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ECC Brainpool Curves for Transport Layer Security (TLS)
secunet Security Networks
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65760 Eschborn
Germany
+49 201 5454 3091
johannes.merkle@secunet.com
Bundesamt fuer Sicherheit in der Informationstechnik (BSI)
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53133 Bonn
Germany
+49 228 9582 5643
manfred.lochter@bsi.bund.de
TLS, Elliptic Curve Cryptography
This document specifies the use of several ECC Brainpool elliptic curves for authentication and key exchange in the Transport Layer Security (TLS) protocol.
In , a new set of elliptic curve groups over
finite prime fields for use in cryptographic applications was specified. 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 .
Usage of elliptic curves for authentication and key agreement in TLS 1.0 and TLS 1.1 is defined in . 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 , 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 Supported Elliptic Curve Extension allows the negotiation of elliptic curve groups during a handshake starting a new TLS session. A client that proposes ECC cipher suites in its ClientHello message SHOULD include this extension to indicate the elliptic curves it supports through NamedCurve IDs, and a server that receives a ClientHello containing this extension MUST use the client's enumerated capabilities to guide its selection of an appropriate cipher suite. Furthermore, the server SHOULD use a NamedCurve ID, if applicable, to specify the elliptic curve corresponding to its ephemeral ECDH public key.
The values of NamedCurve, by which the elliptic curves are identified, are governed by the EC Named Curve Registry of IANA . In , new values for NamedCurve for three of the ECC Brainpool curves defined in are specified. All three curves are suitable for usage in DTLS .
Curve
Value
brainpoolP256r1
TBD1
brainpoolP384r1
TBD2
brainpoolP512r1
TBD3
Test vectors for a Diffie-Hellman key exchange using these ECC Brainpool curves are provided in
The security considerations of 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 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 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 resistence of elliptic curve cryptography implementations against side-channel-attacks is given in and .
Before this document can become an RFC, IANA is required to assign numbers for the elliptic curves specified in to the NamedCurve name space in its Transport Layer Security (TLS) Parameters registry . For all these elliptic curves, suitability with DTLS shall be indicated in the registry.
Although, the authors have no knowledge about any intellectual property rights
which cover the general usage of the ECP groups defined herein, implementations based on these
domain parameters may require use of inventions covered by patent rights. In particular, techniques for an efficient arithmetic exploiting the special parameters of the twisted curves (see ) may be covered by patents.
Transport Layer Security (TLS) Parameters
Internet Assigned Numbers Authority
Key words for use in RFCs to Indicate Requirement Levels
Harvard University
1350 Mass. Ave.
Cambridge
MA 02138
- +1 617 495 3864
sob@harvard.edu
General
keyword
Elliptic Curve Cryptography (ECC) Cipher Suites for Transport Layer Security (TLS)
This document describes new key exchange algorithms based on Elliptic Curve Cryptography (ECC) for the Transport Layer Security (TLS) protocol. In particular, it specifies the use of Elliptic Curve Diffie-Hellman (ECDH) key agreement in a TLS handshake and the use of Elliptic Curve Digital Signature Algorithm (ECDSA) as a new authentication mechanism. This memo provides information for the Internet community.
The Transport Layer Security (TLS) Protocol Version 1.2
This document specifies Version 1.2 of the Transport Layer Security (TLS) protocol. The TLS protocol provides communications security over the Internet. The protocol allows client/server applications to communicate in a way that is designed to prevent eavesdropping, tampering, or message forgery. [STANDARDS-TRACK]
Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation
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.
Datagram Transport Layer Security Version 1.2
This document specifies version 1.2 of the Datagram Transport Layer Security (DTLS) protocol. The DTLS protocol provides communications privacy for datagram protocols. The protocol allows client/server applications to communicate in a way that is designed to prevent eavesdropping, tampering, or message forgery. The DTLS protocol is based on the Transport Layer Security (TLS) protocol and provides equivalent security guarantees. Datagram semantics of the underlying transport are preserved by the DTLS protocol. This document updates DTLS 1.0 to work with TLS version 1.2. [STANDARDS-TRACK]
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 für 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
Algorithms and Identifiers for the Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile
This document specifies algorithm identifiers and ASN.1 encoding formats for digital signatures and subject public keys used in the Internet X.509 Public Key Infrastructure (PKI). Digital signatures are used to sign certificates and certificate revocation list (CRLs). Certificates include the public key of the named subject. [STANDARDS-TRACK]
Fundamental Elliptic Curve Cryptography Algorithms
This note describes the fundamental algorithms of Elliptic Curve Cryptography (ECC) as they were defined in some seminal references from 1994 and earlier. These descriptions may be useful for implementing the fundamental algorithms without using any of the specialized methods that were developed in following years. Only elliptic curves defined over fields of characteristic greater than three are in scope; these curves are those used in Suite B. This document is not an Internet Standards Track specification; it is published for informational purposes.
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 . In all
of the following sections the following notation is used:
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, 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