting an extension that uses periodical key
       transformation with a master key.  This extension can be considered
       as a new mode of operation.

       Additional parameters that define the functioning of modes of
       operation that use the ACPKM-Master re-keying mechanism are the
       section size N, the change frequency T* of the master keys K*_1,
       K*_2, ... (see Figure 10), and the size d of the section key
       material.  The values of N and T* are measured in bits and are fixed
       within a specific protocol based on the requirements of the system
       capacity and the key lifetime.  The section size N MUST be divisible
       by the block size n.  The master key frequency T* MUST be divisible
       by d and by n.

    Smyshlyaev                    Informational                    [Page 29]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       The main idea behind internal re-keying with a master key is
       presented in Figure 10:

       Master key frequency T*,
       section size N,
       maximum message size = m_max.
       _____________________________________________________________________

                               ACPKM                 ACPKM
                    K*_1 = K----------> K*_2 ---------...-----> K*_l_max
                   ___|___            ___|___                 ___|___
                  |       |          |       |               |       |
                  v  ...  v          v  ...  v               v  ...  v
                K[1]     K[t]     K[t+1]  K[2*t]  K[(l_max-1)t+1] K[l_max*t]
                  |       |          |       |               |       |
                  |       |          |       |               |       |
                  v       v          v       v               v       v
       M^{1}||======|...|======||======|...|======||...||======|...|==  : ||
       M^{2}||======|...|======||======|...|======||...||======|...|====: ||
        ... ||      |   |      ||      |   |      ||   ||      |   |    : ||
       M^{q}||======|...|======||====  |...|      ||...||      |...|    : ||
              section                                                   :
             <------>                                                   :
               N bit                                                  m_max
       _____________________________________________________________________
       |K[i]| = d,
       t = T* / d,
       l_max = ceil(m_max / (N * t)).

                  Figure 10: Internal Re-keying with a Master Key

       During the processing of the input message M with the length m in
       some mode of operation that uses ACPKM-Master key transformation with
       the initial key K and the master key frequency T*, the message M is
       divided into l = ceil(m / N) sections (denoted as M = M_1 | M_2 |
       ... | M_l, where M_i is in V_N for i in {1, 2, ... , l - 1} and M_l
       is in V_r, r <= N).  The j-th section of each message is processed
       with the key material K[j], j in {1, ... , l}, |K[j]| = d, which is
       calculated with the ACPKM-Master algorithm as follows:

          K[1] | ... | K[l] = ACPKM-Master(T*, K, d, l) = CTR-ACPKM-Encrypt
          (T*, K, 1^{n/2}, 0^{d*l}).

       Note: The parameters d and l MUST be such that d * l <= n *
       2^{n/2-1}.

    Smyshlyaev                    Informational                    [Page 30]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

    6.3.2.  CTR-ACPKM-Master Encryption Mode

       This section defines a CTR-ACPKM-Master encryption mode that uses the
       ACPKM-Master internal re-keying mechanism for the periodical key
       transformation.

       The CTR-ACPKM-Master encryption mode can be considered as the base
       encryption mode CTR (see [MODES]) extended by the ACPKM-Master
       re-keying mechanism.

       The CTR-ACPKM-Master encryption mode can be used with the following
       parameters:

       o  64 <= n <= 512.

       o  128 <= k <= 512.

       o  The number c of bits in a specific part of the block to be
          incremented is such that 32 <= c <= 3 / 4 n, c is a multiple of 8.

       o  The maximum message size m_max = min{N * (n * 2^{n/2-1} / k), n *
          2^c}.

       The key material K[j] that is used for one-section processing is
       equal to K^j, where |K^j| = k bits.

    Smyshlyaev                    Informational                    [Page 31]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       The CTR-ACPKM-Master mode encryption and decryption procedures are
       defined as follows:

       +----------------------------------------------------------------+
       |  CTR-ACPKM-Master-Encrypt(N, K, T*, ICN, P)                    |
       |----------------------------------------------------------------|
       |  Input:                                                        |
       |  - section size N,                                             |
       |  - initial key K,                                              |
       |  - master key frequency T*,                                    |
       |  - initial counter nonce ICN in V_{n-c},                       |
       |  - plaintext P = P_1 | ... | P_b, |P| <= m_max.                |
       |  Output:                                                       |
       |  - ciphertext C.                                               |
       |----------------------------------------------------------------|
       |  1. CTR_1 = ICN | 0^c                                          |
       |  2. For j = 2, 3, ... , b do                                   |
       |         CTR_{j} = Inc_c(CTR_{j-1})                             |
       |  3. l = ceil(|P| / N)                                          |
       |  4. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                |
       |  5. For j = 1, 2, ... , b do                                   |
       |         i = ceil(j * n / N),                                   |
       |         G_j = E_{K^i}(CTR_j)                                   |
       |  6. C = P (xor) MSB_{|P|}(G_1 | ... |G_b)                      |
       |  7. Return C                                                   |
       |----------------------------------------------------------------+

       +----------------------------------------------------------------+
       |  CTR-ACPKM-Master-Decrypt(N, K, T*, ICN, C)                    |
       |----------------------------------------------------------------|
       |  Input:                                                        |
       |  - section size N,                                             |
       |  - initial key K,                                              |
       |  - master key frequency T*,                                    |
       |  - initial counter nonce ICN in V_{n-c},                       |
       |  - ciphertext C = C_1 | ... | C_b, |C| <= m_max.               |
       |  Output:                                                       |
       |  - plaintext P.                                                |
       |----------------------------------------------------------------|
       |  1. P = CTR-ACPKM-Master-Encrypt(N, K, T*, ICN, C)             |
       |  1. Return P                                                   |
       +----------------------------------------------------------------+

       The initial counter nonce ICN value for each message that is
       encrypted under the given initial key must be chosen in a unique
       manner.

    Smyshlyaev                    Informational                    [Page 32]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

    6.3.3.  GCM-ACPKM-Master Authenticated Encryption Mode

       This section defines a GCM-ACPKM-Master authenticated encryption mode
       that uses the ACPKM-Master internal re-keying mechanism for the
       periodical key transformation.

       The GCM-ACPKM-Master authenticated encryption mode can be considered
       as the base authenticated encryption mode GCM (see [GCM]) extended by
       the ACPKM-Master re-keying mechanism.

       The GCM-ACPKM-Master authenticated encryption mode can be used with
       the following parameters:

       o  n in {128, 256}.

       o  128 <= k <= 512.

       o  The number c of bits in a specific part of the block to be
          incremented is such that 1 / 4 n <= c <= 1 / 2 n, c is a multiple
          of 8.

       o  authentication tag length t.

       o  the maximum message size m_max = min{N * ( n * 2^{n/2-1} / k), n *
          (2^c - 2), 2^{n/2} - 1}.

       The key material K[j] that is used for the j-th section processing is
       equal to K^j, |K^j| = k bits.

       The GCM-ACPKM-Master mode encryption and decryption procedures are
       defined as follows:

       +-------------------------------------------------------------------+
       |  GHASH(X, H)                                                      |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - bit string X = X_1 | ... | X_m, X_i in V_n for i in {1, ... ,m}|
       |  Output:                                                          |
       |  - block GHASH(X, H) in V_n                                       |
       |-------------------------------------------------------------------|
       |  1. Y_0 = 0^n                                                     |
       |  2. For i = 1, ... , m do                                         |
       |         Y_i = (Y_{i-1} (xor) X_i) * H                             |
       |  3. Return Y_m                                                    |
       +-------------------------------------------------------------------+

    Smyshlyaev                    Informational                    [Page 33]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       +-------------------------------------------------------------------+
       |  GCTR(N, K, T*, ICB, X)                                           |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - section size N,                                                |
       |  - initial key K,                                                 |
       |  - master key frequency T*,                                       |
       |  - initial counter block ICB,                                     |
       |  - X = X_1 | ... | X_b.                                           |
       |  Output:                                                          |
       |  - Y in V_{|X|}.                                                  |
       |-------------------------------------------------------------------|
       |  1. If X in V_0, then return Y, where Y in V_0                    |
       |  2. GCTR_1 = ICB                                                  |
       |  3. For i = 2, ... , b do                                         |
       |         GCTR_i = Inc_c(GCTR_{i-1})                                |
       |  4. l = ceil(|X| / N)                                             |
       |  5. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                   |
       |  6. For j = 1, ... , b do                                         |
       |         i = ceil(j * n / N),                                      |
       |         G_j = E_{K^i}(GCTR_j)                                     |
       |  7. Y = X (xor) MSB_{|X|}(G_1 | ... | G_b)                        |
       |  8. Return Y                                                      |
       +-------------------------------------------------------------------+

    Smyshlyaev                    Informational                    [Page 34]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       +-------------------------------------------------------------------+
       |  GCM-ACPKM-Master-Encrypt(N, K, T*, ICN, P, A)                    |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - section size N,                                                |
       |  - initial key K,                                                 |
       |  - master key frequency T*,                                       |
       |  - initial counter nonce ICN in V_{n-c},                          |
       |  - plaintext P = P_1 | ... | P_b, |P| <= m_max.                   |
       |  - additional authenticated data A.                               |
       |  Output:                                                          |
       |  - ciphertext C,                                                  |
       |  - authentication tag T.                                          |
       |-------------------------------------------------------------------|
       |  1. K^1 = ACPKM-Master(T*, K, k, 1)                               |
       |  2. H = E_{K^1}(0^n)                                              |
       |  3. ICB_0 = ICN | 0^{c-1} | 1                                     |
       |  4. C = GCTR(N, K, T*, Inc_c(ICB_0), P)                           |
       |  5. u = n * ceil(|C| / n) - |C|                                   |
       |     v = n * ceil(|A| / n) - |A|                                   |
       |  6. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                |
       |               | Vec_{n/2}(|C|), H)                                |
       |  7. T = MSB_t(E_{K^1}(ICB_0) (xor) S)                             |
       |  8. Return C | T                                                  |
       +-------------------------------------------------------------------+

    Smyshlyaev                    Informational                    [Page 35]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       +-------------------------------------------------------------------+
       |  GCM-ACPKM-Master-Decrypt(N, K, T*, ICN, A, C, T)                 |
       |-------------------------------------------------------------------|
       |  Input:                                                           |
       |  - section size N,                                                |
       |  - initial key K,                                                 |
       |  - master key frequency T*,                                       |
       |  - initial counter nonce ICN in V_{n-c},                          |
       |  - additional authenticated data A.                               |
       |  - ciphertext C = C_1 | ... | C_b, |C| <= m_max,                  |
       |  - authentication tag T.                                          |
       |  Output:                                                          |
       |  - plaintext P or FAIL.                                           |
       |-------------------------------------------------------------------|
       |  1. K^1 = ACPKM-Master(T*, K, k, 1)                               |
       |  2. H = E_{K^1}(0^n)                                              |
       |  3. ICB_0 = ICN | 0^{c-1} | 1                                     |
       |  4. P = GCTR(N, K, T*, Inc_c(ICB_0), C)                           |
       |  5. u = n * ceil(|C| / n) - |C|                                   |
       |     v = n * ceil(|A| / n) - |A|                                   |
       |  6. S = GHASH(A | 0^v | C | 0^u | Vec_{n/2}(|A|) |                |
       |               | Vec_{n/2}(|C|), H)                                |
       |  7. T' = MSB_t(E_{K^1}(ICB_0) (xor) S)                            |
       |  8. If T = T', then return P; else return FAIL.                   |
       +-------------------------------------------------------------------+

       The * operation on (pairs of) the 2^n possible blocks corresponds to
       the multiplication operation for the binary Galois (finite) field of
       2^n elements defined by the polynomial f as follows (by analogy with
       [GCM]):

       n = 128:  f = a^128 + a^7 + a^2 + a^1 + 1,

       n = 256:  f = a^256 + a^10 + a^5 + a^2 + 1.

       The initial counter nonce ICN value for each message that is
       encrypted under the given initial key must be chosen in a unique
       manner.

    Smyshlyaev                    Informational                    [Page 36]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

    6.3.4.  CBC-ACPKM-Master Encryption Mode

       This section defines a CBC-ACPKM-Master encryption mode that uses the
       ACPKM-Master internal re-keying mechanism for the periodical key
       transformation.

       The CBC-ACPKM-Master encryption mode can be considered as the base
       encryption mode CBC (see [MODES]) extended by the ACPKM-Master
       re-keying mechanism.

       The CBC-ACPKM-Master encryption mode can be used with the following
       parameters:

       o  64 <= n <= 512.

       o  128 <= k <= 512.

       o  The maximum message size m_max = N * (n * 2^{n/2-1} / k).

       In the specification of the CBC-ACPKM-Master mode, the plaintext and
       ciphertext must be a sequence of one or more complete data blocks.
       If the data string to be encrypted does not initially satisfy this
       property, then it MUST be padded to form complete data blocks.  The
       padding methods are out of the scope of this document.  An example of
       a padding method can be found in Appendix A of [MODES].

       The key material K[j] that is used for the j-th section processing is
       equal to K^j, |K^j| = k bits.

       We use D_{K} to denote the decryption function that is a permutation
       inverse to E_{K}.

    Smyshlyaev                    Informational                    [Page 37]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       The CBC-ACPKM-Master mode encryption and decryption procedures are
       defined as follows:

       +----------------------------------------------------------------+
       |  CBC-ACPKM-Master-Encrypt(N, K, T*, IV, P)                     |
       |----------------------------------------------------------------|
       |  Input:                                                        |
       |  - section size N,                                             |
       |  - initial key K,                                              |
       |  - master key frequency T*,                                    |
       |  - initialization vector IV in V_n,                            |
       |  - plaintext P = P_1 | ... | P_b, |P_b| = n, |P| <= m_max.     |
       |  Output:                                                       |
       |  - ciphertext C.                                               |
       |----------------------------------------------------------------|
       |  1. l = ceil(|P| / N)                                          |
       |  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                |
       |  3. C_0 = IV                                                   |
       |  4. For j = 1, 2, ... , b do                                   |
       |         i = ceil(j * n / N),                                   |
       |         C_j = E_{K^i}(P_j (xor) C_{j-1})                       |
       |  5. Return C = C_1 | ... | C_b                                 |
       |----------------------------------------------------------------+

       +----------------------------------------------------------------+
       |  CBC-ACPKM-Master-Decrypt(N, K, T*, IV, C)                     |
       |----------------------------------------------------------------|
       |  Input:                                                        |
       |  - section size N,                                             |
       |  - initial key K,                                              |
       |  - master key frequency T*,                                    |
       |  - initialization vector IV in V_n,                            |
       |  - ciphertext C = C_1 | ... | C_b, |C_b| = n, |C| <= m_max.    |
       |  Output:                                                       |
       |  - plaintext P.                                                |
       |----------------------------------------------------------------|
       |  1. l = ceil(|C| / N)                                          |
       |  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)                |
       |  3. C_0 = IV                                                   |
       |  4. For j = 1, 2, ... , b do                                   |
       |         i = ceil(j * n / N)                                    |
       |         P_j = D_{K^i}(C_j) (xor) C_{j-1}                       |
       |  5. Return P = P_1 | ... | P_b                                 |
       +----------------------------------------------------------------+

       The initialization vector IV for any particular execution of the
       encryption process must be unpredictable.

    Smyshlyaev                    Informational                    [Page 38]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

    6.3.5.  CFB-ACPKM-Master Encryption Mode

       This section defines a CFB-ACPKM-Master encryption mode that uses the
       ACPKM-Master internal re-keying mechanism for the periodical key
       transformation.

       The CFB-ACPKM-Master encryption mode can be considered as the base
       encryption mode CFB (see [MODES]) extended by the ACPKM-Master
       re-keying mechanism.

       The CFB-ACPKM-Master encryption mode can be used with the following
       parameters:

       o  64 <= n <= 512.

       o  128 <= k <= 512.

       o  The maximum message size m_max = N * (n * 2^{n/2-1} / k).

       The key material K[j] that is used for the j-th section processing is
       equal to K^j, |K^j| = k bits.

       The CFB-ACPKM-Master mode encryption and decryption procedures are
       defined as follows:

       +-------------------------------------------------------------+
       |  CFB-ACPKM-Master-Encrypt(N, K, T*, IV, P)                  |
       |-------------------------------------------------------------|
       |  Input:                                                     |
       |  - section size N,                                          |
       |  - initial key K,                                           |
       |  - master key frequency T*,                                 |
       |  - initialization vector IV in V_n,                         |
       |  - plaintext P = P_1 | ... | P_b, |P| <= m_max.             |
       |  Output:                                                    |
       |  - ciphertext C.                                            |
       |-------------------------------------------------------------|
       |  1. l = ceil(|P| / N)                                       |
       |  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)             |
       |  3. C_0 = IV                                                |
       |  4. For j = 1, 2, ... , b - 1 do                            |
       |         i = ceil(j * n / N),                                |
       |         C_j = E_{K^i}(C_{j-1}) (xor) P_j                    |
       |  5. C_b = MSB_{|P_b|}(E_{K^l}(C_{b-1})) (xor) P_b           |
       |  6. Return C = C_1 | ... | C_b                              |
       |-------------------------------------------------------------+

    Smyshlyaev                    Informational                    [Page 39]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       +-------------------------------------------------------------+
       |  CFB-ACPKM-Master-Decrypt(N, K, T*, IV, C)                  |
       |-------------------------------------------------------------|
       |  Input:                                                     |
       |  - section size N,                                          |
       |  - initial key K,                                           |
       |  - master key frequency T*,                                 |
       |  - initialization vector IV in V_n,                         |
       |  - ciphertext C = C_1 | ... | C_b, |C| <= m_max.            |
       |  Output:                                                    |
       |  - plaintext P.                                             |
       |-------------------------------------------------------------|
       |  1. l = ceil(|C| / N)                                       |
       |  2. K^1 | ... | K^l = ACPKM-Master(T*, K, k, l)             |
       |  3. C_0 = IV                                                |
       |  4. For j = 1, 2, ... , b - 1 do                            |
       |         i = ceil(j * n / N),                                |
       |         P_j = E_{K^i}(C_{j-1}) (xor) C_j                    |
       |  5. P_b = MSB_{|C_b|}(E_{K^l}(C_{b-1})) (xor) C_b           |
       |  6. Return P = P_1 | ... | P_b                              |
       +-------------------------------------------------------------+

       The initialization vector IV for any particular execution of the
       encryption process must be unpredictable.

    6.3.6.  OMAC-ACPKM-Master Authentication Mode

       This section defines an OMAC-ACPKM-Master message authentication code
       calculation mode that uses the ACPKM-Master internal re-keying
       mechanism for the periodical key transformation.

       The OMAC-ACPKM-Master mode can be considered as the base message
       authentication code calculation mode OMAC1, which is also known as
       CMAC (see [RFC4493]), extended by the ACPKM-Master re-keying
       mechanism.

       The OMAC-ACPKM-Master message authentication code calculation mode
       can be used with the following parameters:

       o  n in {64, 128, 256}.

       o  128 <= k <= 512.

       o  The maximum message size m_max = N * (n * 2^{n/2-1} / (k + n)).

       The key material K[j] that is used for one-section processing is
       equal to K^j | K^j_1, where |K^j| = k bits and |K^j_1| = n bits.

    Smyshlyaev                    Informational                    [Page 40]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       The following is a specification of the subkey generation process of
       OMAC:

       +-------------------------------------------------------------------+
       | Generate_Subkey(K1, r)                                            |
       |-------------------------------------------------------------------|
       | Input:                                                            |
       |  - key K1.                                                        |
       |  Output:                                                          |
       |  - key SK.                                                        |
       |-------------------------------------------------------------------|
       |   1. If r = n, then return K1                                     |
       |   2. If r < n, then                                               |
       |          if MSB_1(K1) = 0                                         |
       |              return K1 << 1                                       |
       |          else                                                     |
       |              return (K1 << 1) (xor) R_n                           |
       +-------------------------------------------------------------------+

       Here, R_n takes the following values:

       o  n = 64: R_{64} = 0^{59} | 11011.

       o  n = 128: R_{128} = 0^{120} | 10000111.

       o  n = 256: R_{256} = 0^{145} | 10000100101.

    Smyshlyaev                    Informational                    [Page 41]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       The OMAC-ACPKM-Master message authentication code calculation mode is
       defined as follows:

       +-------------------------------------------------------------------+
       | OMAC-ACPKM-Master(K, N, T*, M)                                    |
       |-------------------------------------------------------------------|
       | Input:                                                            |
       |  - section size N,                                                |
       |  - initial key K,                                                 |
       |  - master key frequency T*,                                       |
       |  - plaintext M = M_1 | ... | M_b, |M| <= m_max.                   |
       |  Output:                                                          |
       |  - message authentication code T.                                 |
       |-------------------------------------------------------------------|
       | 1. C_0 = 0^n                                                      |
       | 2. l = ceil(|M| / N)                                              |
       | 3. K^1 | K^1_1 | ... | K^l | K^l_1 =                              |
                         = ACPKM-Master(T*, K, (k + n), l)                 |
       | 4. For j = 1, 2, ... , b - 1 do                                   |
       |        i = ceil(j * n / N),                                       |
       |        C_j = E_{K^i}(M_j (xor) C_{j-1})                           |
       | 5. SK = Generate_Subkey(K^l_1, |M_b|)                             |
       | 6. If |M_b| = n, then M*_b = M_b                                  |
       |                  else M*_b = M_b | 1 | 0^{n - 1 -|M_b|}           |
       | 7. T = E_{K^l}(M*_b (xor) C_{b-1} (xor) SK)                       |
       | 8. Return T                                                       |
       +-------------------------------------------------------------------+

    7.  Joint Usage of External and Internal Re-keying

       Both external re-keying and internal re-keying have their own
       advantages and disadvantages, which are discussed in Section 1.  For
       instance, using external re-keying can essentially limit the message
       length, while in the case of internal re-keying, the section size,
       which can be chosen as the maximal possible for operational
       properties, limits the number of separate messages.  Therefore, the
       choice of re-keying mechanism (either external or internal) depends
       on particular protocol features.  However, some protocols may have
       features that require the advantages of both the external and
       internal re-keying mechanisms: for example, the protocol mainly
       transmits short messages, but it must additionally support processing
       of very long messages.  In such situations, it is necessary to use
       external and internal re-keying jointly, since these techniques
       negate each other's disadvantages.

       For composition of external and internal re-keying techniques, any
       mechanism described in Section 5 can be used with any mechanism
       described in Section 6.

    Smyshlyaev                    Informational                    [Page 42]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       For example, consider the GCM-ACPKM mode with external serial
       re-keying based on a KDF on a hash function.  Denote the number of
       messages in each frame (in the case of the implicit approach to the
       key lifetime control) for external re-keying as a frame size.

       Let L be a key lifetime limitation.  The section size N for internal
       re-keying and the frame size q for external re-keying must be chosen
       in such a way that q * N must not exceed L.

       Suppose that t messages (ICN_i, P_i, A_i), with initial counter nonce
       ICN_i, plaintext P_i, and additional authenticated data A_i will be
       processed before renegotiation.

       For authenticated encryption of each message (ICN_i, P_i, A_i), i =
       1, ..., t, the following algorithm can be applied:

       1. j = ceil(i / q),
       2. K^j = ExtSerialH(K, j),
       3. C_i | T_i = GCM-ACPKM-Encrypt(N, K^j, ICN_i, P_i, A_i).

       Note that nonces ICN_i that are used under the same frame key must be
       unique for each message.

    8.  Security Considerations

       Re-keying should be used to increase a priori security properties of
       ciphers in hostile environments (e.g., with side-channel
       adversaries).  If efficient attacks on a cipher are known, the cipher
       must not be used.  Thus, re-keying cannot be used as a patch for
       vulnerable ciphers.  Base cipher properties must be well analyzed
       because the security of re-keying mechanisms is based on the security
       of a block cipher as a pseudorandom function.

       Re-keying is not intended to solve any postquantum security issues
       for symmetric cryptography, since the reduction of security caused by
       Grover's algorithm is not connected with a size of plaintext
       transformed by a cipher -- only a negligible (sufficient for key
       uniqueness) material is needed -- and the aim of re-keying is to
       limit the size of plaintext transformed under one initial key.

       Re-keying can provide backward security only if previous key material
       is securely deleted after usage by all parties.

    9.  IANA Considerations

       This document has no IANA actions.

    Smyshlyaev                    Informational                    [Page 43]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

    10.  References

    10.1.  Normative References

       [CMS]      Housley, R., "Cryptographic Message Syntax (CMS)", STD 70,
                  RFC 5652, DOI 10.17487/RFC5652, September 2009,
                  <https://www.rfc-editor.org/info/rfc5652>.

       [DTLS]     Rescorla, E. and N. Modadugu, "Datagram Transport Layer
                  Security Version 1.2", RFC 6347, DOI 10.17487/RFC6347,
                  January 2012, <https://www.rfc-editor.org/info/rfc6347>.

       [ESP]      Kent, S., "IP Encapsulating Security Payload (ESP)",
                  RFC 4303, DOI 10.17487/RFC4303, December 2005,
                  <https://www.rfc-editor.org/info/rfc4303>.

       [GCM]      Dworkin, M., "Recommendation for Block Cipher Modes of
                  Operation: Galois/Counter Mode (GCM) and GMAC", NIST
                  Special Publication 800-38D, DOI 10.6028/NIST.SP.800-38D,
                  November 2007,
                  <http://nvlpubs.nist.gov/nistpubs/Legacy/SP/
                  nistspecialpublication800-38d.pdf>.

       [MODES]    Dworkin, M., "Recommendation for Block Cipher Modes of
                  Operation: Methods and Techniques", NIST Special
                  Publication 800-38A, DOI 10.6028/NIST.SP.800-38A, December
                  2001.

       [NISTSP800-108]
                  National Institute of Standards and Technology,
                  "Recommendation for Key Derivation Using Pseudorandom
                  Functions", NIST Special Publication 800-108, October
                  2009, <http://nvlpubs.nist.gov/nistpubs/Legacy/SP/
                  nistspecialpublication800-108.pdf>.

       [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
                  Requirement Levels", BCP 14, RFC 2119,
                  DOI 10.17487/RFC2119, March 1997,
                  <https://www.rfc-editor.org/info/rfc2119>.

       [RFC4493]  Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
                  AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
                  2006, <https://www.rfc-editor.org/info/rfc4493>.

       [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
                  Key Derivation Function (HKDF)", RFC 5869,
                  DOI 10.17487/RFC5869, May 2010,
                  <https://www.rfc-editor.org/info/rfc5869>.

    Smyshlyaev                    Informational                    [Page 44]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       [RFC7836]  Smyshlyaev, S., Ed., Alekseev, E., Oshkin, I., Popov, V.,
                  Leontiev, S., Podobaev, V., and D. Belyavsky, "Guidelines
                  on the Cryptographic Algorithms to Accompany the Usage of
                  Standards GOST R 34.10-2012 and GOST R 34.11-2012",
                  RFC 7836, DOI 10.17487/RFC7836, March 2016,
                  <https://www.rfc-editor.org/info/rfc7836>.

       [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
                  2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
                  May 2017, <https://www.rfc-editor.org/info/rfc8174>.

       [SSH]      Ylonen, T. and C. Lonvick, Ed., "The Secure Shell (SSH)
                  Transport Layer Protocol", RFC 4253, DOI 10.17487/RFC4253,
                  January 2006, <https://www.rfc-editor.org/info/rfc4253>.

       [TLS]      Rescorla, E., "The Transport Layer Security (TLS) Protocol
                  Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
                  <https://www.rfc-editor.org/info/rfc8446>.

    10.2.  Informative References

       [AAOS2017] Ahmetzyanova, L., Alekseev, E., Oshkin, I., and S.
                  Smyshlyaev, "Increasing the Lifetime of Symmetric Keys for
                  the GCM Mode by Internal Re-keying", Cryptology ePrint
                  Archive, Report 2017/697, 2017,
                  <https://eprint.iacr.org/2017/697.pdf>.

       [AbBell]   Abdalla, M. and M. Bellare, "Increasing the Lifetime of a
                  Key: A Comparative Analysis of the Security of Re-keying
                  Techniques", ASIACRYPT 2000, Lecture Notes in Computer
                  Science, Volume 1976, pp. 546-559,
                  DOI 10.1007/3-540-44448-3_42, October 2000.

       [AESDUKPT] American National Standards Institute, "Retail Financial
                  Services Symmetric Key Management - Part 3: Derived Unique
                  Key Per Transaction", ANSI X9.24-3-2017, October 2017.

       [FKK2005]  Fu, K., Kamara, S., and T. Kohno, "Key Regression:
                  Enabling Efficient Key Distribution for Secure Distributed
                  Storage", November 2005, <https://homes.cs.washington.edu/
                  ~yoshi/papers/KR/NDSS06.pdf>.

    Smyshlyaev                    Informational                    [Page 45]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       [FPS2012]  Faust, S., Pietrzak, K., and J. Schipper, "Practical
                  Leakage-Resilient Symmetric Cryptography", Cryptographic
                  Hardware and Embedded Systems (CHES), Lecture Notes in
                  Computer Science, Volume 7428, pp. 213-232,
                  DOI 10.1007/978-3-642-33027-8_13, 2012,
                  <https://link.springer.com/content/
                  pdf/10.1007%2F978-3-642-33027-8_13.pdf>.

       [FRESHREKEYING]
                  Dziembowski, S., Faust, S., Herold, G., Journault, A.,
                  Masny, D., and F. Standaert, "Towards Sound Fresh
                  Re-Keying with Hard (Physical) Learning Problems",
                  Cryptology ePrint Archive, Report 2016/573, June 2016,
                  <https://eprint.iacr.org/2016/573>.

       [GGM]      Goldreich, O., Goldwasser, S., and S. Micali, "How to
                  Construct Random Functions", Journal of the Association
                  for Computing Machinery, Volume 33, No. 4, pp. 792-807,
                  DOI 10.1145/6490.6503, October 1986,
                  <https://dl.acm.org/citation.cfm?doid=6490.6503>.

       [KMNT2003] Kim, Y., Maino, F., Narasimha, M., and G. Tsudik, "Secure
                  Group Services for Storage Area Networks",
                  IEEE Communications Magazine 41, Number 8, pp. 92-99,
                  DOI 10.1109/SISW.2002.1183514, August 2003,
                  <https://ieeexplore.ieee.org/document/1183514>.

       [LDC]      Heys, H., "A Tutorial on Linear and Differential
                  Cryptanalysis", 2001, <https://citeseerx.ist.psu.edu/
                  viewdoc/citations?doi=10.1.1.2.2759>.

       [OWT]      Joye, M. and S. Yen, "One-Way Cross-Trees and Their
                  Applications", Public Key Cryptography (PKC), Lecture
                  Notes in Computer Science, Volume 2274,
                  DOI 10.1007/3-540-45664-3_25, February 2002,
                  <https://link.springer.com/content/
                  pdf/10.1007%2F3-540-45664-3_25.pdf>.

       [P3]       Alexander, P., "Subject: [Cfrg] Dynamic Key Changes on
                  Encrypted Sessions. - Draft I-D Attached", message to
                  the CFRG mailing list, 4 November 2017,
                  <https://mailarchive.ietf.org/arch/msg/cfrg/
                  ecTR3Hb-DFfrPCVmY0ghyYOEcxU>.

    Smyshlyaev                    Informational                    [Page 46]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       [Pietrzak2009]
                  Pietrzak, K., "A Leakage-Resilient Mode of Operation",
                  EUROCRYPT 2009, Lecture Notes in Computer Science, Volume
                  5479, pp. 462-482, DOI 10.1007/978-3-642-01001-9_27, April
                  2009, <https://iacr.org/archive/eurocrypt2009/
                  54790461/54790461.pdf>.

       [SIGNAL]   Perrin, T., Ed. and M. Marlinspike, "The Double Ratchet
                  Algorithm", November 2016, <https://signal.org/docs/
                  specifications/doubleratchet/doubleratchet.pdf>.

       [Sweet32]  Bhargavan, K. and G. Leurent, "On the Practical
                  (In-)Security of 64-bit Block Ciphers: Collision Attacks
                  on HTTP over TLS and OpenVPN", Proceedings of the 2016 ACM
                  SIGSAC Conference on Computer and Communications
                  Security, pp. 456-467, DOI 10.1145/2976749.2978423,
                  October 2016, <https://sweet32.info/SWEET32_CCS16.pdf>.

       [TAHA]     Taha, M. and P. Schaumont, "Key Updating for Leakage
                  Resiliency With Application to AES Modes of Operation",
                  IEEE Transactions on Information Forensics and Security,
                  DOI 10.1109/TIFS.2014.2383359, December 2014,
                  <http://ieeexplore.ieee.org/document/6987331/>.

       [TEMPEST]  Ramsay, C. and J. Lohuis, "TEMPEST attacks against AES.
                  Covertly stealing keys for 200 euro", June 2017,
                  <https://www.fox-it.com/en/wp-content/uploads/sites/11/
                  Tempest_attacks_against_AES.pdf>.

       [U2F]      Chang, D., Mishra, S., Sanadhya, S., and A. Singh, "On
                  Making U2F Protocol Leakage-Resilient via Re-keying",
                  Cryptology ePrint Archive, Report 2017/721, August 2017,
                  <https://eprint.iacr.org/2017/721.pdf>.

    Smyshlyaev                    Informational                    [Page 47]

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    Appendix A.  Test Examples

    A.1.  Test Examples for External Re-keying

    A.1.1.  External Re-keying with a Parallel Construction

       External re-keying with a parallel construction based on AES-256
       ****************************************************************
       k = 256
       t = 128

       Initial key:
       00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
       0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00

       K^1:
       51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86
       64 7D 5C D5 1C 3D 62 98 BC 09 B1 D8 64 EC D9 B1

       K^2:
       6F ED F5 D3 77 57 48 75 35 2B 5F 4D B6 5B E0 15
       B8 02 92 32 D8 D3 8D 73 FE DC DD C6 C8 36 78 BD

       K^3:
       B6 40 24 85 A4 24 BD 35 B4 26 43 13 76 26 70 B6
       5B F3 30 3D 3B 20 EB 14 D1 3B B7 91 74 E3 DB EC

       ...

       K^126:
       2F 3F 15 1B 53 88 23 CD 7D 03 FC 3D FD B3 57 5E
       23 E4 1C 4E 46 FF 6B 33 34 12 27 84 EF 5D 82 23

       K^127:
       8E 51 31 FB 0B 64 BB D0 BC D4 C5 7B 1C 66 EF FD
       97 43 75 10 6C AF 5D 5E 41 E0 17 F4 05 63 05 ED

       K^128:
       77 4F BF B3 22 60 C5 3B A3 8E FE B1 96 46 76 41
       94 49 AF 84 2D 84 65 A7 F4 F7 2C DC A4 9D 84 F9

       External re-keying with a parallel construction based on SHA-256
       ****************************************************************
       k = 256
       t = 128

       label:
       SHA2label

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       Initial key:
       00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
       0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00

       K^1:
       C1 A1 4C A0 30 29 BE 43 9F 35 3C 79 1A 51 48 57
       26 7A CD 5A E8 7D E7 D1 B2 E2 C7 AF A4 29 BD 35

       K^2:
       03 68 BB 74 41 2A 98 ED C4 7B 94 CC DF 9C F4 9E
       A9 B8 A9 5F 0E DC 3C 1E 3B D2 59 4D D1 75 82 D4

       K^3:
       2F D3 68 D3 A7 8F 91 E6 3B 68 DC 2B 41 1D AC 80
       0A C3 14 1D 80 26 3E 61 C9 0D 24 45 2A BD B1 AE

       ...

       K^126:
       55 AC 2B 25 00 78 3E D4 34 2B 65 0E 75 E5 8B 76
       C8 04 E9 D3 B6 08 7D C0 70 2A 99 A4 B5 85 F1 A1

       K^127:
       77 4D 15 88 B0 40 90 E5 8C 6A D7 5D 0F CF 0A 4A
       6C 23 F1 B3 91 B1 EF DF E5 77 64 CD 09 F5 BC AF

       K^128:
       E5 81 FF FB 0C 90 88 CD E5 F4 A5 57 B6 AB D2 2E
       94 C3 42 06 41 AB C1 72 66 CC 2F 59 74 9C 86 B3

    A.1.2.  External Re-keying with a Serial Construction

       External re-keying with a serial construction based on AES-256
       **************************************************************
       AES 256 examples:
       k = 256
       t = 128

       Initial key:
       00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
       0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00

       K*_1:
       00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
       0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00

    Smyshlyaev                    Informational                    [Page 49]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       K^1:
       66 B8 BD E5 90 6C EC DF FA 8A B2 FD 92 84 EB F0
       51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86

       K*_2:
       64 7D 5C D5 1C 3D 62 98 BC 09 B1 D8 64 EC D9 B1
       6F ED F5 D3 77 57 48 75 35 2B 5F 4D B6 5B E0 15

       K^2:
       66 B8 BD E5 90 6C EC DF FA 8A B2 FD 92 84 EB F0
       51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86

       K*_3:
       64 7D 5C D5 1C 3D 62 98 BC 09 B1 D8 64 EC D9 B1
       6F ED F5 D3 77 57 48 75 35 2B 5F 4D B6 5B E0 15

       K^3:
       66 B8 BD E5 90 6C EC DF FA 8A B2 FD 92 84 EB F0
       51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86

       ...

       K*_126:
       64 7D 5C D5 1C 3D 62 98 BC 09 B1 D8 64 EC D9 B1
       6F ED F5 D3 77 57 48 75 35 2B 5F 4D B6 5B E0 15

       K^126:
       66 B8 BD E5 90 6C EC DF FA 8A B2 FD 92 84 EB F0
       51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86

       K*_127:
       64 7D 5C D5 1C 3D 62 98 BC 09 B1 D8 64 EC D9 B1
       6F ED F5 D3 77 57 48 75 35 2B 5F 4D B6 5B E0 15

       K^127:
       66 B8 BD E5 90 6C EC DF FA 8A B2 FD 92 84 EB F0
       51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86

       K*_128:
       64 7D 5C D5 1C 3D 62 98 BC 09 B1 D8 64 EC D9 B1
       6F ED F5 D3 77 57 48 75 35 2B 5F 4D B6 5B E0 15

       K^128:
       66 B8 BD E5 90 6C EC DF FA 8A B2 FD 92 84 EB F0
       51 16 8A B6 C8 A8 38 65 54 85 31 A5 D2 BA C3 86

    Smyshlyaev                    Informational                    [Page 50]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       External re-keying with a serial construction based on SHA-256
       **************************************************************
       k = 256
       t = 128

       Initial key:
       00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
       0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00

       label1:
       SHA2label1

       label2:
       SHA2label2

       K*_1:
       00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
       0F 0E 0D 0C 0B 0A 09 08 07 06 05 04 03 02 01 00

       K^1:
       2D A8 D1 37 6C FD 52 7F F7 36 A4 E2 81 C6 0A 9B
       F3 8E 66 97 ED 70 4F B5 FB 10 33 CC EC EE D5 EC

       K*_2:
       14 65 5A D1 7C 19 86 24 9B D3 56 DF CC BE 73 6F
       52 62 4A 9D E3 CC 40 6D A9 48 DA 5C D0 68 8A 04

       K^2:
       2F EA 8D 57 2B EF B8 89 42 54 1B 8C 1B 3F 8D B1
       84 F9 56 C7 FE 01 11 99 1D FB 98 15 FE 65 85 CF

       K*_3:
       18 F0 B5 2A D2 45 E1 93 69 53 40 55 43 70 95 8D
       70 F0 20 8C DF B0 5D 67 CD 1B BF 96 37 D3 E3 EB

       K^3:
       53 C7 4E 79 AE BC D1 C8 24 04 BF F6 D7 B1 AC BF
       F9 C0 0E FB A8 B9 48 29 87 37 E1 BA E7 8F F7 92

       ...

       K*_126:
       A3 6D BF 02 AA 0B 42 4A F2 C0 46 52 68 8B C7 E6
       5E F1 62 C3 B3 2F DD EF E4 92 79 5D BB 45 0B CA

       K^126:
       6C 4B D6 22 DC 40 48 0F 29 C3 90 B8 E5 D7 A7 34
       23 4D 34 65 2C CE 4A 76 2C FE 2A 42 C8 5B FE 9A

    Smyshlyaev                    Informational                    [Page 51]

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       K*_127:
       84 5F 49 3D B8 13 1D 39 36 2B BE D3 74 8F 80 A1
       05 A7 07 37 BA 15 72 E0 73 49 C2 67 5D 0A 28 A1

       K^127:
       57 F0 BD 5A B8 2A F3 6B 87 33 CF F7 22 62 B4 D0
       F0 EE EF E1 50 74 E5 BA 13 C1 23 68 87 36 29 A2

       K*_128:
       52 F2 0F 56 5C 9C 56 84 AF 69 AD 45 EE B8 DA 4E
       7A A6 04 86 35 16 BA 98 E4 CB 46 D2 E8 9A C1 09

       K^128:
       9B DD 24 7D F3 25 4A 75 E0 22 68 25 68 DA 9D D5
       C1 6D 2D 2B 4F 3F 1F 2B 5E 99 82 7F 15 A1 4F A4

    A.2.  Test Examples for Internal Re-keying

    A.2.1.  Internal Re-keying Mechanisms that Do Not Require a Master Key

       CTR-ACPKM mode with AES-256
       ***************************
       k = 256
       n = 128
       c = 64
       N = 256

       Initial key K:
       00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
       00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

       Plaintext P:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88
       00010:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A
       00020:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00
       00030:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11
       00040:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22
       00050:   44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33
       00060:   55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44

       ICN:
       12 34 56 78 90 AB CE F0 A1 B2 C3 D4 E5 F0 01 12
       23 34 45 56 67 78 89 90 12 13 14 15 16 17 18 19

       D_1:
       00000:   80 81 82 83 84 85 86 87 88 89 8A 8B 8C 8D 8E 8F

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    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       D_2:
       00000:   90 91 92 93 94 95 96 97 98 99 9A 9B 9C 9D 9E 9F

       Section_1

       Section key K^1:
       00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
       00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

       Input block CTR_1:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 00

       Output block G_1:
       00000:   FD 7E F8 9A D9 7E A4 B8 8D B8 B5 1C 1C 9D 6D D0

       Input block CTR_2:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 01

       Output block G_2:
       00000:   19 98 C5 71 76 37 FB 17 11 E4 48 F0 0C 0D 60 B2

       Section_2

       Section key K^2:
       00000:   F6 80 D1 21 2F A4 3D F4 EC 3A 91 DE 2A B1 6F 1B
       00010:   36 B0 48 8A 4F C1 2E 09 98 D2 E4 A8 88 E8 4F 3D

       Input block CTR_3:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 02

       Output block G_3:
       00000:   E4 88 89 4F B6 02 87 DB 77 5A 07 D9 2C 89 46 EA

       Input block CTR_4:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 03

       Output block G_4:
       00000:   BC 4F 87 23 DB F0 91 50 DD B4 06 C3 1D A9 7C A4

       Section_3

       Section key K^3:
       00000:   8E B9 7E 43 27 1A 42 F1 CA 8E E2 5F 5C C7 C8 3B
       00010:   1A CE 9E 5E D0 6A A5 3B 57 B9 6A CF 36 5D 24 B8

       Input block CTR_5:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 04

    Smyshlyaev                    Informational                    [Page 53]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Output block G_5:
       00000:   68 6F 22 7D 8F B2 9C BD 05 C8 C3 7D 22 FE 3B B7

       Input block CTR_6:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 05

       Output block G_6:
       00000:   C0 1B F9 7F 75 6E 12 2F 80 59 55 BD DE 2D 45 87

       Section_4

       Section key K^4:
       00000:   C5 71 6C C9 67 98 BC 2D 4A 17 87 B7 8A DF 94 AC
       00010:   E8 16 F8 0B DB BC AD 7D 60 78 12 9C 0C B4 02 F5

       Block number 7:

       Input block CTR_7:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 06

       Output block G_7:
       00000:   03 DE 34 74 AB 9B 65 8A 3B 54 1E F8 BD 2B F4 7D

       The result G = G_1 | G_2 | G_3 | G_4 | G_5 | G_6 | G_7:
       00000:   FD 7E F8 9A D9 7E A4 B8 8D B8 B5 1C 1C 9D 6D D0
       00010:   19 98 C5 71 76 37 FB 17 11 E4 48 F0 0C 0D 60 B2
       00020:   E4 88 89 4F B6 02 87 DB 77 5A 07 D9 2C 89 46 EA
       00030:   BC 4F 87 23 DB F0 91 50 DD B4 06 C3 1D A9 7C A4
       00040:   68 6F 22 7D 8F B2 9C BD 05 C8 C3 7D 22 FE 3B B7
       00050:   C0 1B F9 7F 75 6E 12 2F 80 59 55 BD DE 2D 45 87
       00060:   03 DE 34 74 AB 9B 65 8A 3B 54 1E F8 BD 2B F4 7D

       The result ciphertext C = P (xor) MSB_{|P|}(G):
       00000:   EC 5C CB DE 8C 18 D3 B8 72 56 68 D0 A7 37 F4 58
       00010:   19 89 E7 42 32 62 9D 60 99 7D E2 4B C0 E3 9F B8
       00020:   F5 AA BA 0B E3 64 F0 53 EE F0 BC 15 C2 76 4C EA
       00030:   9E 7C C3 76 BD 87 19 C9 77 0F CA 2D E2 A3 7C B5
       00040:   5B 2B 77 1B F8 3A 05 17 BE 04 2D 82 28 FE 2A 95
       00050:   84 4E 9F 08 FD F7 B8 94 4C B7 AA B7 DE 3C 67 B4
       00060:   56 B8 43 FC 32 31 DE 46 D5 AB 14 F8 AC 09 C7 39

    Smyshlyaev                    Informational                    [Page 54]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       GCM-ACPKM mode with AES-128
       ***************************
       k = 128
       n = 128
       c = 32
       N = 256

       Initial key K:
       00000:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

       Additional data A:
       00000:   11 22 33

       Plaintext:
       00000:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
       00010:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
       00020:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

       ICN:
       00000:   00 00 00 00 00 00 00 00 00 00 00 00

       Number of sections: 2

       Section key K^1:
       00000:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

       Section key K^2:
       00000:   15 1A 9F B0 B6 AC C5 97 6A FB 50 31 D1 DE C8 41

       Encrypted GCTR_1 | GCTR_2 | GCTR_3:
       00000:   03 88 DA CE 60 B6 A3 92 F3 28 C2 B9 71 B2 FE 78
       00010:   F7 95 AA AB 49 4B 59 23 F7 FD 89 FF 94 8B C1 E0
       00020:   D6 B3 12 46 E9 CE 9F F1 3A B3 42 7E E8 91 96 AD

       Ciphertext C:
       00000:   03 88 DA CE 60 B6 A3 92 F3 28 C2 B9 71 B2 FE 78
       00010:   F7 95 AA AB 49 4B 59 23 F7 FD 89 FF 94 8B C1 E0
       00020:   D6 B3 12 46 E9 CE 9F F1 3A B3 42 7E E8 91 96 AD

       GHASH input:
       00000:   11 22 33 00 00 00 00 00 00 00 00 00 00 00 00 00
       00010:   03 88 DA CE 60 B6 A3 92 F3 28 C2 B9 71 B2 FE 78
       00020:   F7 95 AA AB 49 4B 59 23 F7 FD 89 FF 94 8B C1 E0
       00030:   D6 B3 12 46 E9 CE 9F F1 3A B3 42 7E E8 91 96 AD
       00040:   00 00 00 00 00 00 00 18 00 00 00 00 00 00 01 80

       GHASH output S:
       00000:   E8 ED E9 94 9A DD 55 30 B0 F4 4E F5 00 FC 3E 3C

    Smyshlyaev                    Informational                    [Page 55]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Authentication tag  T:
       00000:   B0 0F 15 5A 60 A3 65 51 86 8B 53 A2 A4 1B 7B 66

       The result C | T:
       00000:   03 88 DA CE 60 B6 A3 92 F3 28 C2 B9 71 B2 FE 78
       00010:   F7 95 AA AB 49 4B 59 23 F7 FD 89 FF 94 8B C1 E0
       00020:   D6 B3 12 46 E9 CE 9F F1 3A B3 42 7E E8 91 96 AD
       00030:   B0 0F 15 5A 60 A3 65 51 86 8B 53 A2 A4 1B 7B 66

    A.2.2.  Internal Re-keying Mechanisms with a Master Key

       CTR-ACPKM-Master mode with AES-256
       **********************************
       k = 256
       n = 128
       c for CTR-ACPKM mode = 64
       c for CTR-ACPKM-Master mode = 64
       N = 256
       T* = 512

       Initial key K:
       00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
       00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

       Initial vector ICN:
       00000:   12 34 56 78 90 AB CE F0 A1 B2 C3 D4 E5 F0 01 12

       Plaintext P:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88
       00010:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A
       00020:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00
       00030:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11
       00040:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22
       00050:   44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33
       00060:   55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44

       K^1 | K^2 | K^3 | K^4:
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60
       00020:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0
       00030:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3
       00040:   E8 76 2B 30 8B 08 EB CE 3E 93 9A C2 C0 3E 76 D4
       00050:   60 9A AB D9 15 33 13 D3 CF D3 94 E7 75 DF 3A 94
       00060:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00070:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12

    Smyshlyaev                    Informational                    [Page 56]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Section_1

       K^1:
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60

       Input block CTR_1:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 00

       Output block G_1:
       00000:   8C A2 B6 82 A7 50 65 3F 8E BF 08 E7 9F 99 4D 5C

       Input block CTR_2:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 01

       Output block G_2:
       00000:   F6 A6 A5 BA 58 14 1E ED 23 DC 31 68 D2 35 89 A1

       Section_2

       K^2:
       00000:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0
       00010:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3

       Input block CTR_3:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 02

       Output block G_3:
       00000:   4A 07 5F 86 05 87 72 94 1D 8E 7D F8 32 F4 23 71

       Input block CTR_4:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 03

       Output block G_4:
       00000:   23 35 66 AF 61 DD FE A7 B1 68 3F BA B0 52 4A D7

       Section_3

       K^3:
       00000:   E8 76 2B 30 8B 08 EB CE 3E 93 9A C2 C0 3E 76 D4
       00010:   60 9A AB D9 15 33 13 D3 CF D3 94 E7 75 DF 3A 94

       Input block CTR_5:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 04

    Smyshlyaev                    Informational                    [Page 57]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Output block G_5:
       00000:   A8 09 6D BC E8 BB 52 FC DE 6E 03 70 C1 66 95 E8

       Input block CTR_6:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 05

       Output block G_6:
       00000:   C6 E3 6E 8E 5B 82 AA C4 A6 6C 14 8D B1 F6 9B EF

       Section_4

       K^4:
       00000:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00010:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12

       Input block CTR_7:
       00000:   12 34 56 78 90 AB CE F0 00 00 00 00 00 00 00 06

       Output block G_7:
       00000:   82 2B E9 07 96 37 44 95 75 36 3F A7 07 F8 40 22

       The result G = G_1 | G_2 | G_3 | G_4 | G_5 | G_6 | G_7:
       00000:   8C A2 B6 82 A7 50 65 3F 8E BF 08 E7 9F 99 4D 5C
       00010:   F6 A6 A5 BA 58 14 1E ED 23 DC 31 68 D2 35 89 A1
       00020:   4A 07 5F 86 05 87 72 94 1D 8E 7D F8 32 F4 23 71
       00030:   23 35 66 AF 61 DD FE A7 B1 68 3F BA B0 52 4A D7
       00040:   A8 09 6D BC E8 BB 52 FC DE 6E 03 70 C1 66 95 E8
       00050:   C6 E3 6E 8E 5B 82 AA C4 A6 6C 14 8D B1 F6 9B EF
       00060:   82 2B E9 07 96 37 44 95 75 36 3F A7 07 F8 40 22

       The result ciphertext C = P (xor) MSB_{|P|}(G):
       00000:   9D 80 85 C6 F2 36 12 3F 71 51 D5 2B 24 33 D4 D4
       00010:   F6 B7 87 89 1C 41 78 9A AB 45 9B D3 1E DB 76 AB
       00020:   5B 25 6C C2 50 E1 05 1C 84 24 C6 34 DC 0B 29 71
       00030:   01 06 22 FA 07 AA 76 3E 1B D3 F3 54 4F 58 4A C6
       00040:   9B 4D 38 DA 9F 33 CB 56 65 A2 ED 8F CB 66 84 CA
       00050:   82 B6 08 F9 D3 1B 00 7F 6A 82 EB 87 B1 E7 B9 DC
       00060:   D7 4D 9E 8F 0F 9D FF 59 9B C9 35 A7 16 DA 73 66

    Smyshlyaev                    Informational                    [Page 58]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       GCM-ACPKM-Master mode with AES-256
       **********************************
       k = 192
       n = 128
       c for the CTR-ACPKM mode = 64
       c for the GCM-ACPKM-Master mode = 32
       T* = 384
       N = 256

       Initial key K:
       00000:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
       00010:   00 00 00 00 00 00 00 00

       Additional data A:
       00000:   11 22 33

       Plaintext:
       00000:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
       00010:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
       00020:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
       00030:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
       00040:   00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

       ICN:
       00000:   00 00 00 00 00 00 00 00 00 00 00 00

       Number of sections: 3

       K^1 | K^2 | K^3:
       00000:   93 BA AF FB 35 FB E7 39 C1 7C 6A C2 2E EC F1 8F
       00010:   7B 89 F0 BF 8B 18 07 05 96 48 68 9F 36 A7 65 CC
       00020:   CD 5D AC E2 0D 47 D9 18 D7 86 D0 41 A8 3B AB 99
       00030:   F5 F8 B1 06 D2 71 78 B1 B0 08 C9 99 0B 72 E2 87
       00040:   5A 2D 3C BE F1 6E 67 3C

       Encrypted GCTR_1 | ... | GCTR_5
       00000:   43 FA 71 81 64 B1 E3 D7 1E 7B 65 39 A7 02 1D 52
       00010:   69 9B 9E 1B 43 24 B7 52 95 74 E7 90 F2 BE 60 E8
       00020:   11 62 C9 90 2A 2B 77 7F D9 6A D6 1A 99 E0 C6 DE
       00030:   4B 91 D4 29 E3 1A 8C 11 AF F0 BC 47 F6 80 AF 14
       00040:   40 1C C1 18 14 63 8E 76 24 83 37 75 16 34 70 08

       Ciphertext C:
       00000:   43 FA 71 81 64 B1 E3 D7 1E 7B 65 39 A7 02 1D 52
       00010:   69 9B 9E 1B 43 24 B7 52 95 74 E7 90 F2 BE 60 E8
       00020:   11 62 C9 90 2A 2B 77 7F D9 6A D6 1A 99 E0 C6 DE
       00030:   4B 91 D4 29 E3 1A 8C 11 AF F0 BC 47 F6 80 AF 14
       00040:   40 1C C1 18 14 63 8E 76 24 83 37 75 16 34 70 08

    Smyshlyaev                    Informational                    [Page 59]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       GHASH input:
       00000:   11 22 33 00 00 00 00 00 00 00 00 00 00 00 00 00
       00010:   43 FA 71 81 64 B1 E3 D7 1E 7B 65 39 A7 02 1D 52
       00020:   69 9B 9E 1B 43 24 B7 52 95 74 E7 90 F2 BE 60 E8
       00030:   11 62 C9 90 2A 2B 77 7F D9 6A D6 1A 99 E0 C6 DE
       00040:   4B 91 D4 29 E3 1A 8C 11 AF F0 BC 47 F6 80 AF 14
       00050:   40 1C C1 18 14 63 8E 76 24 83 37 75 16 34 70 08
       00060:   00 00 00 00 00 00 00 18 00 00 00 00 00 00 02 80

       GHASH output S:
       00000:   6E A3 4B D5 6A C5 40 B7 3E 55 D5 86 D1 CC 09 7D

       Authentication tag  T:
       00050:   CC 3A BA 11 8C E7 85 FD 77 78 94 D4 B5 20 69 F8

       The result C | T:
       00000:   43 FA 71 81 64 B1 E3 D7 1E 7B 65 39 A7 02 1D 52
       00010:   69 9B 9E 1B 43 24 B7 52 95 74 E7 90 F2 BE 60 E8
       00020:   11 62 C9 90 2A 2B 77 7F D9 6A D6 1A 99 E0 C6 DE
       00030:   4B 91 D4 29 E3 1A 8C 11 AF F0 BC 47 F6 80 AF 14
       00040:   40 1C C1 18 14 63 8E 76 24 83 37 75 16 34 70 08
       00050:   CC 3A BA 11 8C E7 85 FD 77 78 94 D4 B5 20 69 F8

       CBC-ACPKM-Master mode with AES-256
       **********************************
       k = 256
       n = 128
       c for the CTR-ACPKM mode = 64
       N = 256
       T* = 512

       Initial key K:
       00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
       00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

       Initial vector IV:
       00000:   12 34 56 78 90 AB CE F0 A1 B2 C3 D4 E5 F0 01 12

       Plaintext P:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88
       00010:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A
       00020:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00
       00030:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11
       00040:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22
       00050:   44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33
       00060:   55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44

    Smyshlyaev                    Informational                    [Page 60]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       K^1 | K^2 | K^3 | K^4:
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60
       00020:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0
       00030:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3
       00040:   E8 76 2B 30 8B 08 EB CE 3E 93 9A C2 C0 3E 76 D4
       00050:   60 9A AB D9 15 33 13 D3 CF D3 94 E7 75 DF 3A 94
       00060:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00070:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12

       Section_1

       K^1:
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60

       Plaintext block P_1:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

       Input block P_1 (xor) C_0:
       00000:   03 16 65 3C C5 CD B9 F0 5E 5C 1E 18 5E 5A 98 9A

       Output block C_1:
       00000:   59 CB 5B CA C2 69 2C 60 0D 46 03 A0 C7 40 C9 7C

       Plaintext block P_2:
       00000:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A

       Input block P_2 (xor) C_1:
       00000:   59 DA 79 F9 86 3C 4A 17 85 DF A9 1B 0B AE 36 76

       Output block C_2:
       00000:   80 B6 02 74 54 8B F7 C9 78 1F A1 05 8B F6 8B 42

       Section_2

       K^2:
       00000:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0
       00010:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3

       Plaintext block P_3:
       00000:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00

       Input block P_3 (xor) C_2:
       00000:   91 94 31 30 01 ED 80 41 E1 B5 1A C9 65 09 81 42

       Output block C_3:
       00000:   8C 24 FB CF 68 15 B1 AF 65 FE 47 75 95 B4 97 59

    Smyshlyaev                    Informational                    [Page 61]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Plaintext block P_4:
       00000:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11

       Input block P_4 (xor) C_3:
       00000:   AE 17 BF 9A 0E 62 39 36 CF 45 8B 9B 6A BE 97 48

       Output block C_4:
       00000:   19 65 A5 00 58 0D 50 23 72 1B E9 90 E1 83 30 E9

       Section_3

       K^3:
       00000:   E8 76 2B 30 8B 08 EB CE 3E 93 9A C2 C0 3E 76 D4
       00010:   60 9A AB D9 15 33 13 D3 CF D3 94 E7 75 DF 3A 94

       Plaintext block P_5:
       00000:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22

       Input block P_5 (xor) C_4:
       00000:   2A 21 F0 66 2F 85 C9 89 C9 D7 07 6F EB 83 21 CB

       Output block C_5:
       00000:   56 D8 34 F4 6F 0F 4D E6 20 53 A9 5C B5 F6 3C 14

       Plaintext block P_6:
       00000:   44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33

       Input block P_6 (xor) C_5:
       00000:   12 8D 52 83 E7 96 E7 5D EC BD 56 56 B5 E7 1E 27

       Output block C_6:
       00000:   66 68 2B 8B DD 6E B2 7E DE C7 51 D6 2F 45 A5 45

       Section_4

       K^4:
       00000:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00010:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12

       Plaintext block P_7:
       00000:   55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33 44

       Input block P_7 (xor) C_6:
       00000:   33 0E 5C 03 44 C4 09 B2 30 38 5B D6 3E 67 96 01

       Output block C_7:
       00000:   7F 4D 87 F9 CA E9 56 09 79 C4 FA FE 34 0B 45 34

    Smyshlyaev                    Informational                    [Page 62]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Ciphertext C:
       00000:   59 CB 5B CA C2 69 2C 60 0D 46 03 A0 C7 40 C9 7C
       00010:   80 B6 02 74 54 8B F7 C9 78 1F A1 05 8B F6 8B 42
       00020:   8C 24 FB CF 68 15 B1 AF 65 FE 47 75 95 B4 97 59
       00030:   19 65 A5 00 58 0D 50 23 72 1B E9 90 E1 83 30 E9
       00040:   56 D8 34 F4 6F 0F 4D E6 20 53 A9 5C B5 F6 3C 14
       00050:   66 68 2B 8B DD 6E B2 7E DE C7 51 D6 2F 45 A5 45
       00060:   7F 4D 87 F9 CA E9 56 09 79 C4 FA FE 34 0B 45 34

       CFB-ACPKM-Master mode with AES-256
       **********************************
       k = 256
       n = 128
       c for the CTR-ACPKM mode = 64
       N = 256
       T* = 512

       Initial key K:
       00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
       00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

       Initial vector IV:
       00000:   12 34 56 78 90 AB CE F0 A1 B2 C3 D4 E5 F0 01 12

       Plaintext P:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88
       00010:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A
       00020:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00
       00030:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11
       00040:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22
       00050:   44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33
       00060:   55 66 77 88 99 AA BB CC

       K^1 | K^2 | K^3 | K^4
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60
       00020:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0
       00030:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3
       00040:   E8 76 2B 30 8B 08 EB CE 3E 93 9A C2 C0 3E 76 D4
       00050:   60 9A AB D9 15 33 13 D3 CF D3 94 E7 75 DF 3A 94
       00060:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00070:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12

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    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Section_1

       K^1:
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60

       Plaintext block P_1:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

       Encrypted block E_{K^1}(C_0):
       00000:   1C 39 9D 59 F8 5D 91 91 A9 D2 12 9F 63 15 90 03

       Output block C_1 = E_{K^1}(C_0) (xor) P_1:
       00000:   0D 1B AE 1D AD 3B E6 91 56 3C CF 53 D8 BF 09 8B

       Plaintext block P_2:
       00000:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A

       Encrypted block E_{K^1}(C_1):
       00000:   6B A2 C5 42 52 69 C6 0B 15 14 06 87 90 46 F6 2E

       Output block C_2 = E_{K^1}(C_1) (xor) P_2:
       00000:   6B B3 E7 71 16 3C A0 7C 9D 8D AC 3C 5C A8 09 24

       Section_2

       K^2:
       00000:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0
       00010:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3

       Plaintext block P_3:
       00000:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00

       Encrypted block E_{K^2}(C_2):
       00000:   95 45 5F DB C3 9E 0A 13 9F CB 10 F5 BD 79 A3 88

       Output block C_3 = E_{K^2}(C_2) (xor) P_3:
       00000:   84 67 6C 9F 96 F8 7D 9B 06 61 AB 39 53 86 A9 88

       Plaintext block P_4:
       00000:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11

       Encrypted block E_{K^2}(C_3):
       00000:   E0 AA 32 5D 80 A4 47 95 BA 42 BF 63 F8 4A C8 B2

       Output block C_4 = E_{K^2}(C_3) (xor) P_4:
       00000:   C2 99 76 08 E6 D3 CF 0C 10 F9 73 8D 07 40 C8 A3

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    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Section_3

       K^3:
       00000:   E8 76 2B 30 8B 08 EB CE 3E 93 9A C2 C0 3E 76 D4
       00010:   60 9A AB D9 15 33 13 D3 CF D3 94 E7 75 DF 3A 94

       Plaintext block P_5:
       00000:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22

       Encrypted block E_{K^3}(C_4):
       00000:   FE 42 8C 70 C2 51 CE 13 36 C1 BF 44 F8 49 66 89

       Output block C_5 = E_{K^3}(C_4) (xor) P_5:
       00000:   CD 06 D9 16 B5 D9 57 B9 8D 0D 51 BB F2 49 77 AB

       Plaintext block P_6:
       00000:   44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22 33

       Encrypted block E_{K^3}(C_5):
       00000:   01 24 80 87 86 18 A5 43 11 0A CC B5 0A E5 02 A3

       Output block C_6 = E_{K^3}(C_5) (xor) P_6:
       00000:   45 71 E6 F0 0E 81 0F F8 DD E4 33 BF 0A F4 20 90

       Section_4

       K^4:
       00000:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00010:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12

       Plaintext block P_7:
       00000:   55 66 77 88 99 AA BB CC

       Encrypted block MSB_{|P_7|}(E_{K^4}(C_6)):
       00000:   97 5C 96 37 55 1E 8C 7F

       Output block C_7 = MSB_{|P_7|}(E_{K^4}(C_6)) (xor) P_7
       00000:   C2 3A E1 BF CC B4 37 B3

       Ciphertext C:
       00000:   0D 1B AE 1D AD 3B E6 91 56 3C CF 53 D8 BF 09 8B
       00010:   6B B3 E7 71 16 3C A0 7C 9D 8D AC 3C 5C A8 09 24
       00020:   84 67 6C 9F 96 F8 7D 9B 06 61 AB 39 53 86 A9 88
       00030:   C2 99 76 08 E6 D3 CF 0C 10 F9 73 8D 07 40 C8 A3
       00040:   CD 06 D9 16 B5 D9 57 B9 8D 0D 51 BB F2 49 77 AB
       00050:   45 71 E6 F0 0E 81 0F F8 DD E4 33 BF 0A F4 20 90
       00060:   C2 3A E1 BF CC B4 37 B3

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    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       OMAC-ACPKM-Master mode with AES-256
       ***********************************
       k = 256
       n = 128
       c for the CTR-ACPKM mode = 64
       N = 256
       T* = 768

       Initial key K:
       00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
       00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

       Plaintext M:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88
       00010:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A
       00020:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00
       00030:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11
       00040:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22

       K^1 | K^1_1 | K^2 | K^2_1 | K^3 | K^3_1:
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60
       00020:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0
       00030:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3
       00040:   9D CC 66 42 0D FF 45 5B 21 F3 93 F0 D4 D6 6E 67
       00050:   BB 1B 06 0B 87 66 6D 08 7A 9D A7 49 55 C3 5B 48
       00060:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00070:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12
       00080:   78 21 C7 C7 6C BD 79 63 56 AC F8 8E 69 6A 00 07

       Section_1

       K^1:
       00000:   9F 10 BB F1 3A 79 FB BD 4A 4C A8 64 C4 90 74 64
       00010:   39 FE 50 6D 4B 86 9B 21 03 A3 B6 A4 79 28 3C 60

       K^1_1:
       00000:   77 91 17 50 E0 D1 77 E5 9A 13 78 2B F1 89 08 D0

       Plaintext block M_1:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

       Input block M_1 (xor) C_0:
       00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

       Output block C_1:
       00000:   0B A5 89 BF 55 C1 15 42 53 08 89 76 A0 FE 24 3E

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    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       Plaintext block M_2:
       00000:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A

       Input block M_2 (xor) C_1:
       00000:   0B B4 AB 8C 11 94 73 35 DB 91 23 CD 6C 10 DB 34

       Output block C_2:
       00000:   1C 53 DD A3 6D DC E1 17 ED 1F 14 09 D8 6A F3 2C

       Section_2

       K^2:
       00000:   AB 6B 59 EE 92 49 05 B3 AB C7 A4 E3 69 65 76 C3
       00010:   9D CC 66 42 0D FF 45 5B 21 F3 93 F0 D4 D6 6E 67

       K^2_1:
       00000:   BB 1B 06 0B 87 66 6D 08 7A 9D A7 49 55 C3 5B 48

       Plaintext block M_3:
       00000:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00

       Input block M_3 (xor) C_2:
       00000:   0D 71 EE E7 38 BA 96 9F 74 B5 AF C5 36 95 F9 2C

       Output block C_3:
       00000:   4E D4 BC A6 CE 6D 6D 16 F8 63 85 13 E0 48 59 75

       Plaintext block M_4:
       00000:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11

       Input block M_4 (xor) C_3:
       00000:   6C E7 F8 F3 A8 1A E5 8F 52 D8 49 FD 1F 42 59 64

       Output block C_4:
       00000:   B6 83 E3 96 FD 30 CD 46 79 C1 8B 24 03 82 1D 81

       Section_3

       K^3:
       00000:   F2 EE 91 45 6B DC 3D E4 91 2C 87 C3 29 CF 31 A9
       00010:   2F 20 2E 5A C4 9A 2A 65 31 33 D6 74 8C 4F F9 12

       K^3_1:
       00000:   78 21 C7 C7 6C BD 79 63 56 AC F8 8E 69 6A 00 07

       MSB1(K1) == 0 -> K2 = K1 << 1

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    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

       K1:
       00000:   78 21 C7 C7 6C BD 79 63 56 AC F8 8E 69 6A 00 07

       K2:
       00000:   F0 43 8F 8E D9 7A F2 C6 AD 59 F1 1C D2 D4 00 0E

       Plaintext M_5:
       00000:   33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11 22

       Using K1, padding is not required

       Input block M_5 (xor) C_4:
       00000:   FD E6 71 37 E6 05 2D 8F 94 A1 9D 55 60 E8 0C A4

       Output block C_5:
       00000:   B3 AD B8 92 18 32 05 4C 09 21 E7 B8 08 CF A0 B8

       Message authentication code T:
       00000:   B3 AD B8 92 18 32 05 4C 09 21 E7 B8 08 CF A0 B8

    Smyshlyaev                    Informational                    [Page 68]

    RFC 8645         Re-keying Mechanisms for Symmetric Keys     August 2019

    Acknowledgments

       We thank Mihir Bellare, Scott Fluhrer, Dorothy Cooley, Yoav Nir, Jim
       Schaad, Paul Hoffman, Dmitry Belyavsky, Yaron Sheffer, Alexey
       Melnikov, and Spencer Dawkins for their useful comments.

    Contributors

       Russ Housley
       Vigil Security, LLC
       housley@vigilsec.com

       Evgeny Alekseev
       CryptoPro
       alekseev@cryptopro.ru

       Ekaterina Smyshlyaeva
       CryptoPro
       ess@cryptopro.ru

       Shay Gueron
       University of Haifa, Israel
       Intel Corporation, Israel Development Center, Israel
       shay.gueron@gmail.com

       Daniel Fox Franke
       Akamai Technologies
       dfoxfranke@gmail.com

       Lilia Ahmetzyanova
       CryptoPro
       lah@cryptopro.ru

    Author's Address

       Stanislav Smyshlyaev (editor)
       CryptoPro
       18, Suschevskiy val
       Moscow  127018
       Russian Federation

       Phone: +7 (495) 995-48-20
       Email: svs@cryptopro.ru

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