    Network Working Group                                   D. Eastlake, 3rd
    Request for Comments: 4086                         Motorola Laboratories
    BCP: 106                                                     J. Schiller
    Obsoletes: 1750                                                      MIT
    Category: Best Current Practice                               S. Crocker
                                                                   June 2005

                      Randomness Requirements for Security

    Status of This Memo

       This document specifies an Internet Best Current Practices for the
       Internet Community, and requests discussion and suggestions for
       improvements.  Distribution of this memo is unlimited.

    Copyright Notice

       Copyright (C) The Internet Society (2005).

    Abstract

       Security systems are built on strong cryptographic algorithms that
       foil pattern analysis attempts.  However, the security of these
       systems is dependent on generating secret quantities for passwords,
       cryptographic keys, and similar quantities.  The use of pseudo-random
       processes to generate secret quantities can result in pseudo-
       security.  A sophisticated attacker may find it easier to reproduce
       the environment that produced the secret quantities and to search the
       resulting small set of possibilities than to locate the quantities in
       the whole of the potential number space.

       Choosing random quantities to foil a resourceful and motivated
       adversary is surprisingly difficult.  This document points out many
       pitfalls in using poor entropy sources or traditional pseudo-random
       number generation techniques for generating such quantities.  It
       recommends the use of truly random hardware techniques and shows that
       the existing hardware on many systems can be used for this purpose.
       It provides suggestions to ameliorate the problem when a hardware
       solution is not available, and it gives examples of how large such
       quantities need to be for some applications.

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    Table of Contents

       1. Introduction and Overview .......................................3
       2. General Requirements ............................................4
       3. Entropy Sources .................................................7
          3.1. Volume Required ............................................7
          3.2. Existing Hardware Can Be Used For Randomness ...............8
               3.2.1. Using Existing Sound/Video Input ....................8
               3.2.2. Using Existing Disk Drives ..........................8
          3.3. Ring Oscillator Sources ....................................9
          3.4. Problems with Clocks and Serial Numbers ...................10
          3.5. Timing and Value of External Events .......................11
          3.6. Non-hardware Sources of Randomness ........................12
       4. De-skewing .....................................................12
          4.1. Using Stream Parity to De-Skew ............................13
          4.2. Using Transition Mappings to De-Skew ......................14
          4.3. Using FFT to De-Skew ......................................15
          4.4. Using Compression to De-Skew ..............................15
       5. Mixing .........................................................16
          5.1. A Trivial Mixing Function .................................17
          5.2. Stronger Mixing Functions .................................18
          5.3. Using S-Boxes for Mixing ..................................19
          5.4. Diffie-Hellman as a Mixing Function .......................19
          5.5. Using a Mixing Function to Stretch Random Bits ............20
          5.6. Other Factors in Choosing a Mixing Function ...............20
       6. Pseudo-random Number Generators ................................21
          6.1. Some Bad Ideas ............................................21
               6.1.1. The Fallacy of Complex Manipulation ................21
               6.1.2. The Fallacy of Selection from a Large Database .....22
               6.1.3. Traditional Pseudo-random Sequences ................23
          6.2. Cryptographically Strong Sequences ........................24
               6.2.1. OFB and CTR Sequences ..............................25
               6.2.2. The Blum Blum Shub Sequence Generator ..............26
          6.3. Entropy Pool Techniques ...................................27
       7. Randomness Generation Examples and Standards ...................28
          7.1. Complete Randomness Generators ............................28
               7.1.1. US DoD Recommendations for Password Generation .....28
               7.1.2. The /dev/random Device .............................29
               7.1.3. Windows CryptGenRandom .............................30
          7.2. Generators Assuming a Source of Entropy ...................31
               7.2.1. X9.82 Pseudo-Random Number Generation ..............31
               7.2.2. X9.17 Key Generation ...............................33
               7.2.3. DSS Pseudo-random Number Generation ................34
       8. Examples of Randomness Required ................................34
          8.1. Password Generation .......................................35
          8.2. A Very High Security Cryptographic Key ....................36
       9. Conclusion .....................................................38
      10. Security Considerations ........................................38

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      11. Acknowledgments ................................................39
      Appendix A: Changes from RFC 1750 ..................................40
      Informative References .............................................41

    1.  Introduction and Overview

       Software cryptography is coming into wider use, although there is a
       long way to go until it becomes pervasive.  Systems such as SSH,
       IPSEC, TLS, S/MIME, PGP, DNSSEC, and Kerberos are maturing and
       becoming a part of the network landscape [SSH] [IPSEC] [TLS] [S/MIME]
       [MAIL_PGP*] [DNSSEC*].  For comparison, when the previous version of
       this document [RFC1750] was issued in 1994, the only Internet
       cryptographic security specification in the IETF was the Privacy
       Enhanced Mail protocol [MAIL_PEM*].

       These systems provide substantial protection against snooping and
       spoofing.  However, there is a potential flaw.  At the heart of all
       cryptographic systems is the generation of secret, unguessable (i.e.,
       random) numbers.

       The lack of generally available facilities for generating such random
       numbers (that is, the lack of general availability of truly
       unpredictable sources) forms an open wound in the design of
       cryptographic software.  For the software developer who wants to
       build a key or password generation procedure that runs on a wide
       range of hardware, this is a very real problem.

       Note that the requirement is for data that an adversary has a very
       low probability of guessing or determining.  This can easily fail if
       pseudo-random data is used that meets only traditional statistical
       tests for randomness, or that is based on limited-range sources such
       as clocks.  Sometimes such pseudo-random quantities can be guessed by
       an adversary searching through an embarrassingly small space of
       possibilities.

       This Best Current Practice document describes techniques for
       producing random quantities that will be resistant to attack.  It
       recommends that future systems include hardware random number
       generation or provide access to existing hardware that can be used
       for this purpose.  It suggests methods for use if such hardware is
       not available, and it gives some estimates of the number of random
       bits required for sample applications.

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    2.  General Requirements

       Today, a commonly encountered randomness requirement is to pick a
       user password, usually a simple character string.  Obviously, a
       password that can be guessed does not provide security.  For re-
       usable passwords, it is desirable that users be able to remember the
       password.  This may make it advisable to use pronounceable character
       strings or phrases composed of ordinary words.  But this affects only
       the format of the password information, not the requirement that the
       password be very hard to guess.

       Many other requirements come from the cryptographic arena.
       Cryptographic techniques can be used to provide a variety of
       services, including confidentiality and authentication.  Such
       services are based on quantities, traditionally called "keys", that
       are unknown to and unguessable by an adversary.

       There are even TCP/IP protocol uses for randomness in picking initial
       sequence numbers [RFC1948].

       Generally speaking, the above examples also illustrate two different
       types of random quantities that may be wanted.  In the case of
       human-usable passwords, the only important characteristic is that
       they be unguessable.  It is not important that they may be composed
       of ASCII characters, so the top bit of every byte is zero, for
       example.  On the other hand, for fixed length keys and the like, one
       normally wants quantities that appear to be truly random, that is,
       quantities whose bits will pass statistical randomness tests.

       In some cases, such as the use of symmetric encryption with the one-
       time pads or an algorithm like the US Advanced Encryption Standard
       [AES], the parties who wish to communicate confidentially and/or with
       authentication must all know the same secret key.  In other cases,
       where asymmetric or "public key" cryptographic techniques are used,
       keys come in pairs.  One key of the pair is private and must be kept
       secret by one party; the other is public and can be published to the
       world.  It is computationally infeasible to determine the private key
       from the public key, and knowledge of the public key is of no help to
       an adversary [ASYMMETRIC].  See general references [SCHNEIER,
       FERGUSON, KAUFMAN].

       The frequency and volume of the requirement for random quantities
       differs greatly for different cryptographic systems.  With pure RSA,
       random quantities are required only when a new key pair is generated;
       thereafter, any number of messages can be signed without a further
       need for randomness.  The public key Digital Signature Algorithm
       devised by the US National Institute of Standards and Technology
       (NIST) requires good random numbers for each signature [DSS].  And

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       encrypting with a one-time pad (in principle the strongest possible
       encryption technique) requires randomness of equal volume to all the
       messages to be processed.  See general references [SCHNEIER,
       FERGUSON, KAUFMAN].

       In most of these cases, an adversary can try to determine the
       "secret" key by trial and error.  This is possible as long as the key
       is enough smaller than the message that the correct key can be
       uniquely identified.  The probability of an adversary succeeding at
       this must be made acceptably low, depending on the particular
       application.  The size of the space the adversary must search is
       related to the amount of key "information" present, in an
       information-theoretic sense [SHANNON].  This depends on the number of
       different secret values possible and the probability of each value,
       as follows:

                                  -----
                                  \
            Bits of information =  \     - p   * log  ( p  )
                                   /        i       2    i
                                  /
                                  -----

       where i counts from 1 to the number of possible secret values and p
       sub i is the probability of the value numbered i.  (Because p sub i
       is less than one, the log will be negative, so each term in the sum
       will be non-negative.)

       If there are 2^n different values of equal probability, then n bits
       of information are present and an adversary would have to try, on the
       average, half of the values, or 2^(n-1), before guessing the secret
       quantity.  If the probability of different values is unequal, then
       there is less information present, and fewer guesses will, on
       average, be required by an adversary.  In particular, any values that
       an adversary can know to be impossible or of low probability can be
       initially ignored by the adversary, who will search through the more
       probable values first.

       For example, consider a cryptographic system that uses 128-bit keys.
       If these keys are derived using a fixed pseudo-random number
       generator that is seeded with an 8-bit seed, then an adversary needs
       to search through only 256 keys (by running the pseudo-random number
       generator with every possible seed), not 2^128 keys as may at first
       appear to be the case.  Only 8 bits of "information" are in these
       128-bit keys.

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       While the above analysis is correct on average, it can be misleading
       in some cases for cryptographic analysis where what is really
       important is the work factor for an adversary.  For example, assume
       that there is a pseudo-random number generator generating 128-bit
       keys, as in the previous paragraph, but that it generates zero half
       of the time and a random selection from the remaining 2^128 - 1
       values the rest of the time.  The Shannon equation above says that
       there are 64 bits of information in one of these key values, but an
       adversary, simply by trying the value zero, can break the security of
       half of the uses, albeit a random half.  Thus, for cryptographic
       purposes, it is also useful to look at other measures, such as min-
       entropy, defined as

            Min-entropy =  - log  ( maximum ( p  ) )
                                               i

       where i is as above.  Using this equation, we get 1 bit of min-
       entropy for our new hypothetical distribution, as opposed to 64 bits
       of classical Shannon entropy.

       A continuous spectrum of entropies, sometimes called Renyi entropy,
       has been defined, specified by the parameter r.  Here r = 1 is
       Shannon entropy and r = infinity is min-entropy.  When r = zero, it
       is just log (n), where n is the number of non-zero probabilities.
       Renyi entropy is a non-increasing function of r, so min-entropy is
       always the most conservative measure of entropy and usually the best
       to use for cryptographic evaluation [LUBY].

       Statistically tested randomness in the traditional sense is NOT the
       same as the unpredictability required for security use.

       For example, the use of a widely available constant sequence, such as
       the random table from the CRC Standard Mathematical Tables, is very
       weak against an adversary.  An adversary who learns of or guesses it
       can easily break all security, future and past, based on the sequence
       [CRC].  As another example, using AES with a constant key to encrypt
       successive integers such as 1, 2, 3, ... will produce output that
       also has excellent statistical randomness properties but is
       predictable.  On the other hand, taking successive rolls of a six-
       sided die and encoding the resulting values in ASCII would produce
       statistically poor output with a substantial unpredictable component.
       So note that passing or failing statistical tests doesn't reveal
       whether something is unpredictable or predictable.

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    3.  Entropy Sources

       Entropy sources tend to be very implementation dependent.  Once one
       has gathered sufficient entropy, it can be used as the seed to
       produce the required amount of cryptographically strong pseudo-
       randomness, as described in Sections 6 and 7, after being de-skewed
       or mixed as necessary, as described in Sections 4 and 5.

       Is there any hope for true, strong, portable randomness in the
       future?  There might be.  All that's needed is a physical source of
       unpredictable numbers.

       Thermal noise (sometimes called Johnson noise in integrated circuits)
       or a radioactive decay source and a fast, free-running oscillator
       would do the trick directly [GIFFORD].  This is a trivial amount of
       hardware, and it could easily be included as a standard part of a
       computer system's architecture.  Most audio (or video) input devices
       are usable [TURBID].  Furthermore, any system with a spinning disk or
       ring oscillator and a stable (crystal) time source or the like has an
       adequate source of randomness ([DAVIS] and Section 3.3).  All that's
       needed is the common perception among computer vendors that this
       small additional hardware and the software to access it is necessary
       and useful.

       ANSI X9 is currently developing a standard that includes a part
       devoted to entropy sources.  See Part 2 of [X9.82].

    3.1.  Volume Required

       How much unpredictability is needed?  Is it possible to quantify the
       requirement in terms of, say, number of random bits per second?

       The answer is that not very much is needed.  For AES, the key can be
       128 bits, and, as we show in an example in Section 8, even the
       highest security system is unlikely to require strong keying material
       of much over 200 bits.  If a series of keys is needed, they can be
       generated from a strong random seed (starting value) using a
       cryptographically strong sequence, as explained in Section 6.2.  A
       few hundred random bits generated at start-up or once a day is enough
       if such techniques are used.  Even if the random bits are generated
       as slowly as one per second and it is not possible to overlap the
       generation process, it should be tolerable in most high-security
       applications to wait 200 seconds occasionally.

       These numbers are trivial to achieve.  It could be achieved by a
       person repeatedly tossing a coin, and almost any hardware based
       process is likely to be much faster.

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    3.2.  Existing Hardware Can Be Used For Randomness

       As described below, many computers come with hardware that can, with
       care, be used to generate truly random quantities.

    3.2.1.  Using Existing Sound/Video Input

       Many computers are built with inputs that digitize some real-world
       analog source, such as sound from a microphone or video input from a
       camera.  The "input" from a sound digitizer with no source plugged in
       or from a camera with the lens cap on is essentially thermal noise.
       If the system has enough gain to detect anything, such input can
       provide reasonably high quality random bits.  This method is
       extremely dependent on the hardware implementation.

       For example, on some UNIX-based systems, one can read from the
       /dev/audio device with nothing plugged into the microphone jack or
       with the microphone receiving only low level background noise.  Such
       data is essentially random noise, although it should not be trusted
       without some checking, in case of hardware failure, and it will have
       to be de-skewed.

       Combining this approach with compression to de-skew (see Section 4),
       one can generate a huge amount of medium-quality random data with the
       UNIX-style command line:

            cat /dev/audio | compress - >random-bits-file

       A detailed examination of this type of randomness source appears in
       [TURBID].

    3.2.2.  Using Existing Disk Drives

       Disk drives have small random fluctuations in their rotational speed
       due to chaotic air turbulence [DAVIS, Jakobsson].  The addition of
       low-level disk seek-time instrumentation produces a series of
       measurements that contain this randomness.  Such data is usually
       highly correlated, so significant processing is needed, as described
       in Section 5.2 below.  Nevertheless, experimentation a decade ago
       showed that, with such processing, even slow disk drives on the
       slower computers of that day could easily produce 100 bits a minute
       or more of excellent random data.

       Every increase in processor speed, which increases the resolution
       with which disk motion can be timed or increases the rate of disk
       seeks, increases the rate of random bit generation possible with this
       technique.  At the time of this paper and with modern hardware, a
       more typical rate of random bit production would be in excess of

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       10,000 bits a second.  This technique is used in random number
       generators included in many operating system libraries.

       Note: the inclusion of cache memories in disk controllers has little
       effect on this technique if very short seek times, which represent
       cache hits, are simply ignored.

    3.3.  Ring Oscillator Sources

       If an integrated circuit is being designed or field-programmed, an
       odd number of gates can be connected in series to produce a free-
       running ring oscillator.  By sampling a point in the ring at a fixed
       frequency (for example, one determined by a stable crystal
       oscillator), some amount of entropy can be extracted due to
       variations in the free-running oscillator timing.  It is possible to
       increase the rate of entropy by XOR'ing sampled values from a few
       ring oscillators with relatively prime lengths.  It is sometimes
       recommended that an odd number of rings be used so that, even if the
       rings somehow become synchronously locked to each other, there will
       still be sampled bit transitions.  Another possible source to sample
       is the output of a noisy diode.

       Sampled bits from such sources will have to be heavily de-skewed, as
       disk rotation timings must be (see Section 4).  An engineering study
       would be needed to determine the amount of entropy being produced
       depending on the particular design.  In any case, these can be good
       sources whose cost is a trivial amount of hardware by modern
       standards.

       As an example, IEEE 802.11i suggests the circuit below, with due
       attention in the design to isolation of the rings from each other and
       from clocked circuits to avoid undesired synchronization, etc., and
       with extensive post processing [IEEE_802.11i].

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                 |\     |\                |\
             +-->| >0-->| >0-- 19 total --| >0--+-------+
             |   |/     |/                |/    |       |
             |                                  |       |
             +----------------------------------+       V
                                                     +-----+
                 |\     |\                |\         |     | output
             +-->| >0-->| >0-- 23 total --| >0--+--->| XOR |------>
             |   |/     |/                |/    |    |     |
             |                                  |    +-----+
             +----------------------------------+      ^ ^
                                                       | |
                 |\     |\                |\           | |
             +-->| >0-->| >0-- 29 total --| >0--+------+ |
             |   |/     |/                |/    |        |
             |                                  |        |
             +----------------------------------+        |
                                                         |
                 Other randomness, if available ---------+

    3.4.  Problems with Clocks and Serial Numbers

       Computer clocks and similar operating system or hardware values,
       provide significantly fewer real bits of unpredictability than might
       appear from their specifications.

       Tests have been done on clocks on numerous systems, and it was found
       that their behavior can vary widely and in unexpected ways.  One
       version of an operating system running on one set of hardware may
       actually provide, say, microsecond resolution in a clock, while a
       different configuration of the "same" system may always provide the
       same lower bits and only count in the upper bits at much lower
       resolution.  This means that successive reads of the clock may
       produce identical values even if enough time has passed that the
       value "should" change based on the nominal clock resolution.  There
       are also cases where frequently reading a clock can produce
       artificial sequential values, because of extra code that checks for
       the clock being unchanged between two reads and increases it by one!
       Designing portable application code to generate unpredictable numbers
       based on such system clocks is particularly challenging because the
       system designer does not always know the properties of the system
       clock.

       Use of a hardware serial number (such as an Ethernet MAC address) may
       also provide fewer bits of uniqueness than one would guess.  Such
       quantities are usually heavily structured, and subfields may have
       only a limited range of possible values, or values may be easily
       guessable based on approximate date of manufacture or other data.

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       For example, it is likely that a company that manufactures both
       computers and Ethernet adapters will, at least internally, use its
       own adapters, which significantly limits the range of built-in
       addresses.

       Problems such as those described above make the production of code to
       generate unpredictable quantities difficult if the code is to be
       ported across a variety of computer platforms and systems.

    3.5.  Timing and Value of External Events

       It is possible to measure the timing and content of mouse movement,
       key strokes, and similar user events.  This is a reasonable source of
       unguessable data, with some qualifications.  On some machines, input
       such as key strokes is buffered.  Even though the user's inter-
       keystroke timing may have sufficient variation and unpredictability,
       there might not be an easy way to access that variation.  Another
       problem is that no standard method exists for sampling timing
       details.  This makes it hard to use this technique to build standard
       software intended for distribution to a large range of machines.

       The amount of mouse movement and the actual key strokes are usually
       easier to access than timings, but they may yield less
       unpredictability because the user may provide highly repetitive
       input.

       Other external events, such as network packet arrival times and
       lengths, can also be used, but only with great care.  In particular,
       the possibility of manipulation of such network traffic measurements
       by an adversary and the lack of history at system start-up must be
       carefully considered.  If this input is subject to manipulation, it
       must not be trusted as a source of entropy.

       In principle, almost any external sensor, such as raw radio reception
       or temperature sensing in appropriately equipped computers, can be
       used.  But in each case, careful consideration must be given to how
       much this data is subject to adversarial manipulation and to how much
       entropy it can actually provide.

       The above techniques are quite powerful against attackers that have
       no access to the quantities being measured.  For example, these
       techniques would be powerful against offline attackers who had no
       access to one's environment and who were trying to crack one's random
       seed after the fact.  In all cases, the more accurately one can
       measure the timing or value of an external sensor, the more rapidly
       one can generate bits.

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    3.6.  Non-hardware Sources of Randomness

       The best source of input entropy would be a hardware-based random
       source such as ring oscillators, disk drive timing, thermal noise, or
       radioactive decay.  However, if none of these is available, there are
       other possibilities.  These include system clocks, system or
       input/output buffers, user/system/hardware/network serial numbers or
       addresses and timing, and user input.  Unfortunately, each of these
       sources can produce very limited or predictable values under some
       circumstances.

       Some of the sources listed above would be quite strong on multi-user
       systems, where each user of the system is in essence a source of
       randomness.  However, on a small single-user or embedded system,
       especially at start-up, it might be possible for an adversary to
       assemble a similar configuration.  This could give the adversary
       inputs to the mixing process that were well-enough correlated to
       those used originally to make exhaustive search practical.

       The use of multiple random inputs with a strong mixing function is
       recommended and can overcome weakness in any particular input.  The
       timing and content of requested "random" user keystrokes can yield
       hundreds of random bits, but conservative assumptions need to be
       made.  For example, one reasonably conservative assumption would be
       that an inter-keystroke interval provides at most a few bits of
       randomness, but only when the interval is unique in the sequence of
       intervals up to that point.  A similar assumption would be that a key
       code provides a few bits of randomness, but only when the code is
       unique in the sequence.  Thus, an interval or key code that
       duplicated a previous value would be assumed to provide no additional
       randomness.  The results of mixing these timings with typed
       characters could be further combined with clock values and other
       inputs.

       This strategy may make practical portable code for producing good
       random numbers for security, even if some of the inputs are very weak
       on some of the target systems.  However, it may still fail against a
       high-grade attack on small, single-user, or embedded systems,
       especially if the adversary has ever been able to observe the
       generation process in the past.  A hardware-based random source is
       still preferable.

    4.  De-skewing

       Is there any specific requirement on the shape of the distribution of
       quantities gathered for the entropy to produce the random numbers?
       The good news is that the distribution need not be uniform.  All that
       is needed to bound performance is a conservative estimate of how

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       non-uniform it is.  Simple techniques to de-skew a bit stream are
       given below, and stronger cryptographic techniques are described in
       Section 5.2.

    4.1.  Using Stream Parity to De-Skew

       As a simple but not particularly practical example, consider taking a
       sufficiently long string of bits and mapping the string to "zero" or
       "one".  The mapping will not yield a perfectly uniform distribution,
       but it can be as close as desired.  One mapping that serves the
       purpose is to take the parity of the string.  This has the advantages
       that it is robust across all degrees of skew up to the estimated
       maximum skew and that it is trivial to implement in hardware.

       The following analysis gives the number of bits that must be sampled:

       Suppose that the ratio of ones to zeros is ( 0.5 + E ) to
       ( 0.5 - E ), where E is between 0 and 0.5 and is a measure of the
       "eccentricity" of the distribution.  Consider the distribution of the
       parity function of N bit samples.  The respective probabilities that
       the parity will be one or zero will be the sum of the odd or even
       terms in the binomial expansion of (p + q)^N, where p = 0.5 + E, the
       probability of a one, and q = 0.5 - E, the probability of a zero.

       These sums can be computed easily as

                             N            N
            1/2 * ( ( p + q )  + ( p - q )  )

       and
                             N            N
            1/2 * ( ( p + q )  - ( p - q )  ).

       (Which formula corresponds to the probability that the parity will be
       1 depends on whether N is odd or even.)

       Since p + q = 1 and p - q = 2E, these expressions reduce to

                           N
            1/2 * [1 + (2E)  ]

       and
                           N
            1/2 * [1 - (2E)  ].

       Neither of these will ever be exactly 0.5 unless E is zero, but we
       can bring them arbitrarily close to 0.5.  If we want the
       probabilities to be within some delta d of 0.5, e.g., then

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                                N
            ( 0.5 + ( 0.5 * (2E)  ) )  <  0.5 + d.

       Solving for N yields N > log(2d)/log(2E). (Note that 2E is less than
       1, so its log is negative.  Division by a negative number reverses
       the sense of an inequality.)

       The following table gives the length N of the string that must be
       sampled for various degrees of skew in order to come within 0.001 of
       a 50/50 distribution.

                    +---------+--------+-------+
                    | Prob(1) |    E   |    N  |
                    +---------+--------+-------+
                    |   0.5   |  0.00  |    1  |
                    |   0.6   |  0.10  |    4  |
                    |   0.7   |  0.20  |    7  |
                    |   0.8   |  0.30  |   13  |
                    |   0.9   |  0.40  |   28  |
                    |   0.95  |  0.45  |   59  |
                    |   0.99  |  0.49  |  308  |
                    +---------+--------+-------+

       The last entry shows that even if the distribution is skewed 99% in
       favor of ones, the parity of a string of 308 samples will be within
       0.001 of a 50/50 distribution.  But, as we shall see in section 5.2,
       there are much stronger techniques that extract more of the available
       entropy.

    4.2.  Using Transition Mappings to De-Skew

       Another technique, originally due to von Neumann [VON_NEUMANN], is to
       examine a bit stream as a sequence of non-overlapping pairs.  One
       could then discard any 00 or 11 pairs found, interpret 01 as a 0 and
       10 as a 1.  Assume that the probability of a 1 is 0.5+E and that the
       probability of a 0 is 0.5-E, where E is the eccentricity of the
       source as described in the previous section.  Then the probability of
       each pair is shown in the following table:

                +------+-----------------------------------------+
                | pair |            probability                  |
                +------+-----------------------------------------+
                |  00  | (0.5 - E)^2          =  0.25 - E + E^2  |
                |  01  | (0.5 - E)*(0.5 + E)  =  0.25     - E^2  |
                |  10  | (0.5 + E)*(0.5 - E)  =  0.25     - E^2  |
                |  11  | (0.5 + E)^2          =  0.25 + E + E^2  |
                +------+-----------------------------------------+

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       This technique will completely eliminate any bias but requires an
       indeterminate number of input bits for any particular desired number
       of output bits.  The probability of any particular pair being
       discarded is 0.5 + 2E^2, so the expected number of input bits to
       produce X output bits is X/(0.25 - E^2).

       This technique assumes that the bits are from a stream where each bit
       has the same probability of being a 0 or 1 as any other bit in the
       stream and that bits are uncorrelated, i.e., that the bits come from
       identical independent distributions.  If alternate bits are from two
       correlated sources, for example, the above analysis breaks down.

       The above technique also provides another illustration of how a
       simple statistical analysis can mislead if one is not always on the
       lookout for patterns that could be exploited by an adversary.  If the
       algorithm were misread slightly so that overlapping successive bits
       pairs were used instead of non-overlapping pairs, the statistical
       analysis given would be the same.  However, instead of providing an
       unbiased, uncorrelated series of random 1s and 0s, it would produce a
       totally predictable sequence of exactly alternating 1s and 0s.

    4.3.  Using FFT to De-Skew

       When real-world data consists of strongly correlated bits, it may
       still contain useful amounts of entropy.  This entropy can be
       extracted through various transforms, the most powerful of which are
       described in section 5.2 below.

       Using the Fourier transform of the data or its optimized variant, the
       FFT, is interesting primarily for theoretical reasons.  It can be
       shown that this technique will discard strong correlations.  If
       adequate data is processed and if remaining correlations decay,
       spectral lines that approach statistical independence and normally
       distributed randomness can be produced [BRILLINGER].

    4.4.  Using Compression to De-Skew

       Reversible compression techniques also provide a crude method of de-
       skewing a skewed bit stream.  This follows directly from the
       definition of reversible compression and the formula in Section 2 for
       the amount of information in a sequence.  Since the compression is
       reversible, the same amount of information must be present in the
       shorter output as was present in the longer input.  By the Shannon
       information equation, this is only possible if, on average, the
       probabilities of the different shorter sequences are more uniformly
       distributed than were the probabilities of the longer sequences.
       Therefore, the shorter sequences must be de-skewed relative to the
       input.

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       However, many compression techniques add a somewhat predictable
       preface to their output stream and may insert a similar sequence
       periodically in their output or otherwise introduce subtle patterns
       of their own.  They should be considered only rough techniques
       compared to those described in Section 5.2.  At a minimum, the
       beginning of the compressed sequence should be skipped and only later
       bits should used for applications requiring roughly-random bits.

    5.  Mixing

       What is the best overall strategy for obtaining unguessable random
       numbers in the absence of a strong, reliable hardware entropy source?
       It is to obtain input from a number of uncorrelated sources and to
       mix them with a strong mixing function.  Such a function will
       preserve the entropy present in any of the sources, even if other
       quantities being combined happen to be fixed or easily guessable (low
       entropy).  This approach may be advisable even with a good hardware
       source, as hardware can also fail.  However, this should be weighed
       against a possible increase in the chance of overall failure due to
       added software complexity.

       Once one has used good sources, such as some of those listed in
       Section 3, and mixed them as described in this section, one has a
       strong seed.  This can then be used to produce large quantities of
       cryptographically strong material as described in Sections 6 and 7.

       A strong mixing function is one that combines inputs and produces an
       output in which each output bit is a different complex non-linear
       function of all the input bits.  On average, changing any input bit
       will change about half the output bits.  But because the relationship
       is complex and non-linear, no particular output bit is guaranteed to
       change when any particular input bit is changed.

       Consider the problem of converting a stream of bits that is skewed
       towards 0 or 1 or which has a somewhat predictable pattern to a
       shorter stream which is more random, as discussed in Section 4.  This
       is simply another case where a strong mixing function is desired, to
       mix the input bits and produce a smaller number of output bits.  The
       technique given in Section 4.1, using the parity of a number of bits,
       is simply the result of successively XORing them.  This is examined
       as a trivial mixing function, immediately below.  Use of stronger
       mixing functions to extract more of the randomness in a stream of
       skewed bits is examined in Section 5.2.  See also [NASLUND].

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    5.1.  A Trivial Mixing Function

       For expository purposes we describe a trivial example for single bit
       inputs using the Exclusive Or (XOR) function.  This function is
       equivalent to addition without carry, as show in the table below.
       This is a degenerate case in which the one output bit always changes
       for a change in either input bit.  But, despite its simplicity, it
       provides a useful illustration.

                    +-----------+-----------+----------+
                    |  input 1  |  input 2  |  output  |
                    +-----------+-----------+----------+
                    |     0     |     0     |     0    |
                    |     0     |     1     |     1    |
                    |     1     |     0     |     1    |
                    |     1     |     1     |     0    |
                    +-----------+-----------+----------+

       If inputs 1 and 2 are uncorrelated and combined in this fashion, then
       the output will be an even better (less skewed) random bit than the
       inputs are.  If we assume an "eccentricity" E as defined in Section
       4.1 above, then the output eccentricity relates to the input
       eccentricity as follows:

            E       = 2 * E        * E
             output        input 1    input 2

       Since E is never greater than 1/2, the eccentricity is always
       improved, except in the case in which at least one input is a totally
       skewed constant.  This is illustrated in the following table, where
       the top and left side values are the two input eccentricities and the
       entries are the output eccentricity:

         +--------+--------+--------+--------+--------+--------+--------+
         |    E   |  0.00  |  0.10  |  0.20  |  0.30  |  0.40  |  0.50  |
         +--------+--------+--------+--------+--------+--------+--------+
         |  0.00  |  0.00  |  0.00  |  0.00  |  0.00  |  0.00  |  0.00  |
         |  0.10  |  0.00  |  0.02  |  0.04  |  0.06  |  0.08  |  0.10  |
         |  0.20  |  0.00  |  0.04  |  0.08  |  0.12  |  0.16  |  0.20  |
         |  0.30  |  0.00  |  0.06  |  0.12  |  0.18  |  0.24  |  0.30  |
         |  0.40  |  0.00  |  0.08  |  0.16  |  0.24  |  0.32  |  0.40  |
         |  0.50  |  0.00  |  0.10  |  0.20  |  0.30  |  0.40  |  0.50  |
         +--------+--------+--------+--------+--------+--------+--------+

       However, note that the above calculations assume that the inputs are
       not correlated.  If the inputs were, say, the parity of the number of
       minutes from midnight on two clocks accurate to a few seconds, then
       each might appear random if sampled at random intervals much longer

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       than a minute.  Yet if they were both sampled and combined with XOR,
       the result would be zero most of the time.

    5.2.  Stronger Mixing Functions

       The US Government Advanced Encryption Standard [AES] is an example of
       a strong mixing function for multiple bit quantities.  It takes up to
       384 bits of input (128 bits of "data" and 256 bits of "key") and
       produces 128 bits of output, each of which is dependent on a complex
       non-linear function of all input bits.  Other encryption functions
       with this characteristic, such as [DES], can also be used by
       considering them to mix all of their key and data input bits.

       Another good family of mixing functions is the "message digest" or
       hashing functions such as the US Government Secure Hash Standards
       [SHA*] and the MD4, MD5 [MD4, MD5] series.  These functions all take
       a practically unlimited amount of input and produce a relatively
       short fixed-length output mixing all the input bits.  The MD* series
       produces 128 bits of output, SHA-1 produces 160 bits, and other SHA
       functions produce up to 512 bits.

       Although the message digest functions are designed for variable
       amounts of input, AES and other encryption functions can also be used
       to combine any number of inputs.  If 128 bits of output is adequate,
       the inputs can be packed into a 128-bit data quantity and successive
       AES "keys", padding with zeros if needed; the quantity is then
       successively encrypted by the "keys" using AES in Electronic Codebook
       Mode.  Alternatively, the input could be packed into one 128-bit key
       and multiple data blocks and a CBC-MAC could be calculated [MODES].

       More complex mixing should be used if more than 128 bits of output
       are needed and one wants to employ AES (but note that it is
       absolutely impossible to get more bits of "randomness" out than are
       put in).  For example, suppose that inputs are packed into three
       quantities, A, B, and C.  One may use AES to encrypt A with B and
       then with C as keys to produce the first part of the output, then
       encrypt B with C and then A for more output and, if necessary,
       encrypt C with A and then B for yet more output.  Still more output
       can be produced by reversing the order of the keys given above.  The
       same can be done with the hash functions, hashing various subsets of
       the input data or different copies of the input data with different
       prefixes and/or suffixes to produce multiple outputs.

       For an example of using a strong mixing function, reconsider the case
       of a string of 308 bits, each of which is biased 99% toward zero.
       The parity technique given in Section 4.1 reduces this to one bit,
       with only a 1/1000 deviance from being equally likely a zero or one.
       But, applying the equation for information given in Section 2, this

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       308-bit skewed sequence contains over 5 bits of information.  Thus,
       hashing it with SHA-1 and taking the bottom 5 bits of the result
       would yield 5 unbiased random bits and not the single bit given by
       calculating the parity of the string.  Alternatively, for some
       applications, you could use the entire hash output to retain almost
       all of the 5+ bits of entropy in a 160-bit quantity.

    5.3.  Using S-Boxes for Mixing

       Many modern block encryption functions, including DES and AES,
       incorporate modules known as S-Boxes (substitution boxes).  These
       produce a smaller number of outputs from a larger number of inputs
       through a complex non-linear mixing function that has the effect of
       concentrating limited entropy from the inputs into the output.

       S-Boxes sometimes incorporate bent Boolean functions (functions of an
       even number of bits producing one output bit with maximum non-
       linearity).  Looking at the output for all input pairs differing in
       any particular bit position, exactly half the outputs are different.
       An S-Box in which each output bit is produced by a bent function such
       that any linear combination of these functions is also a bent
       function is called a "perfect S-Box".

       S-boxes and various repeated applications or cascades of such boxes
       can be used for mixing [SBOX1, SBOX2].

    5.4.  Diffie-Hellman as a Mixing Function

       Diffie-Hellman exponential key exchange is a technique that yields a
       shared secret between two parties.  It can be computationally
       infeasible for a third party to determine this secret even if they
       can observe all the messages between the two communicating parties.
       This shared secret is a mixture of initial quantities generated by
       each of the parties [D-H].

       If these initial quantities are random and uncorrelated, then the
       shared secret combines their entropy but, of course, can not produce
       more randomness than the size of the shared secret generated.

       Although this is true if the Diffie-Hellman computation is performed
       privately, an adversary who can observe either of the public keys and
       knows the modulus being used need only search through the space of
       the other secret key in order to be able to calculate the shared
       secret [D-H].  So, conservatively, it would be best to consider
       public Diffie-Hellman to produce a quantity whose guessability
       corresponds to the worse of the two inputs.  Because of this and the
       fact that Diffie-Hellman is computationally intensive, its use as a
       mixing function is not recommended.

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    5.5.  Using a Mixing Function to Stretch Random Bits

       Although it is not necessary for a mixing function to produce the
       same or fewer output bits than its inputs, mixing bits cannot
       "stretch" the amount of random unpredictability present in the
       inputs.  Thus, four inputs of 32 bits each, in which there are 12
       bits worth of unpredictability (such as 4,096 equally probable
       values) in each input, cannot produce more than 48 bits worth of
       unpredictable output.  The output can be expanded to hundreds or
       thousands of bits by, for example, mixing with successive integers,
       but the clever adversary's search space is still 2^48 possibilities.
       Furthermore, mixing to fewer bits than are input will tend to
       strengthen the randomness of the output.

       The last table in Section 5.1 shows that mixing a random bit with a
       constant bit with Exclusive Or will produce a random bit.  While this
       is true, it does not provide a way to "stretch" one random bit into
       more than one.  If, for example, a random bit is mixed with a 0 and
       then with a 1, this produces a two bit sequence but it will always be
       either 01 or 10.  Since there are only two possible values, there is
       still only the one bit of original randomness.

    5.6.  Other Factors in Choosing a Mixing Function

       For local use, AES has the advantages that it has been widely tested
       for flaws, is reasonably efficient in software, and is widely
       documented and implemented with hardware and software implementations
       available all over the world including open source code.  The SHA*
       family have had a little less study and tend to require more CPU
       cycles than AES but there is no reason to believe they are flawed.
       Both SHA* and MD5 were derived from the earlier MD4 algorithm.  They
       all have source code available [SHA*, MD4, MD5].  Some signs of
       weakness have been found in MD4 and MD5.  In particular, MD4 has only
       three rounds and there are several independent breaks of the first
       two or last two rounds.  And some collisions have been found in MD5
       output.

       AES was selected by a robust, public, and international process.  It
       and SHA* have been vouched for by the US National Security Agency
       (NSA) on the basis of criteria that mostly remain secret, as was DES.
       While this has been the cause of much speculation and doubt,
       investigation of DES over the years has indicated that NSA
       involvement in modifications to its design, which originated with
       IBM, was primarily to strengthen it.  There has been no announcement
       of a concealed or special weakness being found in DES.  It is likely
       that the NSA modifications to MD4 to produce the SHA algorithms
       similarly strengthened these algorithms, possibly against threats not
       yet known in the public cryptographic community.

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       Where input lengths are unpredictable, hash algorithms are more
       convenient to use than block encryption algorithms since they are
       generally designed to accept variable length inputs.  Block
       encryption algorithms generally require an additional padding
       algorithm to accommodate inputs that are not an even multiple of the
       block size.

       As of the time of this document, the authors know of no patent claims
       to the basic AES, DES, SHA*, MD4, and MD5 algorithms other than
       patents for which an irrevocable royalty free license has been
       granted to the world.  There may, of course, be essential patents of
       which the authors are unaware or patents on implementations or uses
       or other relevant patents issued or to be issued.

    6.  Pseudo-random Number Generators

       When a seed has sufficient entropy, from input as described in
       Section 3 and possibly de-skewed and mixed as described in Sections 4
       and 5, one can algorithmically extend that seed to produce a large
       number of cryptographically-strong random quantities.  Such
       algorithms are platform independent and can operate in the same
       fashion on any computer.  For the algorithms to be secure, their
       input and internal workings must be protected from adversarial
       observation.

       The design of such pseudo-random number generation algorithms, like
       the design of symmetric encryption algorithms, is not a task for
       amateurs.  Section 6.1 below lists a number of bad ideas that failed
       algorithms have used.  To learn what works, skip Section 6.1 and just
       read the remainder of this section and Section 7, which describes and
       references some standard pseudo random number generation algorithms.
       See Section 7 and Part 3 of [X9.82].

    6.1.  Some Bad Ideas

       The subsections below describe a number of ideas that might seem
       reasonable but that lead to insecure pseudo-random number generation.

    6.1.1.  The Fallacy of Complex Manipulation

       One approach that may give a misleading appearance of
       unpredictability is to take a very complex algorithm (or an excellent
       traditional pseudo-random number generator with good statistical
       properties) and to calculate a cryptographic key by starting with
       limited data such as the computer system clock value as the seed.
       Adversaries who knew roughly when the generator was started would
       have a relatively small number of seed values to test, as they would
       know likely values of the system clock.  Large numbers of pseudo-

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       random bits could be generated, but the search space that an
       adversary would need to check could be quite small.

       Thus, very strong or complex manipulation of data will not help if
       the adversary can learn what the manipulation is and if there is not
       enough entropy in the starting seed value.  They can usually use the
       limited number of results stemming from a limited number of seed
       values to defeat security.

       Another serious strategic error is to assume that a very complex
       pseudo-random number generation algorithm will produce strong random
       numbers, when there has been no theory behind or analysis of the
       algorithm.  There is a excellent example of this fallacy near the
       beginning of Chapter 3 in [KNUTH], where the author describes a
       complex algorithm.  It was intended that the machine language program
       corresponding to the algorithm would be so complicated that a person
       trying to read the code without comments wouldn't know what the
       program was doing.  Unfortunately, actual use of this algorithm
       showed that it almost immediately converged to a single repeated
       value in one case and a small cycle of values in another case.

       Not only does complex manipulation not help you if you have a limited
       range of seeds, but blindly-chosen complex manipulation can destroy
       the entropy in a good seed!

    6.1.2.  The Fallacy of Selection from a Large Database

       Another approach that can give a misleading appearance of
       unpredictability is to randomly select a quantity from a database and
       to assume that its strength is related to the total number of bits in
       the database.  For example, typical USENET servers process many
       megabytes of information per day [USENET_1, USENET_2].  Assume that a
       random quantity was selected by fetching 32 bytes of data from a
       random starting point in this data.  This does not yield 32*8 = 256
       bits worth of unguessability.  Even if much of the data is human
       language that contains no more than 2 or 3 bits of information per
       byte, it doesn't yield 32*2 = 64 bits of unguessability.  For an
       adversary with access to the same Usenet database, the unguessability
       rests only on the starting point of the selection.  That is perhaps a
       little over a couple of dozen bits of unguessability.

       The same argument applies to selecting sequences from the data on a
       publicly available CD/DVD recording or any other large public
       database.  If the adversary has access to the same database, this
       "selection from a large volume of data" step buys little.  However,
       if a selection can be made from data to which the adversary has no
       access, such as system buffers on an active multi-user system, it may
       be of help.

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    6.1.3.  Traditional Pseudo-random Sequences

       This section talks about traditional sources of deterministic or
       "pseudo-random" numbers.  These typically start with a "seed"
       quantity and use simple numeric or logical operations to produce a
       sequence of values.  Note that none of the techniques discussed in
       this section is suitable for cryptographic use.  They are presented
       for general information.

       [KNUTH] has a classic exposition on pseudo-random numbers.
       Applications he mentions are simulations of natural phenomena,
       sampling, numerical analysis, testing computer programs, decision
       making, and games.  None of these have the same characteristics as
       the sorts of security uses we are talking about.  Only in the last
       two could there be an adversary trying to find the random quantity.
       However, in these cases, the adversary normally has only a single
       chance to use a guessed value.  In guessing passwords or attempting
       to break an encryption scheme, the adversary normally has many,
       perhaps unlimited, chances at guessing the correct value.  Sometimes
       the adversary can store the message to be broken and repeatedly
       attack it.  Adversaries are also be assumed to be aided by a
       computer.

       For testing the "randomness" of numbers, Knuth suggests a variety of
       measures, including statistical and spectral.  These tests check
       things like autocorrelation between different parts of a "random"
       sequence or distribution of its values.  But these tests could be met
       by a constant stored random sequence, such as the "random" sequence
       printed in the CRC Standard Mathematical Tables [CRC].  Despite
       meeting all the tests suggested by Knuth, that sequence is unsuitable
       for cryptographic us, as adversaries must be assumed to have copies
       of all commonly published "random" sequences and to be able to spot
       the source and predict future values.

       A typical pseudo-random number generation technique is the linear
       congruence pseudo-random number generator.  This technique uses
       modular arithmetic, where the value numbered N+1 is calculated from
       the value numbered N by

            V    = ( V  * a + b )(Mod c)
             N+1      N

       The above technique has a strong relationship to linear shift
       register pseudo-random number generators, which are well understood
       cryptographically [SHIFT*].  In such generators, bits are introduced
       at one end of a shift register as the Exclusive Or (binary sum
       without carry) of bits from selected fixed taps into the register.
       For example, consider the following:

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          +----+     +----+     +----+                      +----+
          | B  | <-- | B  | <-- | B  | <--  . . . . . . <-- | B  | <-+
          |  0 |     |  1 |     |  2 |                      |  n |   |
          +----+     +----+     +----+                      +----+   |
            |                     |            |                     |
            |                     |            V                  +-----+
            |                     V            +----------------> |     |
            V                     +-----------------------------> | XOR |
            +---------------------------------------------------> |     |
                                                                  +-----+

           V    = ( ( V  * 2 ) + B  XOR  B ... )(Mod 2^n)
            N+1         N         0       2

       The quality of traditional pseudo-random number generator algorithms
       is measured by statistical tests on such sequences.  Carefully-chosen
       values a, b, c, and initial V or carefully-chosen placement of the
       shift register tap in the above simple process can produce excellent
       statistics.

       These sequences may be adequate in simulations (Monte Carlo
       experiments) as long as the sequence is orthogonal to the structure
       of the space being explored.  Even there, subtle patterns may cause
       problems.  However, such sequences are clearly bad for use in
       security applications.  They are fully predictable if the initial
       state is known.  Depending on the form of the pseudo-random number
       generator, the sequence may be determinable from observation of a
       short portion of the sequence [SCHNEIER, STERN].  For example, with
       the generators above, one can determine V(n+1) given knowledge of
       V(n).  In fact, it has been shown that with these techniques, even if
       only one bit of the pseudo-random values are released, the seed can
       be determined from short sequences.

       Not only have linear congruent generators been broken, but techniques
       are now known for breaking all polynomial congruent generators
       [KRAWCZYK].

    6.2.  Cryptographically Strong Sequences

       In cases where a series of random quantities must be generated, an
       adversary may learn some values in the sequence.  In general,
       adversaries should not be able to predict other values from the ones
       that they know.

       The correct technique is to start with a strong random seed, to take
       cryptographically strong steps from that seed [FERGUSON, SCHNEIER],
       and not to reveal the complete state of the generator in the sequence
       elements.  If each value in the sequence can be calculated in a fixed

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       way from the previous value, then when any value is compromised, all
       future values can be determined.  This would be the case, for
       example, if each value were a constant function of the previously
       used values, even if the function were a very strong, non-invertible
       message digest function.

       (Note that if a technique for generating a sequence of key values is
       fast enough, it can trivially be used as the basis for a
       confidentiality system.  If two parties use the same sequence
       generation technique and start with the same seed material, they will
       generate identical sequences.  These could, for example, be XOR'ed at
       one end with data being sent to encrypt it, and XOR'ed with this data
       as received to decrypt it, due to the reversible properties of the
       XOR operation.  This is commonly referred to as a simple stream
       cipher.)

    6.2.1.  OFB and CTR Sequences

       One way to produce a strong sequence is to take a seed value and hash
       the quantities produced by concatenating the seed with successive
       integers, or the like, and then to mask the values obtained so as to
       limit the amount of generator state available to the adversary.

       It may also be possible to use an "encryption" algorithm with a
       random key and seed value to encrypt successive integers, as in
       counter (CTR) mode encryption.  Alternatively, one can feedback all
       of the output value from encryption into the value to be encrypted
       for the next iteration.  This is a particular example of output
       feedback mode (OFB) [MODES].

       An example is shown below in which shifting and masking are used to
       combine part of the output feedback with part of the old input.  This
       type of partial feedback should be avoided for reasons described
       below.

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                +---------------+
                |       V       |
                |  |     n      |--+
                +--+------------+  |
                      |            |     +---------+
                 shift|            +---> |         |      +-----+
                   +--+                  | Encrypt | <--- | Key |
                   |           +-------- |         |      +-----+
                   |           |         +---------+
                   V           V
                +------------+--+
                |      V     |  |
                |       n+1     |
                +---------------+

       Note that if a shift of one is used, this is the same as the shift
       register technique described in Section 6.1.3, but with the all-
       important difference that the feedback is determined by a complex
       non-linear function of all bits rather than by a simple linear or
       polynomial combination of output from a few bit position taps.

       Donald W. Davies showed that this sort of shifted partial output
       feedback significantly weakens an algorithm, compared to feeding all
       the output bits back as input.  In particular, for DES, repeatedly
       encrypting a full 64-bit quantity will give an expected repeat in
       about 2^63 iterations.  Feeding back anything less than 64 (and more
       than 0) bits will give an expected repeat in between 2^31 and 2^32
       iterations!

       To predict values of a sequence from others when the sequence was
       generated by these techniques is equivalent to breaking the
       cryptosystem or to inverting the "non-invertible" hashing with only
       partial information available.  The less information revealed in each
       iteration, the harder it will be for an adversary to predict the
       sequence.  Thus it is best to use only one bit from each value.  It
       has been shown that in some cases this makes it impossible to break a
       system even when the cryptographic system is invertible and could be
       broken if all of each generated value were revealed.

    6.2.2.  The Blum Blum Shub Sequence Generator

       Currently the generator which has the strongest public proof of
       strength is called the Blum Blum Shub generator, named after its
       inventors [BBS].  It is also very simple and is based on quadratic
       residues.  Its only disadvantage is that it is computationally
       intensive compared to the traditional techniques given in Section
       6.1.3.  This is not a major drawback if it is used for moderately-
       infrequent purposes, such as generating session keys.

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       Simply choose two large prime numbers (say, p and q) that each gives
       a remainder of 3 when divided by 4.  Let n = p * q.  Then choose a
       random number, x, that is relatively prime to n.  The initial seed
       for the generator and the method for calculating subsequent values
       are then:

                        2
             s    =  ( x  )(Mod n)
              0
                        2
             s    = ( s   )(Mod n)
              i+1      i

       Be careful to use only a few bits from the bottom of each s.  It is
       always safe to use only the lowest-order bit.  If one uses no more
       than the:

             log  ( log  ( s  ) )
                2      2    i

       low-order bits, then predicting any additional bits from a sequence
       generated in this manner is provably as hard as factoring n.  As long
       as the initial x is secret, n can be made public if desired.

       An interesting characteristic of this generator is that any of the s
       values can be directly calculated.  In particular,

                   ( (2^i) (Mod ((p-1)*(q-1)) ) )
          s  = ( s                                )(Mod n)
           i      0

       This means that in applications where many keys are generated in this
       fashion, it is not necessary to save them all.  Each key can be
       effectively indexed and recovered from that small index and the
       initial s and n.

    6.3.  Entropy Pool Techniques

       Many modern pseudo-random number sources, such as those described in
       Sections 7.1.2 and 7.1.3 utilize the technique of maintaining a
       "pool" of bits and providing operations for strongly mixing input
       with some randomness into the pool and extracting pseudo-random bits
       from the pool.  This is illustrated in the figure below.

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                 +--------+    +------+    +---------+
             --->| Mix In |--->| POOL |--->| Extract |--->
                 |  Bits  |    |      |    |   Bits  |
                 +--------+    +------+    +---------+
                                   ^           V
                                   |           |
                                   +-----------+

       Bits to be fed into the pool can come from any of the various
       hardware, environmental, or user input sources discussed above.  It
       is also common to save the state of the pool on system shutdown and
       to restore it on re-starting, when stable storage is available.

       Care must be taken that enough entropy has been added to the pool to
       support particular output uses desired.  See [RSA_BULL1] for similar
       suggestions.

    7.  Randomness Generation Examples and Standards

       Several public standards and widely deployed examples are now in
       place for the generation of keys or other cryptographically random
       quantities.  Some, in section 7.1, include an entropy source.
       Others, described in section 7.2, provide the pseudo-random number
       strong-sequence generator but assume the input of a random seed or
       input from a source of entropy.

    7.1.  Complete Randomness Generators

       Three standards are described below.  The two older standards use
       DES, with its 64-bit block and key size limit, but any equally strong
       or stronger mixing function could be substituted [DES].  The third is
       a more modern and stronger standard based on SHA-1 [SHA*].  Lastly,
       the widely deployed modern UNIX and Windows random number generators
       are described.

    7.1.1.  US DoD Recommendations for Password Generation

       The United States Department of Defense has specific recommendations
       for password generation [DoD].  It suggests using the US Data
       Encryption Standard [DES] in Output Feedback Mode [MODES] as follows:

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             Use an initialization vector determined from
                  the system clock,
                  system ID,
                  user ID, and
                  date and time;
             use a key determined from
                  system interrupt registers,
                  system status registers, and
                  system counters; and,
             as plain text, use an external randomly generated 64-bit
             quantity such as the ASCII bytes for 8 characters typed
             in by a system administrator.

       The password can then be calculated from the 64 bit "cipher text"
       generated by DES in 64-bit Output Feedback Mode.  As many bits as are
       needed can be taken from these 64 bits and expanded into a
       pronounceable word, phrase, or other format if a human being needs to
       remember the password.

    7.1.2.  The /dev/random Device

       Several versions of the UNIX operating system provide a kernel-
       resident random number generator.  Some of these generators use
       events captured by the Kernel during normal system operation.

       For example, on some versions of Linux, the generator consists of a
       random pool of 512 bytes represented as 128 words of 4 bytes each.
       When an event occurs, such as a disk drive interrupt, the time of the
       event is XOR'ed into the pool, and the pool is stirred via a
       primitive polynomial of degree 128.  The pool itself is treated as a
       ring buffer, with new data being XOR'ed (after stirring with the
       polynomial) across the entire pool.

       Each call that adds entropy to the pool estimates the amount of
       likely true entropy the input contains.  The pool itself contains a
       accumulator that estimates the total over all entropy of the pool.

       Input events come from several sources, as listed below.
       Unfortunately, for server machines without human operators, the first
       and third are not available, and entropy may be added slowly in that
       case.

       1. Keyboard interrupts.  The time of the interrupt and the scan code
          are added to the pool.  This in effect adds entropy from the human
          operator by measuring inter-keystroke arrival times.

       2. Disk completion and other interrupts.  A system being used by a
          person will likely have a hard-to-predict pattern of disk

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          accesses.  (But not all disk drivers support capturing this timing
          information with sufficient accuracy to be useful.)

       3. Mouse motion.  The timing and mouse position are added in.

       When random bytes are required, the pool is hashed with SHA-1 [SHA*]
       to yield the returned bytes of randomness.  If more bytes are
       required than the output of SHA-1 (20 bytes), then the hashed output
       is stirred back into the pool and a new hash is performed to obtain
       the next 20 bytes.  As bytes are removed from the pool, the estimate
       of entropy is correspondingly decremented.

       To ensure a reasonably random pool upon system startup, the standard
       startup and shutdown scripts save the pool to a disk file at shutdown
       and read this file at system startup.

       There are two user-exported interfaces. /dev/random returns bytes
       from the pool but blocks when the estimated entropy drops to zero.
       As entropy is added to the pool from events, more data becomes
       available via /dev/random.  Random data obtained from such a
       /dev/random device is suitable for key generation for long term keys,
       if enough random bits are in the pool or are added in a reasonable
       amount of time.

       /dev/urandom works like /dev/random; however, it provides data even
       when the entropy estimate for the random pool drops to zero.  This
       may be adequate for session keys or for other key generation tasks
       for which blocking to await more random bits is not acceptable.  The
       risk of continuing to take data even when the pool's entropy estimate
       is small in that past output may be computable from current output,
       provided that an attacker can reverse SHA-1.  Given that SHA-1 is
       designed to be non-invertible, this is a reasonable risk.

       To obtain random numbers under Linux, Solaris, or other UNIX systems
       equipped with code as described above, all an application has to do
       is open either /dev/random or /dev/urandom and read the desired
       number of bytes.

       (The Linux Random device was written by Theodore Ts'o.  It was based
       loosely on the random number generator in PGP 2.X and PGP 3.0 (aka
       PGP 5.0).)

    7.1.3.  Windows CryptGenRandom

       Microsoft's recommendation to users of the widely deployed Windows
       operating system is generally to use the CryptGenRandom pseudo-random
       number generation call with the CryptAPI cryptographic service
       provider.  This takes a handle to a cryptographic service provider

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       library, a pointer to a buffer by which the caller can provide
       entropy and into which the generated pseudo-randomness is returned,
       and an indication of how many octets of randomness are desired.

       The Windows CryptAPI cryptographic service provider stores a seed
       state variable with every user.  When CryptGenRandom is called, this
       is combined with any randomness provided in the call and with various
       system and user data such as the process ID, thread ID, system clock,
       system time, system counter, memory status, free disk clusters, and
       hashed user environment block.  This data is all fed to SHA-1, and
       the output is used to seed an RC4 key stream.  That key stream is
       used to produce the pseudo-random data requested and to update the
       user's seed state variable.

       Users of Windows ".NET" will probably find it easier to use the
       RNGCryptoServiceProvider.GetBytes method interface.

       For further information, see [WSC].

    7.2.  Generators Assuming a Source of Entropy

       The pseudo-random number generators described in the following three
       sections all assume that a seed value with sufficient entropy is
       provided to them.  They then generate a strong sequence (see Section
       6.2) from that seed.

    7.2.1.  X9.82 Pseudo-Random Number Generation

       The ANSI X9F1 committee is in the final stages of creating a standard
       for random number generation covering both true randomness generators
       and pseudo-random number generators.  It includes a number of
       pseudo-random number generators based on hash functions, one of which
       will probably be based on HMAC SHA hash constructs [RFC2104].  The
       draft version of this generator is described below, omitting a number
       of optional features [X9.82].

       In the subsections below, the HMAC hash construct is simply referred
       to as HMAC but, of course, a particular standard SHA function must be
       selected in an particular use.  Generally speaking, if the strength
       of the pseudo-random values to be generated is to be N bits, the SHA
       function chosen must generate N or more bits of output, and a source
       of at least N bits of input entropy will be required.  The same hash
       function must be used throughout an instantiation of this generator.

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    7.2.1.1.  Notation

       In the following sections, the notation give below is used:

          hash_length is the output size of the underlying hash function in
             use.

          input_entropy is the input bit string that provides entropy to the
             generator.

          K is a bit string of size hash_length that is part of the state of
             the generator and is updated at least once each time random
             bits are generated.

          V is a bit string of size hash_length and is part of the state of
             the generator.  It is updated each time hash_length bits of
             output are generated.

          "|" represents concatenation.

    7.2.1.2.  Initializing the Generator

       Set V to all zero bytes, except the low-order bit of each byte is set
          to one.

       Set K to all zero bytes, then set:

             K = HMAC ( K, V | 0x00 | input_entropy )

             V = HMAC ( K, V )

             K = HMAC ( K, V | 0x01 | input_entropy )

             V = HMAC ( K, V )

       Note: All SHA algorithms produce an integral number of bytes, so the
       lengths of K and V will be integral numbers of bytes.

    7.2.1.3.  Generating Random Bits

       When output is called for, simply set:

             V = HMAC ( K, V )

       and use the leading bits from V.  If more bits are needed than the
       length of V, set "temp" to a null bit string and then repeatedly
       perform:

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             V = HMAC ( K, V )
             temp = temp | V

       stopping as soon as temp is equal to or longer than the number of
       random bits requested.  Use the requested number of leading bits from
       temp.  The definition of the algorithm prohibits requesting more than
       2^35 bits.

       After extracting and saving the pseudo-random output bits as
       described above, before returning you must also perform two more
       HMACs as follows:

             K = HMAC ( K, V | 0x00 )
             V = HMAC ( K, V )

    7.2.2.  X9.17 Key Generation

             The American National Standards Institute has specified the
             following method for generating a sequence of keys [X9.17]:

          s  is the initial 64 bit seed.
           0

          g  is the sequence of generated 64-bit key quantities
           n

          k is a random key reserved for generating this key sequence.

          t is the time at which a key is generated, to as fine a resolution
             as is available (up to 64 bits).

          DES ( K, Q ) is the DES encryption of quantity Q with key K.

       Then:

             g    = DES ( k, DES ( k, t ) XOR s  )
              n                                n

             s    = DES ( k, DES ( k, t ) XOR  g  )
              n+1                               n

       If g sub n is to be used as a DES key, then every eighth bit should
       be adjusted for parity for that use, but the entire 64 bit unmodified
       g should be used in calculating the next s.

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    7.2.3.  DSS Pseudo-random Number Generation

       Appendix 3 of the NIST Digital Signature Standard [DSS] provides a
       method of producing a sequence of pseudo-random 160 bit quantities
       for use as private keys or the like.  This has been modified by
       Change Notice 1 [DSS_CN1] to produce the following algorithm for
       generating general-purpose pseudo-random numbers:

             t = 0x 67452301 EFCDAB89 98BADCFE 10325476 C3D2E1F0

             XKEY  = initial seed
                 0

             For j = 0 to ...

                 XVAL = ( XKEY  + optional user input ) (Mod 2^512)
                              j

                 X  = G( t, XVAL )
                  j

                 XKEY   = ( 1 + XKEY  + X  ) (Mod 2^512)
                     j+1            j    j

       The quantities X thus produced are the pseudo-random sequence of
       160-bit values.  Two functions can be used for "G" above.  Each
       produces a 160-bit value and takes two arguments, a 160-bit value and
       a 512 bit value.

       The first is based on SHA-1 and works by setting the 5 linking
       variables, denoted H with subscripts in the SHA-1 specification, to
       the first argument divided into fifths.  Then steps (a) through (e)
       of section 7 of the NIST SHA-1 specification are run over the second
       argument as if it were a 512-bit data block.  The values of the
       linking variable after those steps are then concatenated to produce
       the output of G [SHA*].

       As an alternative method, NIST also defined an alternate G function
       based on multiple applications of the DES encryption function [DSS].

    8.  Examples of Randomness Required

       Below are two examples showing rough calculations of randomness
       needed for security.  The first is for moderate security passwords,
       while the second assumes a need for a very high-security
       cryptographic key.

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       In addition, [ORMAN] and [RSA_BULL13] provide information on the
       public key lengths that should be used for exchanging symmetric keys.

    8.1.  Password Generation

       Assume that user passwords change once a year and that it is desired
       that the probability that an adversary could guess the password for a
       particular account be less than one in a thousand.  Further assume
       that sending a password to the system is the only way to try a
       password.  Then the crucial question is how often an adversary can
       try possibilities.  Assume that delays have been introduced into a
       system so that an adversary can make at most one password try every
       six seconds.  That's 600 per hour, or about 15,000 per day, or about
       5,000,000 tries in a year.  Assuming any sort of monitoring, it is
       unlikely that someone could actually try continuously for a year.
       Even if log files are only checked monthly, 500,000 tries is more
       plausible before the attack is noticed and steps are taken to change
       passwords and make it harder to try more passwords.

       To have a one-in-a-thousand chance of guessing the password in
       500,000 tries implies a universe of at least 500,000,000 passwords,
       or about 2^29.  Thus, 29 bits of randomness are needed.  This can
       probably be achieved by using the US DoD-recommended inputs for
       password generation, as it has 8 inputs that probably average over 5
       bits of randomness each (see section 7.1).  Using a list of 1,000
       words, the password could be expressed as a three-word phrase
       (1,000,000,000 possibilities).  By using case-insensitive letters and
       digits, six characters would suffice ((26+10)^6 = 2,176,782,336
       possibilities).

       For a higher-security password, the number of bits required goes up.
       To decrease the probability by 1,000 requires increasing the universe
       of passwords by the same factor, which adds about 10 bits.  Thus, to
       have only a one in a million chance of a password being guessed under
       the above scenario would require 39 bits of randomness and a password
       that was a four-word phrase from a 1,000 word list, or eight
       letters/digits.  To go to a one-in-10^9 chance, 49 bits of randomness
       are needed, implying a five-word phrase or a ten-letter/digit
       password.

       In a real system, of course, there are other factors.  For example,
       the larger and harder to remember passwords are, the more likely
       users will bed to write them down, resulting in an additional risk of
       compromise.

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    8.2.  A Very High Security Cryptographic Key

       Assume that a very high security key is needed for symmetric
       encryption/decryption between two parties.  Assume also that an
       adversary can observe communications and knows the algorithm being
       used.  Within the field of random possibilities, the adversary can
       try key values in hopes of finding the one in use.  Assume further
       that brute force trial of keys is the best the adversary can do.

    8.2.1.  Effort per Key Trial

       How much effort will it take to try each key?  For very high-security
       applications, it is best to assume a low value of effort.  Even if it
       would clearly take tens of thousands of computer cycles or more to
       try a single key, there may be some pattern that enables huge blocks
       of key values to be tested with much less effort per key.  Thus, it
       is probably best to assume no more than a couple of hundred cycles
       per key.  (There is no clear lower bound on this, as computers
       operate in parallel on a number of bits and a poor encryption
       algorithm could allow many keys or even groups of keys to be tested
       in parallel.  However, we need to assume some value and can hope that
       a reasonably strong algorithm has been chosen for our hypothetical
       high-security task.)

       If the adversary can command a highly parallel processor or a large
       network of work stations, 10^11 cycles per second is probably a
       minimum assumption today.  Looking forward a few years, there should
       be at least an order of magnitude improvement.  Thus, it is
       reasonable to assume that 10^10 keys could be checked per second, or
       3.6*10^12 per hour or 6*10^14 per week, or 2.4*10^15 per month.  This
       implies a need for a minimum of 63 bits of randomness in keys, to be
       sure that they cannot be found in a month.  Even then it is possible
       that, a few years from now, a highly determined and resourceful
       adversary could break the key in 2 weeks; on average, they need try
       only half the keys.

       These questions are considered in detail in "Minimal Key Lengths for
       Symmetric Ciphers to Provide Adequate Commercial Security: A Report
       by an Ad Hoc Group of Cryptographers and Computer Scientists"
       [KeyStudy] that was sponsored by the Business Software Alliance.  It
       concluded that a reasonable key length in 1995 for very high security
       is in the range of 75 to 90 bits and, since the cost of cryptography
       does not vary much with the key size, it recommends 90 bits.  To
       update these recommendations, just add 2/3 of a bit per year for
       Moore's law [MOORE].  This translates to a determination, in the year
       2004, a reasonable key length is in the 81- to 96-bit range.  In
       fact, today, it is increasingly common to use keys longer than 96

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       bits, such as 128-bit (or longer) keys with AES and keys with
       effective lengths of 112-bits with triple-DES.

    8.2.2.  Meet-in-the-Middle Attacks

       If chosen or known plain text and the resulting encrypted text are
       available, a "meet-in-the-middle" attack is possible if the structure
       of the encryption algorithm allows it.  (In a known plain text
       attack, the adversary knows all or part (possibly some standard
       header or trailer fields) of the messages being encrypted.  In a
       chosen plain text attack, the adversary can force some chosen plain
       text to be encrypted, possibly by "leaking" an exciting text that is
       sent by the adversary over an encrypted channel because the text is
       so interesting.

       The following is an oversimplified explanation of the meet-in-the-
       middle attack:  the adversary can half-encrypt the known or chosen
       plain text with all possible first half-keys, sort the output, and
       then half-decrypt the encoded text with all the second half-keys.  If
       a match is found, the full key can be assembled from the halves and
       used to decrypt other parts of the message or other messages.  At its
       best, this type of attack can halve the exponent of the work required
       by the adversary while adding a very large but roughly constant
       factor of effort.  Thus, if this attack can be mounted, a doubling of
       the amount of randomness in the very strong key to a minimum of 192
       bits (96*2) is required for the year 2004, based on the [KeyStudy]
       analysis.

       This amount of randomness is well beyond the limit of that in the
       inputs recommended by the US DoD for password generation and could
       require user-typing timing, hardware random number generation, or
       other sources of randomness.

       The meet-in-the-middle attack assumes that the cryptographic
       algorithm can be decomposed in this way.  Hopefully no modern
       algorithm has this weakness, but there may be cases where we are not
       sure of that or even of what algorithm a key will be used with.  Even
       if a basic algorithm is not subject to a meet-in-the-middle attack,
       an attempt to produce a stronger algorithm by applying the basic
       algorithm twice (or two different algorithms sequentially) with
       different keys will gain less added security than would be expected.
       Such a composite algorithm would be subject to a meet-in-the-middle
       attack.

       Enormous resources may be required to mount a meet-in-the-middle
       attack, but they are probably within the range of the national
       security services of a major nation.  Essentially all nations spy on
       other nations' traffic.

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    8.2.3.  Other Considerations

       [KeyStudy] also considers the possibilities of special-purpose code-
       breaking hardware and having an adequate safety margin.

       Note that key length calculations such as those above are
       controversial and depend on various assumptions about the
       cryptographic algorithms in use.  In some cases, a professional with
       a deep knowledge of algorithm-breaking techniques and of the strength
       of the algorithm in use could be satisfied with less than half of the
       192 bit key size derived above.

       For further examples of conservative design principles, see
       [FERGUSON].

    9.  Conclusion

       Generation of unguessable "random" secret quantities for security use
       is an essential but difficult task.

       Hardware techniques for producing the needed entropy would be
       relatively simple.  In particular, the volume and quality would not
       need to be high, and existing computer hardware, such as audio input
       or disk drives, can be used.

       Widely-available computational techniques can process low-quality
       random quantities from multiple sources, or a larger quantity of such
       low-quality input from one source, to produce a smaller quantity of
       higher-quality keying material.  In the absence of hardware sources
       of randomness, a variety of user and software sources can frequently,
       with care, be used instead.  However, most modern systems already
       have hardware, such as disk drives or audio input, that could be used
       to produce high-quality randomness.

       Once a sufficient quantity of high-quality seed key material (a
       couple of hundred bits) is available, computational techniques are
       available to produce cryptographically-strong sequences of
       computationally-unpredictable quantities from this seed material.

    10.  Security Considerations

       The entirety of this document concerns techniques and recommendations
       for generating unguessable "random" quantities for use as passwords,
       cryptographic keys, initialization vectors, sequence numbers, and
       similar security applications.

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    11.  Acknowledgements

       Special thanks to Paul Hoffman and John Kelsey for their extensive
       comments and to Peter Gutmann, who has permitted the incorporation of
       material from his paper "Software Generation of Practically Strong
       Random Numbers".

       The following people (in alphabetic order) have contributed
       substantially to this document:

          Steve Bellovin, Daniel Brown, Don Davis, Peter Gutmann, Tony
          Hansen, Sandy Harris, Paul Hoffman, Scott Hollenback, Russ
          Housley, Christian Huitema, John Kelsey, Mats Naslund, and Damir
          Rajnovic.

       The following people (in alphabetic order) contributed to RFC 1750,
       the predecessor of this document:

          David M.  Balenson, Don T.  Davis, Carl Ellison, Marc Horowitz,
          Christian Huitema, Charlie Kaufman, Steve Kent, Hal Murray, Neil
          Haller, Richard Pitkin, Tim Redmond, and Doug Tygar.

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    RFC 4086         Randomness Requirements for Security          June 2005

    Appendix A: Changes from RFC 1750

       1. Additional acknowledgements have been added.

       2. Insertion of section 5.3 on mixing with S-boxes.

       3. Addition of section 3.3 on Ring Oscillator randomness sources.

       4. Addition of AES and the members of the SHA series producing more
          than 160 bits.  Use of AES has been emphasized and the use of DES
          de-emphasized.

       5. Addition of section 6.3 on entropy pool techniques.

       6. Addition of section 7.2.3 on the pseudo-random number generation
          techniques given in FIPS 186-2 (with Change Notice 1), 7.2.1 on
          those given in X9.82, section 7.1.2 on the random number
          generation techniques of the /dev/random device in Linux and other
          UNIX systems, and section 7.1.3 on random number generation
          techniques in the Windows operating system.

       7. Addition of references to the "Minimal Key Lengths for Symmetric
          Ciphers to Provide Adequate Commercial Security" study published
          in January 1996 [KeyStudy] and to [RFC1948].

       8. Added caveats to using Diffie-Hellman as a mixing function and,
          because of those caveats and its computationally intensive nature,
          recommend against its use.

       9. Addition of references to the X9.82 effort and the [TURBID] and
          [NASLUND] papers.

      10. Addition of discussion of min-entropy and Renyi entropy and
          references to the [LUBY] book.

      11. Major restructuring, minor wording changes, and a variety of
          reference updates.

    Eastlake, et al.            Standards Track                    [Page 40]

    RFC 4086         Randomness Requirements for Security          June 2005

    Informative References

       [AES]          "Specification of the Advanced Encryption Standard
                       (AES)", United States of America, US National
                       Institute of Standards and Technology, FIPS 197,
                       November 2001.

       [ASYMMETRIC]    Simmons, G., Ed., "Secure Communications and
                       Asymmetric Cryptosystems", AAAS Selected Symposium
                       69, ISBN 0-86531-338-5, Westview Press, 1982.

       [BBS]           Blum, L., Blum, M., and M. Shub, "A Simple
                       Unpredictable Pseudo-Random Number Generator", SIAM
                       Journal on Computing, v. 15, n. 2, 1986.

       [BRILLINGER]    Brillinger, D., "Time Series: Data Analysis and
                       Theory", Holden-Day, 1981.

       [CRC]           "C.R.C. Standard Mathematical Tables", Chemical
                       Rubber Publishing Company.

       [DAVIS]         Davis, D., Ihaka, R., and P. Fenstermacher,
                       "Cryptographic Randomness from Air Turbulence in Disk
                       Drives", Advances in Cryptology - Crypto '94,
                       Springer-Verlag Lecture Notes in Computer Science
                       #839, 1984.

       [DES]           "Data Encryption Standard", US National Institute of
                       Standards and Technology, FIPS 46-3, October 1999.
                       Also, "Data Encryption Algorithm", American National
                       Standards Institute, ANSI X3.92-1981.  See also FIPS
                       112, "Password Usage", which includes FORTRAN code
                       for performing DES.

       [D-H]           Rescorla, E., "Diffie-Hellman Key Agreement Method",
                       RFC 2631, June 1999.

       [DNSSEC1]       Arends, R., Austein, R., Larson, M., Massey, D., and
                       S. Rose, "DNS Security Introduction and
                       Requirements", RFC 4033, March 2005.

       [DNSSEC2]       Arends, R., Austein, R., Larson, M., Massey, D., and
                       S. Rose, "Resource Records for the DNS Security
                       Extensions", RFC 4034, March 2005.

       [DNSSEC3]       Arends, R., Austein, R., Larson, M., Massey, D., and
                       S. Rose, "Protocol Modifications for the DNS Security
                       Extensions", RFC 4035, March 2005.

    Eastlake, et al.            Standards Track                    [Page 41]

    RFC 4086         Randomness Requirements for Security          June 2005

       [DoD]           "Password Management Guideline", United States of
                       America, Department of Defense, Computer Security
                       Center, CSC-STD-002-85, April 1885.

                       (See also "Password Usage", FIPS 112, which
                       incorporates CSC-STD-002-85 as one of its appendices.
                       FIPS 112 is currently available at:
                       http://www.idl.nist.gov/fipspubs/fip112.htm.)

       [DSS]           "Digital Signature Standard (DSS)", US National
                       Institute of Standards and Technology, FIPS 186-2,
                       January 2000.

       [DSS_CN1]       "Digital Signature Standard Change Notice 1", US
                       National Institute of Standards and Technology, FIPS
                       186-2 Change Notice 1, 5, October 2001.

       [FERGUSON]      Ferguson, N. and B. Schneier, "Practical
                       Cryptography",  Wiley Publishing Inc., ISBN
                       047122894X, April 2003.

       [GIFFORD]       Gifford, D., "Natural Random Number", MIT/LCS/TM-371,
                       September 1988.

       [IEEE_802.11i]  "Amendment to Standard for Telecommunications and
                       Information Exchange Between Systems - LAN/MAN
                       Specific Requirements - Part 11: Wireless Medium
                       Access Control (MAC) and physical layer (PHY)
                       specifications: Medium Access Control (MAC) Security
                       Enhancements", IEEE, January 2004.

       [IPSEC]         Kent, S. and R. Atkinson, "Security Architecture for
                       the Internet Protocol", RFC 2401, November 1998.

       [Jakobsson]     Jakobsson, M., Shriver, E., Hillyer, B., and A.
                       Juels, "A practical secure random bit generator",
                       Proceedings of the Fifth ACM Conference on Computer
                       and Communications Security, 1998.

       [KAUFMAN]       Kaufman, C., Perlman, R., and M. Speciner, "Network
                       Security:  Private Communication in a Public World",
                       Prentis Hall PTR, ISBN 0-13-046019-2, 2nd Edition
                       2002.

    Eastlake, et al.            Standards Track                    [Page 42]

    RFC 4086         Randomness Requirements for Security          June 2005

       [KeyStudy]      Blaze, M., Diffie, W., Riverst, R., Schneier, B.
                       Shimomura, T., Thompson, E., and M.  Weiner, "Minimal
                       Key Lengths for Symmetric Ciphers to Provide Adequate
                       Commercial Security: A Report by an Ad Hoc Group of
                       Cryptographers and Computer Scientists", January
                       1996.  Currently available at:
                       http://www.crypto.com/papers/keylength.txt and
                       http://www.securitydocs.com/library/441.

       [KNUTH]         Knuth, D., "The Art of Computer Programming", Volume
                       2:  Seminumerical Algorithms, Chapter 3: Random
                       Numbers, Addison-Wesley Publishing Company, 3rd
                       Edition, November 1997.

       [KRAWCZYK]      Krawczyk, H., "How to Predict Congruential
                       Generators", Journal of Algorithms, V. 13, N. 4,
                       December 1992.

       [LUBY]          Luby, M., "Pseudorandomness and Cryptographic
                       Applications", Princeton University Press, ISBN
                       0691025460, 8 January 1996.

       [MAIL_PEM1]     Linn, J., "Privacy Enhancement for Internet
                       Electronic Mail: Part I: Message Encryption and
                       Authentication Procedures", RFC 1421, February 1993.

       [MAIL_PEM2]     Kent, S., "Privacy Enhancement for Internet
                       Electronic Mail: Part II: Certificate-Based Key
                       Management", RFC 1422, February 1993.

       [MAIL_PEM3]     Balenson, D., "Privacy Enhancement for Internet
                       Electronic Mail: Part III: Algorithms, Modes, and
                       Identifiers", RFC 1423, February 1993.

       [MAIL_PEM4]     Kaliski, B., "Privacy Enhancement for Internet
                       Electronic Mail: Part IV: Key Certification and
                       Related Services", RFC 1424, February 1993.

       [MAIL_PGP1]     Callas, J., Donnerhacke, L., Finney, H., and R.
                       Thayer, "OpenPGP Message Format", RFC 2440, November
                       1998.

       [MAIL_PGP2]     Elkins, M., Del Torto, D., Levien, R., and T.
                       Roessler, "MIME Security with OpenPGP", RFC 3156,
                       August 2001.

    Eastlake, et al.            Standards Track                    [Page 43]

    RFC 4086         Randomness Requirements for Security          June 2005

       [S/MIME]        RFCs 2632 through 2634:

                       Ramsdell, B., "S/MIME Version 3 Certificate
                       Handling", RFC 2632, June 1999.

                       Ramsdell, B., "S/MIME Version 3 Message
                       Specification", RFC 2633, June 1999.

                       Hoffman, P., "Enhanced Security Services for S/MIME",
                       RFC 2634, June 1999.

       [MD4]           Rivest, R., "The MD4 Message-Digest Algorithm", RFC
                       1320, April 1992.

       [MD5]           Rivest, R., "The MD5 Message-Digest Algorithm ", RFC
                       1321, April 1992.

       [MODES]         "DES Modes of Operation", US National Institute of
                       Standards and Technology, FIPS 81, December 1980.
                       Also:  "Data Encryption Algorithm - Modes of
                       Operation", American National Standards Institute,
                       ANSI X3.106-1983.

       [MOORE]         Moore's Law: the exponential increase in the logic
                       density of silicon circuits.  Originally formulated
                       by Gordon Moore in 1964 as a doubling every year
                       starting in 1962, in the late 1970s the rate fell to
                       a doubling every 18 months and has remained there
                       through the date of this document.  See "The New
                       Hacker's Dictionary", Third Edition, MIT Press, ISBN
                       0-262-18178-9, Eric S.  Raymond, 1996.

       [NASLUND]       Naslund, M. and A. Russell, "Extraction of Optimally
                       Unbiased Bits from a Biased Source", IEEE
                       Transactions on Information Theory. 46(3), May 2000.

       [ORMAN]         Orman, H. and P. Hoffman, "Determining Strengths For
                       Public Keys Used For Exchanging Symmetric Keys", BCP
                       86, RFC 3766, April 2004.

       [RFC1750]       Eastlake 3rd, D., Crocker, S., and J. Schiller,
                       "Randomness Recommendations for Security", RFC 1750,
                       December 1994.

       [RFC1948]       Bellovin, S., "Defending Against Sequence Number
                       Attacks", RFC 1948, May 1996.

    Eastlake, et al.            Standards Track                    [Page 44]

    RFC 4086         Randomness Requirements for Security          June 2005

       [RFC2104]       Krawczyk, H., Bellare, M., and R. Canetti, "HMAC:
                       Keyed-Hashing for Message Authentication", RFC 2104,
                       February 1997.

       [RSA_BULL1]     "Suggestions for Random Number Generation in
                       Software", RSA Laboratories Bulletin #1, January
                       1996.

       [RSA_BULL13]    Silverman, R., "A Cost-Based Security Analysis of
                       Symmetric and Asymmetric Key Lengths", RSA
                       Laboratories Bulletin #13, April 2000 (revised
                       November 2001).

       [SBOX1]         Mister, S. and C. Adams, "Practical S-box Design",
                       Selected Areas in Cryptography, 1996.

       [SBOX2]         Nyberg, K., "Perfect Non-linear S-boxes", Advances in
                       Cryptography, Eurocrypt '91 Proceedings, Springer-
                       Verland, 1991.

       [SCHNEIER]      Schneier, B., "Applied Cryptography: Protocols,
                       Algorithms, and Source Code in C", 2nd Edition, John
                       Wiley & Sons, 1996.

       [SHANNON]       Shannon, C., "The Mathematical Theory of
                       Communication", University of Illinois Press, 1963.
                       Originally from:  Bell System Technical Journal, July
                       and October, 1948.

       [SHIFT1]        Golub, S., "Shift Register Sequences", Aegean Park
                       Press, Revised Edition, 1982.

       [SHIFT2]        Barker, W., "Cryptanalysis of Shift-Register
                       Generated Stream Cypher Systems", Aegean Park Press,
                       1984.

       [SHA]           "Secure Hash Standard", US National Institute of
                       Science and Technology, FIPS 180-2, 1 August 2002.

       [SHA_RFC]       Eastlake 3rd, D. and P. Jones, "US Secure Hash
                       Algorithm 1 (SHA1)", RFC 3174, September 2001.

       [SSH]           Products of the SECSH Working Group, Works in
                       Progress, 2005.

       [STERN]         Stern, J., "Secret Linear Congruential Generators are
                       not Cryptographically Secure", Proc. IEEE STOC, 1987.

    Eastlake, et al.            Standards Track                    [Page 45]

    RFC 4086         Randomness Requirements for Security          June 2005

       [TLS]           Dierks, T. and C. Allen, "The TLS Protocol Version
                       1.0", RFC 2246, January 1999.

       [TURBID]        Denker, J., "High Entropy Symbol Generator",
                       <http://www.av8n.com/turbid/paper/turbid.htm>, 2003.

       [USENET_1]      Kantor, B. and P. Lapsley, "Network News Transfer
                       Protocol", RFC 977, February 1986.

       [USENET_2]      Barber, S., "Common NNTP Extensions", RFC 2980,
                       October 2000.

       [VON_NEUMANN]   Von Nuemann, J., "Various techniques used in
                       connection with random digits", Von Neumann's
                       Collected Works, Vol. 5, Pergamon Press, 1963.

       [WSC]           Howard, M. and D. LeBlanc, "Writing Secure Code,
                       Second Edition", Microsoft Press, ISBN 0735617228,
                       December 2002.

       [X9.17]         "American National Standard for Financial Institution
                       Key Management (Wholesale)", American Bankers
                       Association, 1985.

       [X9.82]         "Random Number Generation", American National
                       Standards Institute, ANSI X9F1, Work in Progress.
                          Part 1 - Overview and General Principles.
                          Part 2 - Non-Deterministic Random Bit Generators
                          Part 3 - Deterministic Random Bit Generators

    Eastlake, et al.            Standards Track                    [Page 46]

    RFC 4086         Randomness Requirements for Security          June 2005

    Authors' Addresses

       Donald E. Eastlake 3rd
       Motorola Laboratories
       155 Beaver Street
       Milford, MA 01757 USA

       Phone: +1 508-786-7554 (w)
              +1 508-634-2066 (h)
       EMail: Donald.Eastlake@motorola.com

       Jeffrey I. Schiller
       MIT, Room E40-311
       77 Massachusetts Avenue
       Cambridge, MA 02139-4307 USA

       Phone: +1 617-253-0161
       EMail: jis@mit.edu

       Steve Crocker

       EMail: steve@stevecrocker.com

    Eastlake, et al.            Standards Track                    [Page 47]

    RFC 4086         Randomness Requirements for Security          June 2005

    Full Copyright Statement

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    Acknowledgement

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    Eastlake, et al.            Standards Track                    [Page 48]