Independent Submission                                       L. Bruckert
Request for Comments: 8734                                     J. Merkle
Category: Informational                        secunet Security Networks
ISSN: 2070-1721                                               M. Lochter
                                                           February 2020

Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer

Security (TLS) Version 1.3


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.

This document provides the necessary protocol mechanisms for using ECC Brainpool curves in TLS 1.3. This approach is not endorsed by the IETF.

Status of This Memo

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

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.

Information about the current status of this document, any errata, and how to provide feedback on it may be obtained at

Copyright Notice

Copyright © 2020 IETF Trust and the persons identified as the document authors. All rights reserved.

This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents ( 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.

Table of Contents

   1.  Introduction
   2.  Requirements Terminology
   3.  Brainpool NamedGroup Types
   4.  Brainpool SignatureScheme Types
   5.  IANA Considerations
   6.  Security Considerations
   7.  References
     7.1.  Normative References
     7.2.  Informative References
   Appendix A.  Test Vectors
     A.1.  256-Bit Curve
     A.2.  384-Bit Curve
     A.3.  512-Bit Curve
   Authors' Addresses

1. Introduction

[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 [ISO1][ISO2], ANSI [ANSI1], NIST [FIPS], and SECG [SEC2].

[RFC8422] defines the usage of elliptic curves for authentication and key agreement in TLS 1.2 and earlier versions, and [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 [RFC7027] due to the lack of widespread deployment. However, there is some interest in using these curves in TLS 1.3.

The negotiation of ECC Brainpool curves for key exchange in TLS 1.3, according to [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 [RFC5639].

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.

2. Requirements Terminology

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

3. Brainpool NamedGroup Types

According to [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 [RFC5639] to the "supported_groups" extension, as follows.

           enum {
           } NamedGroup;

The encoding of Ephemeral Elliptic Curve Diffie-Hellman (ECDHE) parameters for sec256r1, secp384r1, and secp521r1, as defined in Section of [RFC8446], also applies to this document.

Test vectors for a Diffie-Hellman key exchange using these elliptic curves are provided in Appendix A.

4. Brainpool SignatureScheme Types

According to [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 [RFC5639], as follows.

           enum {
           } SignatureScheme;

5. IANA Considerations

IANA has updated the references for the ECC Brainpool curves listed in the "TLS Supported Groups" [IANA-TLS] subregistry of the "Transport Layer Security (TLS) Parameters" registry to refer to this document.

   | Value |     Description      | DTLS-OK | Recommended | Reference |
   |   31  | brainpoolP256r1tls13 |    Y    |      N      |  RFC 8734 |
   |   32  | brainpoolP384r1tls13 |    Y    |      N      |  RFC 8734 |
   |   33  | brainpoolP512r1tls13 |    Y    |      N      |  RFC 8734 |

Table 1

IANA has updated the references for the ECC Brainpool curves in the "TLS SignatureScheme" subregistry [IANA-TLS] of the "Transport Layer Security (TLS) Parameters" registry to refer to this document.

   |Value |            Description            | Recommended |Reference |
   |0x081A| ecdsa_brainpoolP256r1tls13_sha256 |      N      | RFC 8734 |
   |0x081B| ecdsa_brainpoolP384r1tls13_sha384 |      N      | RFC 8734 |
   |0x081C| ecdsa_brainpoolP512r1tls13_sha512 |      N      | RFC 8734 |

Table 2

6. Security Considerations

The security considerations of [RFC8446] apply accordingly.

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 Table 1 for key exchange and/or one of the signature algorithms from 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 [NIST800-57] and [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 [RFC5639]. The value of the private Diffie-Hellman keys should be less than the order of the group E(GF(p)).

When using ECDHE key agreement with the curves brainpoolP256r1tls13, brainpoolP384r1tls13, or brainpoolP512r1tls13, the peers MUST 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.

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 [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 [BSI1] and [HMV].

7. References

7.1. Normative References

[IANA-TLS] IANA, "Transport Layer Security (TLS) Parameters",


   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
   [RFC5639]  Lochter, M. and J. Merkle, "Elliptic Curve Cryptography
              (ECC) Brainpool Standard Curves and Curve Generation",
              RFC 5639, DOI 10.17487/RFC5639, March 2010,
   [RFC7027]  Merkle, J. and M. Lochter, "Elliptic Curve Cryptography
              (ECC) Brainpool Curves for Transport Layer Security
              (TLS)", RFC 7027, DOI 10.17487/RFC7027, October 2013,
   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <>.
   [RFC8422]  Nir, Y., Josefsson, S., and M. Pegourie-Gonnard, "Elliptic
              Curve Cryptography (ECC) Cipher Suites for Transport Layer
              Security (TLS) Versions 1.2 and Earlier", RFC 8422,
              DOI 10.17487/RFC8422, August 2018,
   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,

7.2. Informative References

   [ANSI1]    American National Standards Institute, "Public Key
              Cryptography For The Financial Services Industry: the
              Elliptic Curve Digital Signature Algorithm (ECDSA)",
              ANSI X9.62, November 2005.
   [BSI1]     Bundesamt fuer Sicherheit in der Informationstechnik,
              "Minimum Requirements for Evaluating Side-Channel Attack
              Resistance of Elliptic Curve Implementations", July 2011.
   [FIPS]     National Institute of Standards and Technology, "Digital
              Signature Standard (DSS)", FIPS PUB 186-4,
              DOI 10.6028/NIST.FIPS.186-4, July 2013,
   [HMV]      Hankerson, D., Menezes, A., and S. Vanstone, "Guide to
              Elliptic Curve Cryptography", Springer Verlag, 2004.
   [ISO1]     International Organization for Standardization,
              "Information Technology - Security Techniques - Digital
              Signatures with Appendix - Part 3: Discrete Logarithm
              Based Mechanisms", ISO/IEC 14888-3, November 2018.
   [ISO2]     International Organization for Standardization,
              "Information Technology - Security techniques -
              Cryptographic techniques based on elliptic curves - Part
              2: Digital signatures", ISO/IEC 15946-2:2002, December


              National Institute of Standards and Technology,
              "Recommendation for Key Management - Part 1: General
              (Revised)", NIST Special Publication 800-57,
              DOI 10.6028/NIST.SP.800-57ptlr4, January 2016,
   [SEC1]     Standards for Efficient Cryptography Group, "SEC1:
              Elliptic Curve Cryptography", May 2019.
   [SEC2]     Standards for Efficient Cryptography Group, "SEC 2:
              Recommended Elliptic Curve Domain Parameters", January

Appendix A. Test Vectors

This non-normative Appendix provides some test vectors (for example, Diffie-Hellman key exchanges using each of the curves defined in Table 1). The following notation is used in all of the subsequent sections:

   d_A:   the secret key of party A
   x_qA:  the x-coordinate of the public key of party A
   y_qA:  the y-coordinate of the public key of party A
   d_B:   the secret key of party B
   x_qB:  the x-coordinate of the public key of party B
   y_qB:  the y-coordinate of the public key of party B
   x_Z:   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
   y_Z:   the y-coordinate of the shared secret that results from
          completion of the Diffie-Hellman computation

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 [SEC1].

A.1. 256-Bit Curve

   Curve brainpoolP256r1

dA = 81DB1EE100150FF2EA338D708271BE38300CB54241D79950F77B063039804F1D

x_qA = 44106E913F92BC02A1705D9953A8414DB95E1AAA49E81D9E85F929A8E3100BE5

y_qA = 8AB4846F11CACCB73CE49CBDD120F5A900A69FD32C272223F789EF10EB089BDC

dB = 55E40BC41E37E3E2AD25C3C6654511FFA8474A91A0032087593852D3E7D76BD3

x_qB = 8D2D688C6CF93E1160AD04CC4429117DC2C41825E1E9FCA0ADDD34E6F1B39F7B

y_qB = 990C57520812BE512641E47034832106BC7D3E8DD0E4C7F1136D7006547CEC6A

x_Z = 89AFC39D41D3B327814B80940B042590F96556EC91E6AE7939BCE31F3A18BF2B

y_Z = 49C27868F4ECA2179BFD7D59B1E3BF34C1DBDE61AE12931648F43E59632504DE

A.2. 384-Bit Curve

   Curve brainpoolP384r1

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

A.3. 512-Bit Curve

   Curve brainpoolP512r1

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

Authors' Addresses

Leonie Bruckert
secunet Security Networks
Ammonstr. 74
01067 Dresden

   Phone: +49 201 5454 3819

Johannes Merkle
secunet Security Networks
Mergenthaler Allee 77
65760 Eschborn

   Phone: +49 201 5454 3091

Manfred Lochter
Postfach 200363
53133 Bonn

   Phone: +49 228 9582 5643