]> Elliptic Curve Cryptography (ECC) Brainpool Curves for Transport Layer Security (TLS) secunet Security Networks
Mergenthaler Allee 77 65760 Eschborn Germany +49 201 5454 3091 johannes.merkle@secunet.com
Bundesamt fuer Sicherheit in der Informationstechnik (BSI)
Postfach 200363 53133 Bonn Germany +49 228 9582 5643 manfred.lochter@bsi.bund.de
TLS, Elliptic Curve Cryptography 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.
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 pseudo-random way and comply with the security requirements of relevant standards from ISO , ANSI , NIST , and SecG . defines the usage of elliptic curves for authentication and key agreement in TLS 1.0 and TLS 1.1; these mechanisms may also be used with TLS 1.2 . While the ASN.1 object identifiers defined in already allow usage of the ECC Brainpool curves for TLS (client or server) authentication through reference in X.509 certificates according to and , their negotiation for key exchange according to requires the definition and assignment of additional NamedCurve IDs. This document specifies such values for three curves from .
According to , the name space NamedCurve is used for the negotiation of elliptic curve groups for key exchange during a handshake starting a new TLS session. This document adds new NamedCurve types to three elliptic curves defined in as follows:
These curves are suitable for use with Datagram TLS . Test vectors for a Diffie-Hellman key exchange using these elliptic curves are provided in .
IANA has assigned numbers for the ECC Brainpool curves listed in in the "EC Named Curve registry of the "Transport Layer Security (TLS) Parameters" registry as follows: Value Description DTLS-OK Reference 26 brainpoolP256r1 Y RFC 7027 27 brainpoolP384r1 Y RFC 7027 28 brainpoolP512r1 Y RFC 7027
The security considerations of apply to the ECC Brainpool curves described in this document. 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 for authentication and/or key exchange in TLS, the key derivation function; the algorithms and key lengths of symmetric encryption; and message authentication (as well as the algorithm, bit length, and hash function used for signature generation) should be chosen according to the recommendations of and . Furthermore, the private Diffie-Hellman keys should be selected with the same bit length as the order of the group generated by the base point G and with approximately maximum entropy. 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 , 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), an 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 and .
Transport Layer Security (TLS) Parameters Internet Assigned Numbers Authority Public Key Cryptography For The Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA) American National Standards Institute Minimum Requirements for Evaluating Side-Channel Attack Resistance of Elliptic Curve Implementations Bundesamt fuer Sicherheit in der Informationstechnik Digital Signature Standard (DSS) National Institute of Standards and Technology Guide to Elliptic Curve Cryptography Information Technology - Security Techniques - Digital Signatures with Appendix - Part 3: Discrete Logarithm Based Mechanisms International Organization for Standardization Information Technology - Security Techniques - Cryptographic Techniques Based on Elliptic Curves - Part 2: Digital signatures International Organization for Standardization Recommendation for Key Management - Part 1: General (Revised) National Institute of Standards and Technology Elliptic Curve Cryptography Certicom Research Recommended Elliptic Curve Domain Parameters Certicom Research
This section provides some test vectors for example Diffie-Hellman key exchanges using each of the curves defined in . The following notation is used in 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 pre-master 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 .
Curve brainpoolP256r1 dA = 81DB1EE100150FF2EA338D708271BE38300CB54241D79950F77B063039804F1D x_qA = 44106E913F92BC02A1705D9953A8414DB95E1AAA49E81D9E85F929A8E3100BE5 y_qA = 8AB4846F11CACCB73CE49CBDD120F5A900A69FD32C272223F789EF10EB089BDC dB = 55E40BC41E37E3E2AD25C3C6654511FFA8474A91A0032087593852D3E7D76BD3 x_qB = 8D2D688C6CF93E1160AD04CC4429117DC2C41825E1E9FCA0ADDD34E6F1B39F7B y_qB = 990C57520812BE512641E47034832106BC7D3E8DD0E4C7F1136D7006547CEC6A x_Z = 89AFC39D41D3B327814B80940B042590F96556EC91E6AE7939BCE31F3A18BF2B y_Z = 49C27868F4ECA2179BFD7D59B1E3BF34C1DBDE61AE12931648F43E59632504DE
Curve brainpoolP384r1 dA = 1E20F5E048A5886F1F157C74E91BDE2B98C8B52D58E5003D57053FC4B0BD65D6F15EB5D1EE1610DF870795143627D042 x_qA = 68B665DD91C195800650CDD363C625F4E742E8134667B767B1B476793588F885AB698C852D4A6E77A252D6380FCAF068 y_qA = 55BC91A39C9EC01DEE36017B7D673A931236D2F1F5C83942D049E3FA20607493E0D038FF2FD30C2AB67D15C85F7FAA59 dB = 032640BC6003C59260F7250C3DB58CE647F98E1260ACCE4ACDA3DD869F74E01F8BA5E0324309DB6A9831497ABAC96670 x_qB = 4D44326F269A597A5B58BBA565DA5556ED7FD9A8A9EB76C25F46DB69D19DC8CE6AD18E404B15738B2086DF37E71D1EB4 y_qB = 62D692136DE56CBE93BF5FA3188EF58BC8A3A0EC6C1E151A21038A42E9185329B5B275903D192F8D4E1F32FE9CC78C48 x_Z = 0BD9D3A7EA0B3D519D09D8E48D0785FB744A6B355E6304BC51C229FBBCE239BBADF6403715C35D4FB2A5444F575D4F42 y_Z = 0DF213417EBE4D8E40A5F76F66C56470C489A3478D146DECF6DF0D94BAE9E598157290F8756066975F1DB34B2324B7BD
Curve brainpoolP512r1 dA = 16302FF0DBBB5A8D733DAB7141C1B45ACBC8715939677F6A56850A38BD87BD59B09E80279609FF333EB9D4C061231FB26F92EEB04982A5F1D1764CAD57665422 x_qA = 0A420517E406AAC0ACDCE90FCD71487718D3B953EFD7FBEC5F7F27E28C6149999397E91E029E06457DB2D3E640668B392C2A7E737A7F0BF04436D11640FD09FD y_qA = 72E6882E8DB28AAD36237CD25D580DB23783961C8DC52DFA2EC138AD472A0FCEF3887CF62B623B2A87DE5C588301EA3E5FC269B373B60724F5E82A6AD147FDE7 dB = 230E18E1BCC88A362FA54E4EA3902009292F7F8033624FD471B5D8ACE49D12CFABBC19963DAB8E2F1EBA00BFFB29E4D72D13F2224562F405CB80503666B25429 x_qB = 9D45F66DE5D67E2E6DB6E93A59CE0BB48106097FF78A081DE781CDB31FCE8CCBAAEA8DD4320C4119F1E9CD437A2EAB3731FA9668AB268D871DEDA55A5473199F y_qB = 2FDC313095BCDD5FB3A91636F07A959C8E86B5636A1E930E8396049CB481961D365CC11453A06C719835475B12CB52FC3C383BCE35E27EF194512B71876285FA x_Z = A7927098655F1F9976FA50A9D566865DC530331846381C87256BAF3226244B76D36403C024D7BBF0AA0803EAFF405D3D24F11A9B5C0BEF679FE1454B21C4CD1F y_Z = 7DB71C3DEF63212841C463E881BDCF055523BD368240E6C3143BD8DEF8B3B3223B95E0F53082FF5E412F4222537A43DF1C6D25729DDB51620A832BE6A26680A2