Network Working Group                                            W. Ladd
Internet-Draft                                   Grad Student UC Berkley
Intended status: Informational                                   R. Salz
Expires: January 30, 2015                                         Akamai
                                                               S. Turner
                                                              IECA, Inc.
                                                           July 29, 2014

                        The Curve25519 Function


   This document specifies the Curve25519 function, an ECDH (Elliptic-
   Curve Diffie-Hellman) key-agreement 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.

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   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 Diffie-Hellman) key-agreement 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 side-channel 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",
   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 elliptic-curve equation E.

   E: An elliptic-curve 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.

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   ^: Denotes exponentiation.

   +, -, *, /: Denotes addition, subtraction, multiplication, and

   Note that all operations are performed mod p.

3.  The Curve25519 Function

   Let p=2^255-19.  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 little-endian 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 Diffie-Hellman

4.  Implementing Curve25519

   Let s be a 255 bits long integer, where s=sum s_i2^i with s_i in

   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.

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   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^(p-1))

   In implementing this procedure, due to the existence of side-channels
   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_2-x_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 all-1 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 little-endian 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

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   t[0]=9, and the remaining bytes are all zero.  The natural wire-
   format representation of the value is in little-endian 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 key-derivation 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:


   Alice's secret key


   Bob's public key:


   Bob's secret key:


   Shared secret:


7.  Security Considerations

   Curve25519 meets all standard assumptions on DH and DLP difficulty.

   In addition, Curve25519 is twist secure: the co-factor 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.

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   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


9.  Acknowledgements

   We would like to thank Tanja Lange (Technische Universiteit
   Eindhoven) for her review and comments.

10.  References

10.1.  Normative References

              Bernstein, D., "Curve25519 - new Diffie-Hellman speed
              records", April 2006,

   [Mont]     Montgomery, P., "Speeding the Pollard and elliptic curve
              methods of factorization", 1983,

   [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,

Authors' Addresses

   Watson Ladd
   Grad Student UC Berkley


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   Rich Salz


   Sean Turner
   IECA, Inc.
   Suite 106
   Fairfax, VA 22031

   Phone: +1-703-628-3180

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