The Curve25519 Function
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Authors  Watson Ladd , Rich Salz , Sean Turner  
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draftturnerthecurve25519function00
Network Working Group W. Ladd InternetDraft Grad Student UC Berkley Intended status: Informational R. Salz Expires: January 30, 2015 Akamai S. Turner IECA, Inc. July 29, 2014 The Curve25519 Function draftturnerthecurve25519function00 Abstract This document specifies the Curve25519 function, an ECDH (Elliptic Curve DiffieHellman) keyagreement scheme for use in cryptographic applications. It was designed with performance and security in mind. This document is based on information in the public domain. Status of This Memo This InternetDraft is submitted in full conformance with the provisions of BCP 78 and BCP 79. InternetDrafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as InternetDrafts. The list of current Internet Drafts is at http://datatracker.ietf.org/drafts/current/. InternetDrafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use InternetDrafts as reference material or to cite them other than as "work in progress." This InternetDraft will expire on January 30, 2015. Copyright Notice Copyright (c) 2014 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 (http://trustee.ietf.org/licenseinfo) 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. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of Ladd, et al. Expires January 30, 2015 [Page 1] InternetDraft The Curve25519 Function July 2014 the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. 1. Introduction This document specifies the Curve25519 function, an ECDH (Elliptic curve DiffieHellman) keyagreement scheme for use in cryptographic applications. It was designed with performance and security in mind. This document is based on information in the public domain. This document provides a stable reference for the Curve25519 function [Curve25519] to which other specifications may refer when defining their use of Curve25519 This document does not specify the use of Curve25519 in any other specific protocol, such as TLS (Transport Layer Security) or IPsec (Internet Protocol Security). This document specifies how to use Curve25519 for key exchange; it does not specify how to use Curve25519 for use with digital signatures. This document defines the algorithm, expected "wire format," and provides some implementation guidance to avoid known sidechannel exposures. Readers are assumed to be familiar with the concepts of elliptic curves, modular arithmetic, group operations, and finite fields [RFC6090] as well as rings [Curve25519]. 1.1. Terminology The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119]. 2. Notation and Definitions The following notation and definitions are used in this document (notation is to the left of the ":"): A: A value used in the ellipticcurve equation E. E: An ellipticcurve equation. p: A prime. GF(p): The field with p elements. mod: An abbreviation for modulo. _#: Subscript notation, where # is a number or letter =: Denotes equal to. Ladd, et al. Expires January 30, 2015 [Page 2] InternetDraft The Curve25519 Function July 2014 ^: Denotes exponentiation. +, , *, /: Denotes addition, subtraction, multiplication, and division. Note that all operations are performed mod p. 3. The Curve25519 Function Let p=2^25519. Let E be the elliptic curve with the equation y^2=x^3+486662*x^2+x over GF(p). Each element x of GF(p) has a unique littleendian representation as 32 bytes s[0] ... s[31], such that s[0]+256_s[1]+256^2_s[2]+...+256^31*s[31] is congruent to x modulo p, and s[31] is minimal. Implementations MUST only produce points in this form, and MUST mask the high bit of byte 31 to zero on receiving a point. The high bit is, following convention, 0x80. Let X denote the projection map from a point (x,y) on E, to x, extended so that X of the point at infinity is zero. X is surjective onto GF(p) if the y coordinate takes on values in GF(p) and in a quadratic extension of GF(p). Then Curve25519(s, X(Q))=X(sQ) is a function defined for all elements of GF(p). The remainder of this document describes how to compute this function quickly and securely, and use it in a DiffieHellman scheme. 4. Implementing Curve25519 Let s be a 255 bits long integer, where s=sum s_i2^i with s_i in {0,1}. Computing Curve25519(s, x) is done by the following procedure, taken from [Curve25519] based on formulas from [Mont]. All calculations are done over GF(p), i.e., they are performed modulo p. The parameter a24 is a24 = (486662  2)/4 = 121665. Ladd, et al. Expires January 30, 2015 [Page 3] InternetDraft The Curve25519 Function July 2014 Let x_1 = 1 x_2 = 1 z_2 = 0 x_3 = x z_3 = 1 For t = 254 to 0: Do constant time conditional swap of: (x_2, z_2) and (x_3, z_3) if s_t is set A = x_2 + z_2 AA = A^2 B = x_2  z_2 BB = B^2 E = AA  BB C = x_3 + z_3 D = x_3  z_3 DA = D * A CB = C * B x_3 = (DA + CB)^2 z_3 = x_1 * (DA  CB)^2 x_2 = AA * BB z_2 = E * (AA + a24 * E) Do constant time conditional swap of: (x_2, z_2) and (x_3, z_3) if s_t is set Return x_2*(z_2^(p1)) In implementing this procedure, due to the existence of sidechannels in commodity hardware, it is vital that the pattern of memory accesses and jumps not depend on the bits of s. It is also essential that the arithmetic used not leak information about words. To compute the conditional swap in constant time (independent of s_t) use dummy = s_t*(x_2x_3) x_2 = x_2  dummy x_3 = x_3 + dummy where s_t is 1 or 0, or dummy = s_t & (x_2 XOR x_3) x_2 = x_2 XOR x_3 x_3 = x_3 XOR x_2 where s_t is regarded as the all1 word of 255 bits. The latter version is more efficient on most architectures. 5. Use of the Curve25519 function The Curve25519 function can be used in an ECDH protocol as follows: Alice takes 32 random bytes in s[0] to s[32]. She masks the lower three bits of s[0] and the top bit of s[31] to zero and sets the second top most bit of s[31] to 1. This means that s is of the form 2^254+8*{0,1, ...., 2^(251)1} as a littleendian integer. Alice then transmits K_A = Curve25519(s, 9) to Bob, where 9 is the number 9. As a sequence of 32 bytes, t, the representation of 9 is Ladd, et al. Expires January 30, 2015 [Page 4] InternetDraft The Curve25519 Function July 2014 t[0]=9, and the remaining bytes are all zero. The natural wire format representation of the value is in littleendian byte order. Bob picks a random g, and computes K_B = Curve25519(g, 9) similarly, and transmits it to Alice. Alice computes Curve25519(s, Curve25519(g, 9)); Bob computes Curve25519(g, Curve25519(s, 9)) using their secret values and the received input. Both of them now share K=Curve25519(s, Curve25519(g, 9))=Curve25519(g, Curve25519(s, 9)) as a shared secret. Alice and Bob use a keyderivation function, such as hashing K, to compute a shared secret. 6. Test Vectors The following test vectors are taken from [NaCl]: Alice's public key: 0x8520f0098930a754748b7ddcb43ef75a0dbf3a0d26381af4eba4a98eaa9b4e6a Alice's secret key 0x77076d0a7318a57d3c16c17251b26645df4c2f87ebc0992ab177fba51db92c2a Bob's public key: 0xde9edb7d7b7dc1b4d35b61c2ece435373f8343c85b78674dadfc7e146f882b4f Bob's secret key: 0x5dab087e624a8a4b79e17f8b83800ee66f3bb1292618b6fd1c2f8b27ff88e0eb Shared secret: 0x4a5d9d5ba4ce2de1728e3bf480350f25e07e21c947d19e3376f09b3c1e161742 7. Security Considerations Curve25519 meets all standard assumptions on DH and DLP difficulty. In addition, Curve25519 is twist secure: the cofactor of the curve is 8, that of the twist is 4. Protocols that require contributory behavior must ban outputs K_A =0, K_B = 0 or K = 0. Ladd, et al. Expires January 30, 2015 [Page 5] InternetDraft The Curve25519 Function July 2014 Curve25519 is designed to enable very high performance software implementations, thus reducing the cost of highly secure cryptography to a point where it can be used more widely. 8. IANA Considerations None. 9. Acknowledgements We would like to thank Tanja Lange (Technische Universiteit Eindhoven) for her review and comments. 10. References 10.1. Normative References [Curve25519] Bernstein, D., "Curve25519  new DiffieHellman speed records", April 2006, <http://www.iacr.org/cryptodb/archive/2006/ PKC/3351/3351.pdf>. [Mont] Montgomery, P., "Speeding the Pollard and elliptic curve methods of factorization", 1983, <http://www.ams.org/journals/mcom/198748177/ S00255718198708661137/S00255718198708661137.pdf>. [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, March 1997. [RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic Curve Cryptography Algorithms", RFC 6090, February 2011. 10.2. Informative References [NaCl] Bernstein, D., "Cryptography in NaCl", 2013, <http://cr.yp.to/highspeed/naclcrypto20090310.pdf>. Authors' Addresses Watson Ladd Grad Student UC Berkley Email: watsonbladd@gmail.com Ladd, et al. Expires January 30, 2015 [Page 6] InternetDraft The Curve25519 Function July 2014 Rich Salz Akamai Email: rsalz@akamai.com Sean Turner IECA, Inc. Suite 106 Fairfax, VA 22031 USA Phone: +17036283180 Email: turners@ieca.com Ladd, et al. Expires January 30, 2015 [Page 7]