<?xml version='1.0' encoding='utf-8'?>
<rfc xmlns:xi="http://www.w3.org/2001/XInclude" version="3" category="info" docName="draft-bruckert-brainpool-for-tls13-07" indexInclude="true" ipr="trust200902" number="8734" prepTime="2020-02-21T14:15:18" scripts="Common,Latin" sortRefs="true" submissionType="independent" symRefs="true" tocDepth="3" tocInclude="true" xml:lang="en">
  <link href="https://datatracker.ietf.org/doc/draft-bruckert-brainpool-for-tls13-07" rel="prev"/>
  <link href="https://dx.doi.org/10.17487/rfc8734" rel="alternate"/>
  <link href="urn:issn:2070-1721" rel="alternate"/>
  <front>
    <title abbrev="ECC Brainpool Curves for TLS 1.3">Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS) Version 1.3</title>
    <seriesInfo name="RFC" value="8734" stream="independent"/>
    <author fullname="Leonie Bruckert" initials="L." surname="Bruckert">
      <organization showOnFrontPage="true">secunet Security Networks</organization>
      <address>
        <postal>
          <street>Ammonstr. 74</street>
          <city>Dresden</city>
          <code>01067</code>
          <country>Germany</country>
        </postal>
        <phone>+49 201 5454 3819</phone>
        <email>leonie.bruckert@secunet.com</email>
      </address>
    </author>
    <author fullname="Johannes Merkle" initials="J." surname="Merkle">
      <organization showOnFrontPage="true">secunet Security Networks</organization>
      <address>
        <postal>
          <street>Mergenthaler Allee 77</street>
          <city>Eschborn</city>
          <code>65760</code>
          <country>Germany</country>
        </postal>
        <phone>+49 201 5454 3091</phone>
        <email>johannes.merkle@secunet.com</email>
      </address>
    </author>
    <author fullname="Manfred Lochter" initials="M." surname="Lochter">
      <organization showOnFrontPage="true">BSI</organization>
      <address>
        <postal>
          <street>Postfach 200363</street>
          <city>Bonn</city>
          <code>53133</code>
          <country>Germany</country>
        </postal>
        <phone>+49 228 9582 5643</phone>
        <email>manfred.lochter@bsi.bund.de</email>
      </address>
    </author>
    <date month="02" year="2020"/>
    <workgroup/>
    <keyword>TLS</keyword>
    <keyword>Elliptic Curve Cryptography</keyword>
    <abstract pn="section-abstract">
      <t pn="section-abstract-1">Elliptic Curve Cryptography (ECC) Brainpool curves were an option 
      for authentication and key
      exchange in the Transport Layer Security (TLS) protocol version 1.2 but
      were deprecated by the IETF for use with TLS version 1.3 because they
      had little usage. However, these curves have not been shown to have
      significant cryptographical weaknesses, and there is some interest in
      using several of these curves in TLS 1.3.</t>
      <t pn="section-abstract-2">This document provides the necessary protocol mechanisms for using
      ECC Brainpool curves in TLS 1.3. This approach is not endorsed by the
      IETF.</t>
    </abstract>
    <boilerplate>
      <section anchor="status-of-memo" numbered="false" removeInRFC="false" toc="exclude" pn="section-boilerplate.1">
        <name slugifiedName="name-status-of-this-memo">Status of This Memo</name>
        <t pn="section-boilerplate.1-1">
            This document is not an Internet Standards Track specification; it is
            published for informational purposes.
        </t>
        <t pn="section-boilerplate.1-2">
            This is a contribution to the RFC Series, independently of any
            other RFC stream.  The RFC Editor has chosen to publish this
            document at its discretion and makes no statement about its value
            for implementation or deployment.  Documents approved for
            publication by the RFC Editor are not candidates for any level of
            Internet Standard; see Section 2 of RFC 7841.
        </t>
        <t pn="section-boilerplate.1-3">
            Information about the current status of this document, any
            errata, and how to provide feedback on it may be obtained at
            <eref target="https://www.rfc-editor.org/info/rfc8734" brackets="none"/>.
        </t>
      </section>
      <section anchor="copyright" numbered="false" removeInRFC="false" toc="exclude" pn="section-boilerplate.2">
        <name slugifiedName="name-copyright-notice">Copyright Notice</name>
        <t pn="section-boilerplate.2-1">
            Copyright (c) 2020 IETF Trust and the persons identified as the
            document authors. All rights reserved.
        </t>
        <t pn="section-boilerplate.2-2">
            This document is subject to BCP 78 and the IETF Trust's Legal
            Provisions Relating to IETF Documents
            (<eref target="https://trustee.ietf.org/license-info" brackets="none"/>) in effect on the date of
            publication of this document. Please review these documents
            carefully, as they describe your rights and restrictions with
            respect to this document.
        </t>
      </section>
    </boilerplate>
    <toc>
      <section anchor="toc" numbered="false" removeInRFC="false" toc="exclude" pn="section-toc.1">
        <name slugifiedName="name-table-of-contents">Table of Contents</name>
        <ul bare="true" empty="true" indent="2" spacing="compact" pn="section-toc.1-1">
          <li pn="section-toc.1-1.1">
            <t keepWithNext="true" pn="section-toc.1-1.1.1"><xref derivedContent="1" format="counter" sectionFormat="of" target="section-1"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-introduction">Introduction</xref></t>
          </li>
          <li pn="section-toc.1-1.2">
            <t keepWithNext="true" pn="section-toc.1-1.2.1"><xref derivedContent="2" format="counter" sectionFormat="of" target="section-2"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-requirements-terminology">Requirements Terminology</xref></t>
          </li>
          <li pn="section-toc.1-1.3">
            <t keepWithNext="true" pn="section-toc.1-1.3.1"><xref derivedContent="3" format="counter" sectionFormat="of" target="section-3"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-brainpool-namedgroup-types">Brainpool NamedGroup Types</xref></t>
          </li>
          <li pn="section-toc.1-1.4">
            <t keepWithNext="true" pn="section-toc.1-1.4.1"><xref derivedContent="4" format="counter" sectionFormat="of" target="section-4"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-brainpool-signaturescheme-t">Brainpool SignatureScheme Types</xref></t>
          </li>
          <li pn="section-toc.1-1.5">
            <t keepWithNext="true" pn="section-toc.1-1.5.1"><xref derivedContent="5" format="counter" sectionFormat="of" target="section-5"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-iana-considerations">IANA Considerations</xref></t>
          </li>
          <li pn="section-toc.1-1.6">
            <t keepWithNext="true" pn="section-toc.1-1.6.1"><xref derivedContent="6" format="counter" sectionFormat="of" target="section-6"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-security-considerations">Security Considerations</xref></t>
          </li>
          <li pn="section-toc.1-1.7">
            <t keepWithNext="true" pn="section-toc.1-1.7.1"><xref derivedContent="7" format="counter" sectionFormat="of" target="section-7"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-references">References</xref></t>
            <ul bare="true" empty="true" indent="2" spacing="compact" pn="section-toc.1-1.7.2">
              <li pn="section-toc.1-1.7.2.1">
                <t keepWithNext="true" pn="section-toc.1-1.7.2.1.1"><xref derivedContent="7.1" format="counter" sectionFormat="of" target="section-7.1"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-normative-references">Normative References</xref></t>
              </li>
              <li pn="section-toc.1-1.7.2.2">
                <t keepWithNext="true" pn="section-toc.1-1.7.2.2.1"><xref derivedContent="7.2" format="counter" sectionFormat="of" target="section-7.2"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-informative-references">Informative References</xref></t>
              </li>
            </ul>
          </li>
          <li pn="section-toc.1-1.8">
            <t keepWithNext="true" pn="section-toc.1-1.8.1"><xref derivedContent="Appendix A" format="default" sectionFormat="of" target="section-appendix.a"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-test-vectors">Test Vectors</xref></t>
            <ul bare="true" empty="true" indent="2" spacing="compact" pn="section-toc.1-1.8.2">
              <li pn="section-toc.1-1.8.2.1">
                <t keepWithNext="true" pn="section-toc.1-1.8.2.1.1"><xref derivedContent="A.1" format="counter" sectionFormat="of" target="section-a.1"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-256-bit-curve">256-Bit Curve</xref></t>
              </li>
              <li pn="section-toc.1-1.8.2.2">
                <t keepWithNext="true" pn="section-toc.1-1.8.2.2.1"><xref derivedContent="A.2" format="counter" sectionFormat="of" target="section-a.2"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-384-bit-curve">384-Bit Curve</xref></t>
              </li>
              <li pn="section-toc.1-1.8.2.3">
                <t keepWithNext="true" pn="section-toc.1-1.8.2.3.1"><xref derivedContent="A.3" format="counter" sectionFormat="of" target="section-a.3"/>.  <xref derivedContent="" format="title" sectionFormat="of" target="name-512-bit-curve">512-Bit Curve</xref></t>
              </li>
            </ul>
          </li>
          <li pn="section-toc.1-1.9">
            <t keepWithNext="true" pn="section-toc.1-1.9.1"><xref derivedContent="" format="none" sectionFormat="of" target="section-appendix.b"/><xref derivedContent="" format="title" sectionFormat="of" target="name-authors-addresses">Authors' Addresses</xref></t>
          </li>
        </ul>
      </section>
    </toc>
  </front>
  <middle>
    <section numbered="true" toc="include" removeInRFC="false" pn="section-1">
      <name slugifiedName="name-introduction">Introduction</name>
      <t pn="section-1-1"><xref target="RFC5639" format="default" sectionFormat="of" derivedContent="RFC5639"/> specifies a new set of
      elliptic curve groups over finite prime fields for use in cryptographic
      applications. These groups, denoted as ECC Brainpool curves, were
      generated in a verifiably pseudorandom way and comply with the security
      requirements of relevant standards from ISO <xref target="ISO1" format="default" sectionFormat="of" derivedContent="ISO1"/><xref target="ISO2" format="default" sectionFormat="of" derivedContent="ISO2"/>, ANSI <xref target="ANSI1" format="default" sectionFormat="of" derivedContent="ANSI1"/>, NIST <xref target="FIPS" format="default" sectionFormat="of" derivedContent="FIPS"/>, and SECG <xref target="SEC2" format="default" sectionFormat="of" derivedContent="SEC2"/>. </t>
      <t pn="section-1-2"><xref target="RFC8422" format="default" sectionFormat="of" derivedContent="RFC8422"/> defines the usage of
      elliptic curves for authentication and key agreement in TLS 1.2 and
      earlier versions, and <xref target="RFC7027" format="default" sectionFormat="of" derivedContent="RFC7027"/> defines
      the usage of the ECC Brainpool curves for authentication and key
      exchange in TLS. The latter is applicable to TLS 1.2 and earlier
      versions but not to TLS 1.3, which deprecates the ECC Brainpool curve IDs defined in <xref target="RFC7027" format="default" sectionFormat="of" derivedContent="RFC7027"/> due to the lack of widespread deployment. However, there is some interest in using these curves in TLS 1.3.</t>
      <t pn="section-1-3">The negotiation of ECC Brainpool curves for key exchange in TLS 1.3, according to <xref target="RFC8446" format="default" sectionFormat="of" derivedContent="RFC8446"/>, requires the definition and assignment of additional NamedGroup IDs. This document provides the necessary definition and assignment of additional SignatureScheme IDs for using three ECC Brainpool curves from <xref target="RFC5639" format="default" sectionFormat="of" derivedContent="RFC5639"/>.</t>
      <t pn="section-1-4">This approach is not endorsed by the IETF. Implementers and deployers need to be aware of the strengths and weaknesses of all security mechanisms that they use.</t>
    </section>
    <section numbered="true" toc="include" removeInRFC="false" pn="section-2">
      <name slugifiedName="name-requirements-terminology">Requirements Terminology</name>
      <t pn="section-2-1">
    The key words "<bcp14>MUST</bcp14>", "<bcp14>MUST NOT</bcp14>",
    "<bcp14>REQUIRED</bcp14>", "<bcp14>SHALL</bcp14>", "<bcp14>SHALL NOT</bcp14>", "<bcp14>SHOULD</bcp14>", "<bcp14>SHOULD NOT</bcp14>",
    "<bcp14>RECOMMENDED</bcp14>", "<bcp14>NOT RECOMMENDED</bcp14>",
    "<bcp14>MAY</bcp14>", and "<bcp14>OPTIONAL</bcp14>" in this document are
    to be interpreted as described in BCP 14 <xref target="RFC2119" format="default" sectionFormat="of" derivedContent="RFC2119"/>
        <xref target="RFC8174" format="default" sectionFormat="of" derivedContent="RFC8174"/> when, and only when, they appear in all capitals,
    as shown here.
      </t>
    </section>
    <section anchor="Main" numbered="true" toc="include" removeInRFC="false" pn="section-3">
      <name slugifiedName="name-brainpool-namedgroup-types">Brainpool NamedGroup Types</name>
      <t pn="section-3-1">According to <xref target="RFC8446" format="default" sectionFormat="of" derivedContent="RFC8446"/>, the
      "supported_groups" extension is used for the negotiation of
      Diffie-Hellman groups and elliptic curve groups for key exchange during
      a handshake starting a new TLS session. This document adds new named
      groups for three elliptic curves defined in <xref target="RFC5639" format="default" sectionFormat="of" derivedContent="RFC5639"/> to the "supported_groups" extension, as follows.</t>
      <sourcecode type="tls-presentation" markers="false" pn="section-3-2">
        enum {
             brainpoolP256r1tls13(31),
             brainpoolP384r1tls13(32),
             brainpoolP512r1tls13(33)
        } NamedGroup;
</sourcecode>
      <t pn="section-3-3">The encoding of Ephemeral Elliptic Curve Diffie-Hellman (ECDHE) 
      parameters for sec256r1, secp384r1, and
      secp521r1, as defined in <xref target="RFC8446" sectionFormat="of" section="4.2.8.2" format="default" derivedLink="https://rfc-editor.org/rfc/rfc8446#section-4.2.8.2" derivedContent="RFC8446"/>, also applies to this document. </t>
      <t pn="section-3-4">Test vectors for a Diffie-Hellman key exchange using these elliptic
      curves are provided in <xref target="test_vectors" format="default" sectionFormat="of" derivedContent="Appendix A"/>.</t>
    </section>
    <section anchor="Main2" numbered="true" toc="include" removeInRFC="false" pn="section-4">
      <name slugifiedName="name-brainpool-signaturescheme-t">Brainpool SignatureScheme Types</name>
      <t pn="section-4-1">According to <xref target="RFC8446" format="default" sectionFormat="of" derivedContent="RFC8446"/>, the name space SignatureScheme is used for the negotiation of elliptic curve groups for authentication via the "signature_algorithms" extension. Besides, it is required to specify the hash function that is used to hash the message before signing. This document adds new SignatureScheme types for three elliptic curves defined in <xref target="RFC5639" format="default" sectionFormat="of" derivedContent="RFC5639"/>, as follows.</t>
      <sourcecode type="tls-presentation" markers="false" pn="section-4-2">
        enum {
             ecdsa_brainpoolP256r1tls13_sha256(0x081A),
             ecdsa_brainpoolP384r1tls13_sha384(0x081B),
             ecdsa_brainpoolP512r1tls13_sha512(0x081C)
        } SignatureScheme;
</sourcecode>
    </section>
    <section numbered="true" toc="include" removeInRFC="false" pn="section-5">
      <name slugifiedName="name-iana-considerations">IANA Considerations</name>
      <t pn="section-5-1">IANA has updated the references for the ECC Brainpool
      curves listed in the "TLS Supported Groups" <xref target="IANA-TLS" format="default" sectionFormat="of" derivedContent="IANA-TLS"/> subregistry of the "Transport Layer Security (TLS)
      Parameters" registry to refer to this document. </t>
      <table anchor="namedGroups" align="left" pn="table-1">
        <thead>
          <tr>
            <th align="center" colspan="1" rowspan="1">Value</th>
            <th align="center" colspan="1" rowspan="1">Description</th>
            <th align="center" colspan="1" rowspan="1">DTLS-OK</th>
            <th align="center" colspan="1" rowspan="1">Recommended</th>
            <th align="center" colspan="1" rowspan="1">Reference</th>
          </tr>
        </thead>
        <tbody>
          <tr>
            <td align="center" colspan="1" rowspan="1">31</td>
            <td align="center" colspan="1" rowspan="1">brainpoolP256r1tls13</td>
            <td align="center" colspan="1" rowspan="1">Y</td>
            <td align="center" colspan="1" rowspan="1">N</td>
            <td align="center" colspan="1" rowspan="1">RFC 8734</td>
          </tr>
          <tr>
            <td align="center" colspan="1" rowspan="1">32</td>
            <td align="center" colspan="1" rowspan="1">brainpoolP384r1tls13</td>
            <td align="center" colspan="1" rowspan="1">Y</td>
            <td align="center" colspan="1" rowspan="1">N</td>
            <td align="center" colspan="1" rowspan="1">RFC 8734</td>
          </tr>
          <tr>
            <td align="center" colspan="1" rowspan="1">33</td>
            <td align="center" colspan="1" rowspan="1">brainpoolP512r1tls13</td>
            <td align="center" colspan="1" rowspan="1">Y</td>
            <td align="center" colspan="1" rowspan="1">N</td>
            <td align="center" colspan="1" rowspan="1">RFC 8734</td>
          </tr>
        </tbody>
      </table>
      <t pn="section-5-3">IANA has updated the references for the ECC Brainpool curves in the
      "TLS SignatureScheme" subregistry <xref target="IANA-TLS" format="default" sectionFormat="of" derivedContent="IANA-TLS"/> of the "Transport Layer Security
(TLS) Parameters" registry to refer to this document.</t>
      <table anchor="SignatureSchemes" align="left" pn="table-2">
        <thead>
          <tr>
            <th align="center" colspan="1" rowspan="1">Value</th>
            <th align="center" colspan="1" rowspan="1">Description</th>
            <th align="center" colspan="1" rowspan="1">Recommended</th>
            <th align="center" colspan="1" rowspan="1">Reference</th>
          </tr>
        </thead>
        <tbody>
          <tr>
            <td align="center" colspan="1" rowspan="1">0x081A</td>
            <td align="center" colspan="1" rowspan="1">ecdsa_brainpoolP256r1tls13_sha256</td>
            <td align="center" colspan="1" rowspan="1">N</td>
            <td align="center" colspan="1" rowspan="1">RFC 8734</td>
          </tr>
          <tr>
            <td align="center" colspan="1" rowspan="1">0x081B</td>
            <td align="center" colspan="1" rowspan="1">ecdsa_brainpoolP384r1tls13_sha384</td>
            <td align="center" colspan="1" rowspan="1">N</td>
            <td align="center" colspan="1" rowspan="1">RFC 8734</td>
          </tr>
          <tr>
            <td align="center" colspan="1" rowspan="1">0x081C</td>
            <td align="center" colspan="1" rowspan="1">ecdsa_brainpoolP512r1tls13_sha512</td>
            <td align="center" colspan="1" rowspan="1">N</td>
            <td align="center" colspan="1" rowspan="1">RFC 8734</td>
          </tr>
        </tbody>
      </table>
    </section>
    <section anchor="Security" numbered="true" toc="include" removeInRFC="false" pn="section-6">
      <name slugifiedName="name-security-considerations">Security Considerations</name>
      <t pn="section-6-1">The security considerations of <xref target="RFC8446" format="default" sectionFormat="of" derivedContent="RFC8446"/> apply accordingly. </t>
      <t pn="section-6-2">The confidentiality, authenticity, and integrity of the TLS
      communication is limited by the weakest cryptographic primitive
      applied. 
In order to achieve a maximum security level when using one of the elliptic
curves from <xref target="namedGroups" format="default" sectionFormat="of" derivedContent="Table 1"/> for key exchange
and/or one of the signature algorithms from <xref target="SignatureSchemes" format="default" sectionFormat="of" derivedContent="Table 2"/> for authentication in TLS, parameters of other deployed 
cryptographic schemes should be chosen at commensurate strengths, for example,
according to the recommendations of <xref target="NIST800-57" format="default" sectionFormat="of" derivedContent="NIST800-57"/> and <xref target="RFC5639" format="default" sectionFormat="of" derivedContent="RFC5639"/>.  In
particular, this applies to (a) the key derivation function, (b) the
algorithms and key length of symmetric encryption and message authentication,
and (c) the algorithm, bit length, and hash function for signature generation.
      Furthermore, the private
      Diffie-Hellman keys should be generated from a random keystream with a
      length equal to the length of the order of the group E(GF(p)) defined in
      <xref target="RFC5639" format="default" sectionFormat="of" derivedContent="RFC5639"/>. The value of the private
      Diffie-Hellman keys should be less than the order of the group
      E(GF(p)).</t>
      <t pn="section-6-3">When using ECDHE key agreement with the curves brainpoolP256r1tls13,
      brainpoolP384r1tls13, or brainpoolP512r1tls13, the peers
      <bcp14>MUST</bcp14> validate each other's public value Q by ensuring
      that the point is a valid point on the elliptic curve. If this check is
      not conducted, an attacker can force the key exchange into a small
      subgroup, and the resulting shared secret can be guessed with
      significantly less effort. </t>
      <t pn="section-6-4">Implementations of elliptic curve cryptography for TLS may be
      susceptible to side-channel attacks. Particular care should be taken for
      implementations that internally transform curve points to points on the
      corresponding "twisted curve", using the map (x',y') = (x*Z^2, y*Z^3)
      with the coefficient Z specified for that curve in <xref target="RFC5639" format="default" sectionFormat="of" derivedContent="RFC5639"/>, in order to take advantage of an
      efficient arithmetic based on the twisted curve's special parameters (A
      = -3). Although the twisted curve itself offers the same level of
      security as the corresponding random curve (through mathematical
      equivalence), arithmetic based on small curve parameters may be harder
      to protect against side-channel attacks. General guidance on resistance
      of elliptic curve cryptography implementations against side-channel attacks is given in <xref target="BSI1" format="default" sectionFormat="of" derivedContent="BSI1"/> and <xref target="HMV" format="default" sectionFormat="of" derivedContent="HMV"/>.</t>
    </section>
  </middle>
  <back>
    <references pn="section-7">
      <name slugifiedName="name-references">References</name>
      <references pn="section-7.1">
        <name slugifiedName="name-normative-references">Normative References</name>
        <reference anchor="IANA-TLS" target="https://www.iana.org/assignments/tls-parameters" quoteTitle="true" derivedAnchor="IANA-TLS">
          <front>
            <title>Transport Layer Security (TLS) Parameters</title>
            <author>
              <organization showOnFrontPage="true">IANA</organization>
            </author>
          </front>
        </reference>
        <reference anchor="RFC2119" target="https://www.rfc-editor.org/info/rfc2119" quoteTitle="true" derivedAnchor="RFC2119">
          <front>
            <title>Key words for use in RFCs to Indicate Requirement Levels</title>
            <author initials="S." surname="Bradner" fullname="S. Bradner">
              <organization showOnFrontPage="true"/>
            </author>
            <date year="1997" month="March"/>
            <abstract>
              <t>In many standards track documents several words are used to signify the requirements in the specification.  These words are often capitalized. This document defines these words as they should be interpreted in IETF documents.  This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements.</t>
            </abstract>
          </front>
          <seriesInfo name="BCP" value="14"/>
          <seriesInfo name="RFC" value="2119"/>
          <seriesInfo name="DOI" value="10.17487/RFC2119"/>
        </reference>
        <reference anchor="RFC5639" target="https://www.rfc-editor.org/info/rfc5639" quoteTitle="true" derivedAnchor="RFC5639">
          <front>
            <title>Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation</title>
            <author initials="M." surname="Lochter" fullname="M. Lochter">
              <organization showOnFrontPage="true"/>
            </author>
            <author initials="J." surname="Merkle" fullname="J. Merkle">
              <organization showOnFrontPage="true"/>
            </author>
            <date year="2010" month="March"/>
            <abstract>
              <t>This memo proposes several elliptic curve domain parameters over finite prime fields for use in cryptographic applications.  The domain parameters are consistent with the relevant international standards, and can be used in X.509 certificates and certificate revocation lists (CRLs), for Internet Key Exchange (IKE), Transport Layer Security (TLS), XML signatures, and all applications or protocols based on the cryptographic message syntax (CMS).  This document is  not an Internet Standards Track specification; it is published for  informational purposes.</t>
            </abstract>
          </front>
          <seriesInfo name="RFC" value="5639"/>
          <seriesInfo name="DOI" value="10.17487/RFC5639"/>
        </reference>
        <reference anchor="RFC7027" target="https://www.rfc-editor.org/info/rfc7027" quoteTitle="true" derivedAnchor="RFC7027">
          <front>
            <title>Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS)</title>
            <author initials="J." surname="Merkle" fullname="J. Merkle">
              <organization showOnFrontPage="true"/>
            </author>
            <author initials="M." surname="Lochter" fullname="M. Lochter">
              <organization showOnFrontPage="true"/>
            </author>
            <date year="2013" month="October"/>
            <abstract>
              <t>This document specifies the use of several Elliptic Curve Cryptography (ECC) Brainpool curves for authentication and key exchange in the Transport Layer Security (TLS) protocol.</t>
            </abstract>
          </front>
          <seriesInfo name="RFC" value="7027"/>
          <seriesInfo name="DOI" value="10.17487/RFC7027"/>
        </reference>
        <reference anchor="RFC8174" target="https://www.rfc-editor.org/info/rfc8174" quoteTitle="true" derivedAnchor="RFC8174">
          <front>
            <title>Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words</title>
            <author initials="B." surname="Leiba" fullname="B. Leiba">
              <organization showOnFrontPage="true"/>
            </author>
            <date year="2017" month="May"/>
            <abstract>
              <t>RFC 2119 specifies common key words that may be used in protocol  specifications.  This document aims to reduce the ambiguity by clarifying that only UPPERCASE usage of the key words have the  defined special meanings.</t>
            </abstract>
          </front>
          <seriesInfo name="BCP" value="14"/>
          <seriesInfo name="RFC" value="8174"/>
          <seriesInfo name="DOI" value="10.17487/RFC8174"/>
        </reference>
        <reference anchor="RFC8422" target="https://www.rfc-editor.org/info/rfc8422" quoteTitle="true" derivedAnchor="RFC8422">
          <front>
            <title>Elliptic Curve Cryptography (ECC) Cipher Suites for Transport Layer Security (TLS) Versions 1.2 and Earlier</title>
            <author initials="Y." surname="Nir" fullname="Y. Nir">
              <organization showOnFrontPage="true"/>
            </author>
            <author initials="S." surname="Josefsson" fullname="S. Josefsson">
              <organization showOnFrontPage="true"/>
            </author>
            <author initials="M." surname="Pegourie-Gonnard" fullname="M. Pegourie-Gonnard">
              <organization showOnFrontPage="true"/>
            </author>
            <date year="2018" month="August"/>
            <abstract>
              <t>This document describes key exchange algorithms based on Elliptic Curve Cryptography (ECC) for the Transport Layer Security (TLS) protocol.  In particular, it specifies the use of Ephemeral Elliptic Curve Diffie-Hellman (ECDHE) key agreement in a TLS handshake and the use of the Elliptic Curve Digital Signature Algorithm (ECDSA) and Edwards-curve Digital Signature Algorithm (EdDSA) as authentication mechanisms.</t>
              <t>This document obsoletes RFC 4492.</t>
            </abstract>
          </front>
          <seriesInfo name="RFC" value="8422"/>
          <seriesInfo name="DOI" value="10.17487/RFC8422"/>
        </reference>
        <reference anchor="RFC8446" target="https://www.rfc-editor.org/info/rfc8446" quoteTitle="true" derivedAnchor="RFC8446">
          <front>
            <title>The Transport Layer Security (TLS) Protocol Version 1.3</title>
            <author initials="E." surname="Rescorla" fullname="E. Rescorla">
              <organization showOnFrontPage="true"/>
            </author>
            <date year="2018" month="August"/>
            <abstract>
              <t>This document specifies version 1.3 of the Transport Layer Security (TLS) protocol.  TLS allows client/server applications to communicate over the Internet in a way that is designed to prevent eavesdropping, tampering, and message forgery.</t>
              <t>This document updates RFCs 5705 and 6066, and obsoletes RFCs 5077, 5246, and 6961.  This document also specifies new requirements for TLS 1.2 implementations.</t>
            </abstract>
          </front>
          <seriesInfo name="RFC" value="8446"/>
          <seriesInfo name="DOI" value="10.17487/RFC8446"/>
        </reference>
      </references>
      <references pn="section-7.2">
        <name slugifiedName="name-informative-references">Informative References</name>
        <reference anchor="ANSI1" quoteTitle="true" derivedAnchor="ANSI1">
          <front>
            <title>Public Key Cryptography For The Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA)</title>
            <seriesInfo name="ANSI" value="X9.62"/>
            <author>
              <organization showOnFrontPage="true">American National Standards Institute</organization>
            </author>
            <date month="November" year="2005"/>
          </front>
        </reference>
        <reference anchor="BSI1" quoteTitle="true" derivedAnchor="BSI1">
          <front>
            <title>Minimum Requirements for Evaluating Side-Channel Attack Resistance of Elliptic Curve Implementations</title>
            <author>
              <organization showOnFrontPage="true">Bundesamt fuer Sicherheit in der Informationstechnik</organization>
            </author>
            <date month="July" year="2011"/>
          </front>
        </reference>
        <reference anchor="FIPS" quoteTitle="true" target="https://doi.org/10.6028/NIST.FIPS.186-4" derivedAnchor="FIPS">
          <front>
            <title>Digital Signature Standard (DSS)</title>
            <author>
              <organization showOnFrontPage="true">National Institute of Standards and Technology</organization>
            </author>
            <date month="July" year="2013"/>
          </front>
          <seriesInfo name="FIPS" value="PUB 186-4"/>
          <seriesInfo name="DOI" value="10.6028/NIST.FIPS.186-4"/>
        </reference>
        <reference anchor="HMV" quoteTitle="true" derivedAnchor="HMV">
          <front>
            <title>Guide to Elliptic Curve Cryptography</title>
            <seriesInfo name="Springer" value="Verlag"/>
            <author initials="D" surname="Hankerson">
              <organization showOnFrontPage="true"></organization>
            </author>
            <author initials="A" surname="Menezes">
              <organization showOnFrontPage="true"></organization>
            </author>
            <author initials="S" surname="Vanstone">
              <organization showOnFrontPage="true"></organization>
            </author>
            <date month="" year="2004"/>
          </front>
        </reference>
        <reference anchor="ISO1" quoteTitle="true" derivedAnchor="ISO1">
          <front>
            <title>Information Technology - Security Techniques - Digital Signatures with Appendix - Part 3: Discrete Logarithm Based Mechanisms</title>
            <seriesInfo name="ISO/IEC" value="14888-3"/>
            <author>
              <organization showOnFrontPage="true">International Organization for Standardization</organization>
            </author>
            <date month="November" year="2018"/>
          </front>
        </reference>
        <reference anchor="ISO2" quoteTitle="true" derivedAnchor="ISO2">
          <front>
            <title>Information Technology - Security techniques - Cryptographic techniques based on elliptic curves - Part 2: Digital signatures</title>
            <author>
              <organization showOnFrontPage="true">International Organization for Standardization</organization>
            </author>
            <date month="December" year="2002"/>
          </front>
          <seriesInfo name="ISO/IEC" value="15946-2:2002"/>
        </reference>
        <reference anchor="NIST800-57" quoteTitle="true" target="https://doi.org/10.6028/NIST.SP.800-57ptlr4" derivedAnchor="NIST800-57">
          <front>
            <title>Recommendation for Key Management - Part 1: General (Revised)</title>
            <author>
              <organization showOnFrontPage="true">National Institute of Standards and Technology</organization>
            </author>
            <date month="January" year="2016"/>
          </front>
          <seriesInfo name="NIST Special Publication" value="800-57"/>
          <seriesInfo name="DOI" value="10.6028/NIST.SP.800-57ptlr4"/>
        </reference>
        <reference anchor="SEC1" quoteTitle="true" derivedAnchor="SEC1">
          <front>
            <title>SEC1: Elliptic Curve Cryptography</title>
            <author>
              <organization showOnFrontPage="true">Standards for Efficient Cryptography Group</organization>
            </author>
            <date month="May" year="2019"/>
          </front>
        </reference>
        <reference anchor="SEC2" quoteTitle="true" derivedAnchor="SEC2">
          <front>
            <title>SEC 2: Recommended Elliptic Curve Domain Parameters</title>
            <author>
              <organization showOnFrontPage="true">Standards for Efficient Cryptography Group</organization>
            </author>
            <date month="January" year="2010"/>
          </front>
        </reference>
      </references>
    </references>
    <section anchor="test_vectors" numbered="true" toc="include" removeInRFC="false" pn="section-appendix.a">
      <name slugifiedName="name-test-vectors">Test Vectors</name>
      <t pn="section-appendix.a-1">This non-normative Appendix provides some test vectors (for example, Diffie-Hellman
   key exchanges using each of the curves defined in <xref target="namedGroups" format="default" sectionFormat="of" derivedContent="Table 1"/>). The following notation is used in all 
   of the subsequent sections:
      </t>
      <dl newline="false" indent="7" spacing="normal" pn="section-appendix.a-2">
        <dt pn="section-appendix.a-2.1">d_A:</dt>
        <dd pn="section-appendix.a-2.2">the secret key of party A </dd>
        <dt pn="section-appendix.a-2.3">x_qA:</dt>
        <dd pn="section-appendix.a-2.4">the x-coordinate of the public key of party A </dd>
        <dt pn="section-appendix.a-2.5">y_qA:</dt>
        <dd pn="section-appendix.a-2.6">the y-coordinate of the public key of party A </dd>
        <dt pn="section-appendix.a-2.7">d_B:</dt>
        <dd pn="section-appendix.a-2.8">the secret key of party B </dd>
        <dt pn="section-appendix.a-2.9">x_qB:</dt>
        <dd pn="section-appendix.a-2.10">the x-coordinate of the public key of party B </dd>
        <dt pn="section-appendix.a-2.11">y_qB:</dt>
        <dd pn="section-appendix.a-2.12">the y-coordinate of the public key of party B </dd>
        <dt pn="section-appendix.a-2.13">x_Z:</dt>
        <dd pn="section-appendix.a-2.14">the x-coordinate of the shared secret that results from
      completion of the Diffie-Hellman computation, i.e., the hex representation of the premaster secret</dd>
        <dt pn="section-appendix.a-2.15">y_Z:</dt>
        <dd pn="section-appendix.a-2.16">the y-coordinate of the shared secret that results from
      completion of the Diffie-Hellman computation </dd>
      </dl>
      <t pn="section-appendix.a-3"> 
The field elements x_qA, y_qA, x_qB, y_qB, x_Z, and y_Z are represented as hexadecimal values using the FieldElement-to-OctetString conversion method specified in <xref target="SEC1" format="default" sectionFormat="of" derivedContent="SEC1"/>.
</t>
      <section numbered="true" toc="include" removeInRFC="false" pn="section-a.1">
        <name slugifiedName="name-256-bit-curve">256-Bit Curve</name>
        <t pn="section-a.1-1">Curve brainpoolP256r1
        </t>
        <sourcecode type="test-vectors" markers="false" pn="section-a.1-2">
dA =
81DB1EE100150FF2EA338D708271BE38300CB54241D79950F77B063039804F1D

x_qA =
44106E913F92BC02A1705D9953A8414DB95E1AAA49E81D9E85F929A8E3100BE5

y_qA =
8AB4846F11CACCB73CE49CBDD120F5A900A69FD32C272223F789EF10EB089BDC

dB =
55E40BC41E37E3E2AD25C3C6654511FFA8474A91A0032087593852D3E7D76BD3

x_qB =
8D2D688C6CF93E1160AD04CC4429117DC2C41825E1E9FCA0ADDD34E6F1B39F7B

y_qB =
990C57520812BE512641E47034832106BC7D3E8DD0E4C7F1136D7006547CEC6A

x_Z =
89AFC39D41D3B327814B80940B042590F96556EC91E6AE7939BCE31F3A18BF2B

y_Z =
49C27868F4ECA2179BFD7D59B1E3BF34C1DBDE61AE12931648F43E59632504DE
</sourcecode>
      </section>
      <section numbered="true" toc="include" removeInRFC="false" pn="section-a.2">
        <name slugifiedName="name-384-bit-curve">384-Bit Curve</name>
        <t pn="section-a.2-1">Curve brainpoolP384r1
        </t>
        <sourcecode type="test-vectors" markers="false" pn="section-a.2-2">
dA = 1E20F5E048A5886F1F157C74E91BDE2B98C8B52D58E5003D57053FC4B0BD6
5D6F15EB5D1EE1610DF870795143627D042

x_qA = 68B665DD91C195800650CDD363C625F4E742E8134667B767B1B47679358
8F885AB698C852D4A6E77A252D6380FCAF068

y_qA = 55BC91A39C9EC01DEE36017B7D673A931236D2F1F5C83942D049E3FA206
07493E0D038FF2FD30C2AB67D15C85F7FAA59

dB = 032640BC6003C59260F7250C3DB58CE647F98E1260ACCE4ACDA3DD869F74E
01F8BA5E0324309DB6A9831497ABAC96670

x_qB = 4D44326F269A597A5B58BBA565DA5556ED7FD9A8A9EB76C25F46DB69D19
DC8CE6AD18E404B15738B2086DF37E71D1EB4

y_qB = 62D692136DE56CBE93BF5FA3188EF58BC8A3A0EC6C1E151A21038A42E91
85329B5B275903D192F8D4E1F32FE9CC78C48

x_Z = 0BD9D3A7EA0B3D519D09D8E48D0785FB744A6B355E6304BC51C229FBBCE2
39BBADF6403715C35D4FB2A5444F575D4F42

y_Z = 0DF213417EBE4D8E40A5F76F66C56470C489A3478D146DECF6DF0D94BAE9
E598157290F8756066975F1DB34B2324B7BD
</sourcecode>
      </section>
      <section numbered="true" toc="include" removeInRFC="false" pn="section-a.3">
        <name slugifiedName="name-512-bit-curve">512-Bit Curve</name>
        <t pn="section-a.3-1">Curve brainpoolP512r1
        </t>
        <sourcecode type="test-vectors" markers="false" pn="section-a.3-2">
dA = 16302FF0DBBB5A8D733DAB7141C1B45ACBC8715939677F6A56850A38BD87B
D59B09E80279609FF333EB9D4C061231FB26F92EEB04982A5F1D1764CAD5766542
2

x_qA = 0A420517E406AAC0ACDCE90FCD71487718D3B953EFD7FBEC5F7F27E28C6
149999397E91E029E06457DB2D3E640668B392C2A7E737A7F0BF04436D11640FD0
9FD

y_qA = 72E6882E8DB28AAD36237CD25D580DB23783961C8DC52DFA2EC138AD472
A0FCEF3887CF62B623B2A87DE5C588301EA3E5FC269B373B60724F5E82A6AD147F
DE7

dB = 230E18E1BCC88A362FA54E4EA3902009292F7F8033624FD471B5D8ACE49D1
2CFABBC19963DAB8E2F1EBA00BFFB29E4D72D13F2224562F405CB80503666B2542
9

x_qB = 9D45F66DE5D67E2E6DB6E93A59CE0BB48106097FF78A081DE781CDB31FC
E8CCBAAEA8DD4320C4119F1E9CD437A2EAB3731FA9668AB268D871DEDA55A54731
99F

y_qB = 2FDC313095BCDD5FB3A91636F07A959C8E86B5636A1E930E8396049CB48
1961D365CC11453A06C719835475B12CB52FC3C383BCE35E27EF194512B7187628
5FA

x_Z = A7927098655F1F9976FA50A9D566865DC530331846381C87256BAF322624
4B76D36403C024D7BBF0AA0803EAFF405D3D24F11A9B5C0BEF679FE1454B21C4CD
1F

y_Z = 7DB71C3DEF63212841C463E881BDCF055523BD368240E6C3143BD8DEF8B3
B3223B95E0F53082FF5E412F4222537A43DF1C6D25729DDB51620A832BE6A26680
A2
</sourcecode>
      </section>
    </section>
    <section anchor="authors-addresses" numbered="false" removeInRFC="false" toc="include" pn="section-appendix.b">
      <name slugifiedName="name-authors-addresses">Authors' Addresses</name>
      <author fullname="Leonie Bruckert" initials="L." surname="Bruckert">
        <organization showOnFrontPage="true">secunet Security Networks</organization>
        <address>
          <postal>
            <street>Ammonstr. 74</street>
            <city>Dresden</city>
            <code>01067</code>
            <country>Germany</country>
          </postal>
          <phone>+49 201 5454 3819</phone>
          <email>leonie.bruckert@secunet.com</email>
        </address>
      </author>
      <author fullname="Johannes Merkle" initials="J." surname="Merkle">
        <organization showOnFrontPage="true">secunet Security Networks</organization>
        <address>
          <postal>
            <street>Mergenthaler Allee 77</street>
            <city>Eschborn</city>
            <code>65760</code>
            <country>Germany</country>
          </postal>
          <phone>+49 201 5454 3091</phone>
          <email>johannes.merkle@secunet.com</email>
        </address>
      </author>
      <author fullname="Manfred Lochter" initials="M." surname="Lochter">
        <organization showOnFrontPage="true">BSI</organization>
        <address>
          <postal>
            <street>Postfach 200363</street>
            <city>Bonn</city>
            <code>53133</code>
            <country>Germany</country>
          </postal>
          <phone>+49 228 9582 5643</phone>
          <email>manfred.lochter@bsi.bund.de</email>
        </address>
      </author>
    </section>
  </back>
</rfc>
