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Versions: 00 draft-irtf-cfrg-curves

CFRG                                                          A. Langley
Internet-Draft                                                    Google
Intended status: Informational                           January 6, 2015
Expires: July 10, 2015


                      Elliptic Curves for Security
                         draft-agl-cfrgcurve-00

Abstract

   This memo describes an algorithm for deterministically generating
   parameters for elliptic curves over prime fields offering high
   practical security in cryptographic applications, including Transport
   Layer Security (TLS) and X.509 certificates.  It also specifies a
   specific curve at the ~128-bit security level.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at http://datatracker.ietf.org/drafts/current/.

   Internet-Drafts 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 Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on July 10, 2015.

Copyright Notice

   Copyright (c) 2015 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/license-info) 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
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.



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Table of Contents

   1.  Note on authorship  . . . . . . . . . . . . . . . . . . . . .   2
   2.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   3.  Requirements Language . . . . . . . . . . . . . . . . . . . .   3
   4.  Security Requirements . . . . . . . . . . . . . . . . . . . .   3
   5.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   3
   6.  Parameter Generation  . . . . . . . . . . . . . . . . . . . .   4
     6.1.  Edwards Curves  . . . . . . . . . . . . . . . . . . . . .   4
     6.2.  Twisted Edwards Curves  . . . . . . . . . . . . . . . . .   5
     6.3.  Generators  . . . . . . . . . . . . . . . . . . . . . . .   6
   7.  Recommended Curves  . . . . . . . . . . . . . . . . . . . . .   7
   8.  Wire-format of field elements . . . . . . . . . . . . . . . .   8
   9.  Elliptic Curve Diffie-Hellman . . . . . . . . . . . . . . . .   9
     9.1.  Diffie-Hellman protocol . . . . . . . . . . . . . . . . .  11
   10. Test vectors  . . . . . . . . . . . . . . . . . . . . . . . .  11
   11. References  . . . . . . . . . . . . . . . . . . . . . . . . .  12
     11.1.  Normative References . . . . . . . . . . . . . . . . . .  12
     11.2.  Informative References . . . . . . . . . . . . . . . . .  12
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  13

1.  Note on authorship

   This document is a merging of "draft-black-rpgecc-01" (by Benjamin
   Black, Joppe W.  Bos, Craig Costello, Patrick Longa and Michael
   Naehrig) and "draft-turner-thecurve25519function-01" (by Watson Ladd,
   Rich Salz and Sean Turner).  They are the actual authors of the words
   and figures, but authorship also implies support and so are not
   listed as authors until they have confirmed that they support this
   document.  None the less, they deserve any credit for the contents.

2.  Introduction

   Since the initial standardization of elliptic curve cryptography
   (ECC) in [SEC1] there has been significant progress related to both
   efficiency and security of curves and implementations.  Notable
   examples are algorithms protected against certain side-channel
   attacks, different 'special' prime shapes which allow faster modular
   arithmetic, and a larger set of curve models from which to choose.
   There is also concern in the community regarding the generation and
   potential weaknesses of the curves defined in [NIST].

   This memo describes a deterministic algorithm for generation of
   elliptic curves for cryptography.  The constraints in the generation
   process produce curves that support constant-time, exception-free
   scalar multiplications that are resistant to a wide range of side-
   channel attacks including timing and cache attacks, thereby offering
   high practical security in cryptographic applications.  The



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   deterministic algorithm operates without any hidden parameters,
   reliance on randomness or any other processes offering opportunities
   for manipulation of the resulting curves.  The selection between
   curve models is determined by choosing the curve form that supports
   the fastest (currently known) complete formulas for each modularity
   option of the underlying field prime.  Specifically, the Edwards
   curve x^2 + y^2 = 1 + dx^2y^2 is used with primes p with p = 3 mod 4,
   and the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2 is used for
   primes p with p = 1 mod 4.

3.  Requirements Language

   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 RFC 2119 [RFC2119].

4.  Security Requirements

   For each curve at a specific security level:

   1.  The domain parameters SHALL be generated in a simple,
       deterministic manner, without any secret or random inputs.  The
       derivation of the curve parameters is defined in Section 6.

   2.  The trace of Frobenius MUST NOT be in {0, 1} in order to rule out
       the attacks described in [Smart], [AS], and [S], as in [EBP].

   3.  MOV Degree: the embedding degree k MUST be greater than (r - 1) /
       100, as in [EBP].

   4.  CM Discriminant: discriminant D MUST be greater than 2^100, as in
       [SC].

5.  Notation

   Throughout this document, the following notation is used:

   p  Denotes the prime number defining the underlying field.

   GF(p)  The finite field with p elements.

   d  An element in the finite field GF(p), not equal to -1 or zero.

   Ed An Edwards curve: an elliptic curve over GF(p) with equation x^2 +
      y^2 = 1 + dx^2y^2.

   tEd  A twisted Edwards curve where a=-1: an elliptic curve over GF(p)
      with equation -x^2 + y^2 = 1 + dx^2y^2.



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   oddDivisor  The largest odd divisor of the number of GF(p)-rational
      points on a (twisted) Edwards curve.

   oddDivisor'  The largest odd divisor of the number of GF(p)-rational
      points on the non-trivial quadratic twist of a (twisted) Edwards
      curve.

   cofactor  The cofactor of the subgroup of order oddDivisor in the
      group of GF(p)-rational points of a (twisted) Edwards curve.

   cofactor'  The cofactor of the subgroup of order oddDivisor in the
      group of GF(p)-rational points on the non-trivial quadratic twist
      of a (twisted) Edwards curve.

   trace  The trace of Frobenius of Ed or tEd such that #Ed(GF(p)) = p +
      1 - trace or #tEd(GF(p)) = p + 1 - trace, respectively.

   P  A generator point defined over GF(p) of prime order oddDivisor on
      Ed or tEd.

   X(P)  The x-coordinate of the elliptic curve point P.

   Y(P)  The y-coordinate of the elliptic curve point P.

6.  Parameter Generation

   This section describes the generation of the curve parameter, namely
   d, of the elliptic curve.  The input to this process is p, the prime
   that defines the underlying field.  The size of p determines the
   amount of work needed to compute a discrete logarithm in the elliptic
   curve group and choosing a precise p depends on many implementation
   concerns.  The performance of the curve will be dominated by
   operations in GF(p) and thus carefully choosing a value that allows
   for easy reductions on the intended architecture is critical for
   performance.  This document does not attempt to articulate all these
   considerations.

6.1.  Edwards Curves

   For p = 3 mod 4, the elliptic curve Ed in Edwards form is determined
   by the non-square element d from GF(p) (not equal to -1 or zero) with
   smallest absolute value such that #Ed(GF(p)) = cofactor * oddDivisor,
   #Ed'(GF(p)) = cofactor' * oddDivisor', cofactor = cofactor' = 4, and
   both subgroup orders oddDivisor and oddDivisor' are prime.  In
   addition, care must be taken to ensure the MOV degree and CM
   discriminant requirements from Section 4 are met.

   These cofactors are chosen because they are minimal.



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Input: a prime p, with p = 3 mod 4
Output: the parameter d defining the curve Ed
1. Set d = 0
2. repeat
     repeat
       if (d > 0) then
         d = -d
       else
         d = -d + 1
       end if
     until d is not a square in GF(p)

     Compute oddDivisor, oddDivisor', cofactor and cofactor' where #Ed(GF(p)) =
     cofactor * oddDivisor, #Ed'(GF(p)) = cofactor' * oddDivisor', cofactor and
     cofactor' are powers of 2 and oddDivisor, oddDivisor' are odd.
   until ((cofactor = cofactor' = 4), oddDivisor is prime and oddDivisor' is prime)
3. Output d

                           GenerateCurveEdwards

6.2.  Twisted Edwards Curves

   For a prime p = 1 mod 4, the elliptic curve tEd in twisted Edwards
   form is determined by the non-square element d from GF(p) (not equal
   to -1 or zero) with smallest absolute value such that #tEd(GF(p)) =
   cofactor * oddDivisor, #tEd'(GF(p)) = cofactor' * oddDivisor',
   cofactor = 8, cofactor' = 4 and both subgroup orders oddDivisor and
   oddDivisor' are prime.  In addition, care must be taken to ensure the
   MOV degree and CM discriminant requirements from Section 4 are met.

   These cofactors are chosen so that they are minimal such that the
   cofactor of the main curve is greater than the cofactor of the twist.
   It's not possible in this case for the cofactors to be equal, but it
   is possible for the twist cofactor to be larger.  The latter is
   considered dangerous because algorithms that depend on the cofactor
   of the curve may be vulnerable if a point on the twist is accepted.















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Input: a prime p, with p = 1 mod 4
Output: the parameter d defining the curve tEd
1. Set d = 0
2. repeat
     repeat
       if (d > 0) then
         d = -d
       else
         d = -d + 1
       end if
     until d is not a square in GF(p)

     Compute oddDivisor, oddDivisor', cofactor, cofactor' where #tEd(GF(p)) =
     cofactor * oddDivisor, #tEd'(GF(p)) = cofactor' * oddDivisor', cofactor
     and cofactor' are powers of 2 and oddDivisor, oddDivisor' are odd.
   until (cofactor = 8 and cofactor' = 4 and rd is prime and rd' is prime)
3. Output d

                           GenerateCurveTEdwards

6.3.  Generators

   Any point with the correct order will serve as a generator for the
   group.  The following algorithm computes a possible generator by
   taking the smallest positive value x in GF(p) (when represented as an
   integer) such that (x, y) is on the curve and such that (X(P),Y(P)) =
   8 * (x, y) has large prime order oddDivisor.
























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Input: a prime p and curve parameters non-square d and
       a = -1 for twisted Edwards (p = 1 mod 4) or
       a = 1 for Edwards (p = 3 mod 4)
Output: a generator point P = (X(P), Y(P)) of order oddDivisor
1. Set x = 0 and found_gen = false
2. while (not found_gen) do
    x = x + 1
    while ((1 - a * x^2) * (1 - d * x^2) is not a quadratic
           residue mod p) do
      x = x + 1
    end while
    Compute an integer s, 0 < s < p, such that
           s^2 * (1 - d * x^2) = 1 - a * x^2 mod p
    Set y = min(s, p - s)

    (X(P), Y(P)) = 8 * (x, y)

    if ((X(P), Y(P)) has order oddDivisor on Ed or tEd, respectively) then
      found_gen = true
    end if
  end while
3. Output (X(P),Y(P))

                                GenerateGen

7.  Recommended Curves

   For the ~128-bit security level, the prime 2^255-19 is recommended
   for performance over a wide-range of architectures.  This prime is
   congruent to 1 mod 4 and the above procedure results in the following
   twisted Edwards curve, called "intermediate25519":

   p  2^255-19

   d  121665

   order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

   cofactor  8

   In order to be compatible with widespread existing practice, the
   recommended curve is an isogeny of this curve.  An isogeny is a
   "renaming" of the points on the curve and thus cannot affect the
   security of the curve:

   p  2^255-19





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   d  370957059346694393431380835087545651895421138798432190163887855330
      85940283555

   order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

   cofactor  8

   X(P)  151122213495354007725011514095885315114540126930418572060461132
      83949847762202

   Y(P)  463168356949264781694283940034751631413079938662562256157830336
      03165251855960

   The d value in the this curve is much larger than the generated curve
   and this might slow down some implementations.  If this is a problem
   then implementations are free to calculate on the original curve,
   with small d as the isogeny map can be merged into the affine
   transform without any performance impact.

   The latter curve is isomorphic to a Montgomery curve defined by v^2 =
   u^3 + 486662u^2 + u where the maps are:

   (u, v) = ((1+y)/(1-y), sqrt(-1)*sqrt(486664)*u/x)
   (x, y) = (sqrt(-1)*sqrt(486664)*u/v, (u-1)/(u+1)

   The base point maps onto the Montgomery curve such that u = 9, v = 14
   781619447589544791020593568409986887264606134616475288964881837755586
   237401.

   The Montgomery curve defined here is equal to the one defined in
   [curve25519] and the isomorphic twisted Edwards curve is equal to the
   one defined in [ed25519].

8.  Wire-format of field elements

   When transmitting field elements in the Diffie-Hellman protocol
   below, they MUST be encoded as an array of bytes, x, in little-endian
   order such that x[0] + 256 * x[1] + 256^2 * x[2] + ... + 256^n * x[n]
   is congruent to the value modulo p and x[n] is minimal.  On receiving
   such an array, implementations MUST mask the (8-log2(p)%8)%8 most-
   significant bits in the final byte.  This is done to preserve
   compatibility with point formats which reserve the sign bit for use
   in other protocols and to increase resistance to implementation
   fingerprinting.

   (NOTE: draft-turner-thecurve25519function also says "Implementations
   MUST reject numbers in the range [2^255-19, 2^255-1], inclusive." but
   I'm not aware of any implementations that do so.)



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9.  Elliptic Curve Diffie-Hellman

   This section describes how to perform Diffie-Hellman using curves
   generated by the above procedure.  For safety reasons, Diffie-Hellman
   is performed on the Montgomery isomorphism of the curve and the
   public values transmitted are u coordinates.

   Let U denote the projection map from a point (u,v) on E, to u,
   extended so that U of the point at infinity is zero.  U is surjective
   onto GF(p) if the v coordinate takes on values in GF(p) and in a
   quadratic extension of GF(p).

   Then DH(s, U(Q)) = U(sQ) is a function defined for all integers s and
   elements U(Q) of GF(p).  Proper implementations use a restricted set
   of integers for s and only u-coordinates of points Q defined over
   GF(p).  The remainder of this section describes how to compute this
   function quickly and securely, and use it in a Diffie- Hellman
   scheme.

   Let s be a 255 bits long integer, where s = sum s_i * 2^i with s_i in
   {0, 1}.

   Computing DH(s, u) is done by the following procedure, taken from
   [curve25519] based on formulas from [montgomery].  All calculations
   are performed in GF(p), i.e., they are performed modulo p.  The
   parameter a24 is a24 = (486662 - 2) / 4 = 121665.

























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   x_1 = u
   x_2 = 0
   z_2 = 1
   x_3 = u
   z_3 = 1
   For t = 254 down to 0:
       // Conditional swap; see text below.
       (x_2, x_3) = cswap (s_t, x_2, x_3)
       (z_2, z_3) = cswap (s_t, z_2, z_3)
       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)
       // Conditional swap; see text below.
       (x_2, x_3) = cswap (s_t, x_2, x_3)
       (z_2, z_3) = cswap (s_t, z_2, z_3)
   Return x_2 * (z_2^(p - 1))

   In implementing this procedure, due to the existence of side-channels
   in commodity hardware, it is important that the pattern of memory
   accesses and jumps not depend on the values of any of the bits of s.
   It is also important that the arithmetic used not leak information
   about the integers modulo p (such as having b * c distinguishable
   from c * c).

   The cswap instruction SHOULD be implemented in constant time
   (independent of s_t) as follows:

   cswap(s_t, x_2, x_3)
         dummy = s_t * (x_2 - x_3)
         x_2 = x_2 - dummy
         x_3 = x_3 + dummy
   Return (x_2, x_3)

   where s_t is 1 or 0.  Alternatively, an implementation MAY use the
   following:






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   cswap(s_t, x_2, x_3)
         dummy = mask(s_t) AND (x_2 XOR x_3)
         x_2 = x_2 XOR dummy
         x_3 = x_3 XOR dummy
   Return (x_2, x_3)

   where mask(s_t) is the all-1 or all-0 word of the same length as x_2
   and x_3, computed, e.g., as mask(s_t) = 1 - s_t.  The latter version
   is often more efficient.

9.1.  Diffie-Hellman protocol

   The DH function can be used in an ECDH protocol with the recommended
   curve as follows:

   Alice generates 32 random bytes in f[0] to f[31].  She masks the
   three rightmost bits of f[0] and the leftmost bit of f[31] to zero
   and sets the second leftmost bit of f[31] to 1.  This means that f is
   of the form 2^254 + 8 * {0, 1, ..., 2^(251) - 1} as a little-endian
   integer.

   Alice then transmits K_A = DH(f, 9) to Bob, where 9 is the number 9.

   Bob similarly generates 32 random bytes in g[0] to g[31], applies the
   same masks, computes K_B = DH(g, 9) and transmits it to Alice.

   Alice computes DH(f, DH(g, 9)); Bob computes DH(g, DH(f, 9)) using
   their generated values and the received input.

   Both of them now share K = DH(f, DH(g, 9)) = DH(g, DH(f, 9)) as a
   shared secret.  Alice and Bob can then use a key-derivation function,
   such as hashing K, to compute a key.

10.  Test vectors

   The following test vectors are taken from [nacl].  All numbers are
   shown as little-endian hexadecimal byte strings:














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   Alice's private key, f:

        77 07 6d 0a 73 18 a5 7d 3c 16 c1 72 51 b2 66 45
        df 4c 2f 87 eb c0 99 2a b1 77 fb a5 1d b9 2c 2a

   Alice's public key, DH(f, 9):

        85 20 f0 09 89 30 a7 54 74 8b 7d dc b4 3e f7 5a
        0d bf 3a 0d 26 38 1a f4 eb a4 a9 8e aa 9b 4e 6a

   Bob's private key, g:

        5d ab 08 7e 62 4a 8a 4b 79 e1 7f 8b 83 80 0e e6
        6f 3b b1 29 26 18 b6 fd 1c 2f 8b 27 ff 88 e0 eb

   Bob's public key, DH(g, 9):

        de 9e db 7d 7b 7d c1 b4 d3 5b 61 c2 ec e4 35 37
        3f 83 43 c8 5b 78 67 4d ad fc 7e 14 6f 88 2b 4f

   Their shared secret, K:

        4a 5d 9d 5b a4 ce 2d e1 72 8e 3b f4 80 35 0f 25
        e0 7e 21 c9 47 d1 9e 33 76 f0 9b 3c 1e 16 17 42

11.  References

11.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

11.2.  Informative References

   [AS]       Satoh, T. and K. Araki, "Fermat quotients and the
              polynomial time discrete log algorithm for anomalous
              elliptic curves", 1998.

   [EBP]      ECC Brainpool, "ECC Brainpool Standard Curves and Curve
              Generation", October 2005, <http://www.ecc-
              brainpool.org/download/Domain-parameters.pdf>.

   [NIST]     National Institute of Standards, "Recommended Elliptic
              Curves for Federal Government Use", July 1999,
              <http://csrc.nist.gov/groups/ST/toolkit/documents/dss/
              NISTReCur.pdf>.





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   [S]        Semaev, I., "Evaluation of discrete logarithms on some
              elliptic curves", 1998.

   [SC]       Bernstein, D. and T. Lange, "SafeCurves: choosing safe
              curves for elliptic-curve cryptography", June 2014,
              <http://safecurves.cr.yp.to/>.

   [SEC1]     Certicom Research, "SEC 1: Elliptic Curve Cryptography",
              September 2000,
              <http://www.secg.org/collateral/sec1_final.pdf>.

   [Smart]    Smart, N., "The discrete logarithm problem on elliptic
              curves of trace one", 1999.

   [curve25519]
              Bernstein, D., "Curve25519 -- new Diffie-Hellman speed
              records", 2006,
              <http://www.iacr.org/cryptodb/archive/2006/
              PKC/3351/3351.pdf>.

   [ed25519]  Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B.
              Yang, "High-speed high-security signatures", 2011,
              <http://ed25519.cr.yp.to/ed25519-20110926.pdf>.

   [montgomery]
              Montgomery, P., "Speeding the Pollard and elliptic curve
              methods of factorization", 1983,
              <http://www.ams.org/journals/mcom/1987-48-177/
              S0025-5718-1987-0866113-7/S0025-5718-1987-0866113-7.pdf>.

   [nacl]     Bernstein, D., "Cryptography in NaCl", 2009,
              <http://cr.yp.to/highspeed/naclcrypto-20090310.pdf>.

Author's Address

   Adam Langley
   Google
   345 Spear St
   San Francisco, CA  94105
   US

   Email: agl@google.com









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