Elliptic Curves for Security
draft-irtf-cfrg-curves-02

CFRG                                                          A. Langley
Internet-Draft                                                    Google
Intended status: Informational                                   R. Salz
Expires: September 25, 2015                          Akamai Technologies
                                                               S. Turner
                                                              IECA, Inc.
                                                          March 24, 2015

                      Elliptic Curves for Security
                       draft-irtf-cfrg-curves-02

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 and a specific curve at
   the ~224-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 September 25, 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

Langley, et al.        Expires September 25, 2015               [Page 1]
Internet-Draft                  cfrgcurve                     March 2015

   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.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Requirements Language . . . . . . . . . . . . . . . . . . . .   3
   3.  Security Requirements . . . . . . . . . . . . . . . . . . . .   3
   4.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   3
   5.  Parameter Generation  . . . . . . . . . . . . . . . . . . . .   4
     5.1.  p = 1 mod 4 . . . . . . . . . . . . . . . . . . . . . . .   4
     5.2.  p = 3 mod 4 . . . . . . . . . . . . . . . . . . . . . . .   5
     5.3.  Base points . . . . . . . . . . . . . . . . . . . . . . .   5
   6.  Recommended Curves  . . . . . . . . . . . . . . . . . . . . .   6
     6.1.  Curve25519  . . . . . . . . . . . . . . . . . . . . . . .   6
     6.2.  Goldilocks  . . . . . . . . . . . . . . . . . . . . . . .   7
   7.  The curve25519 and curve448 functions . . . . . . . . . . . .   8
     7.1.  Test vectors  . . . . . . . . . . . . . . . . . . . . . .  12
   8.  Diffie-Hellman  . . . . . . . . . . . . . . . . . . . . . . .  13
     8.1.  Test vectors  . . . . . . . . . . . . . . . . . . . . . .  14
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  15
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  15
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  15
     10.2.  Informative References . . . . . . . . . . . . . . . . .  15
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  16

1.  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, various '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 generating
   cryptographic elliptic curves over a given prime field.  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 deterministic algorithm operates
   without any input parameters that would permit manipulation of the
   resulting curves.  The selection between curve models is determined

Langley, et al.        Expires September 25, 2015               [Page 2]
Internet-Draft                  cfrgcurve                     March 2015

   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 when p = 1 mod 4.

2.  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].

3.  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 5.

   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].

4.  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.

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

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

   P  A generator point defined over GF(p) of prime order.

   X(P)  The x-coordinate of the elliptic curve point P on a (twisted)
      Edwards curve.

   Y(P)  The y-coordinate of the elliptic curve point P on a (twisted)
      Edwards curve.

Langley, et al.        Expires September 25, 2015               [Page 3]
Internet-Draft                  cfrgcurve                     March 2015

5.  Parameter Generation

   This section describes the generation of the curve parameter, namely
   A, of the Montgomery curve y^2 = x^3 + Ax^2 + x.  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.  This document does not attempt to
   articulate all these considerations.

   The value (A-2)/4 is used in several of the elliptic curve point
   arithmetic formulas.  For simplicity and performance reasons, it is
   beneficial to make this constant small, i.e. to choose A so that
   (A-2) is a small integer which is divisible by four.

5.1.  p = 1 mod 4

   For primes congruent to 1 mod 4, the minimal cofactors of the curve
   and its twist are either {4, 8} or {8, 4}. We choose a curve with the
   latter cofactors so that any algorithms that take the cofactor into
   account don't have to worry about checking for points on the twist,
   because the twist cofactor is larger.

   To generate the Montgomery curve we find the minimal, positive A
   value such (A-2) is divisible by four and where the cofactors are as
   desired.  The "find1Mod4" function in the following Sage script
   returns this value given p:

Langley, et al.        Expires September 25, 2015               [Page 4]
Internet-Draft                  cfrgcurve                     March 2015

def findCurve(prime, curveCofactor, twistCofactor):
    F = GF(prime)

    for A in xrange(1, 1e9):
        if (A-2) % 4 != 0:
          continue

        try:
          E = EllipticCurve(F, [0, A, 0, 1, 0])
        except:
          continue

        order = E.order()
        twistOrder = 2*(prime+1)-order

        if (order % curveCofactor == 0 and is_prime(order // curveCofactor) and
            twistOrder % twistCofactor == 0 and is_prime(twistOrder // twistCofactor)):
            return A

def find1Mod4(prime):
    assert((prime % 4) == 1)
    return findCurve(prime, 8, 4)

                   Generating a curve where p = 1 mod 4

5.2.  p = 3 mod 4

   For a prime congruent to 3 mod 4, both the curve and twist cofactors
   can be 4 and this is minimal.  Thus we choose the curve with these
   cofactors and minimal, positive A such that (A-2) is divisible by
   four.  The "find3Mod4" function in the following Sage script returns
   this value given p:

   def find3Mod4(prime):
       assert((prime % 4) == 3)
       return findCurve(prime, 4, 4)

                   Generating a curve where p = 3 mod 4

5.3.  Base points

   The base point for a curve is the point with minimal, positive u
   value that is in the correct subgroup.  The "findBasepoint" function
   in the following Sage script returns this value given p and A:

Langley, et al.        Expires September 25, 2015               [Page 5]
Internet-Draft                  cfrgcurve                     March 2015

   def findBasepoint(prime, A):
       F = GF(prime)
       E = EllipticCurve(F, [0, A, 0, 1, 0])

       for uInt in range(1, 1e3):
         u = F(uInt)
         v2 = u^3 + A*u^2 + u
         if not v2.is_square():
           continue
         v = v2.sqrt()

         point = E(u, v)
         order = point.order()
         if order > 8 and order.is_prime():
           return point

                         Generating the base point

6.  Recommended Curves

6.1.  Curve25519

   For the ~128-bit security level, the prime 2^255-19 is recommended
   for performance on a wide-range of architectures.  This prime is
   congruent to 1 mod 4 and the above procedure results in the following
   Montgomery curve, called "curve25519":

   p  2^255-19

   A  486662

   order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

   cofactor  8

   The base point is u = 9, v = 1478161944758954479102059356840998688726
   4606134616475288964881837755586237401.

   This curve is isomorphic to the twisted Edwards curve -x^2 + y^2 = 1
   - d*x^2*y^2 where:

   p  2^255-19

   d  370957059346694393431380835087545651895421138798432190163887855330
      85940283555

   order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

Langley, et al.        Expires September 25, 2015               [Page 6]
Internet-Draft                  cfrgcurve                     March 2015

   cofactor  8

   X(P)  151122213495354007725011514095885315114540126930418572060461132
      83949847762202

   Y(P)  463168356949264781694283940034751631413079938662562256157830336
      03165251855960

   The isomorphism 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 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].

6.2.  Goldilocks

   For the ~224-bit security level, the prime 2^448-2^224-1 is
   recommended for performance on a wide-range of architectures.  This
   prime is congruent to 3 mod 4 and the above procedure results in the
   following Montgomery curve, called "curve448":

   p  2^448-2^224-1

   A  156326

   order  2^446 -
      0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

   cofactor  4

   The base point is u = 5, v =
   3552939267855681752641275020637833348089763993877142718318808984351 \
   69088786967410002932673765864550910142774147268105838985595290606362.

   This curve is isomorphic to the Edwards curve x^2 + y^2 = 1 +
   d*x^2*y^2 where:

   p  2^448-2^224-1

   d  611975850744529176160423220965553317543219696871016626328968936415
      0 \ 87860042636474891785599283666020414768678979989378147065462815
      545017

   order  2^446 -
      0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

Langley, et al.        Expires September 25, 2015               [Page 7]
Internet-Draft                  cfrgcurve                     March 2015

   cofactor  4

   X(P)  345397493039729516374008604150537410266655260075183290216406970
      2816 \ 45695073672344430481787759340633221708391583424041788924124
      567700732

   Y(P)  363419362147803445274661903944002267176820680343659030140745099
      5903 \ 06164083365386343198191849338272965044442230921818680526749
      009182718

   The isomorphism maps are:

     (u, v) = ((y-1)/(y+1), sqrt(156324)*u/x)
     (x, y) = (sqrt(156324)*u/v, (1+u)/(1-u)

   That curve is also 4-isogenous to the following Edward's curve x^2 +
   y^2 = 1 + d*x^2*y^2, called "Ed448-Goldilocks":

   p  2^448-2^224-1

   d  -39081

   order  2^446 -
      0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

   cofactor  4

   The 4-isogeny maps between the Montgomery curve and the Edward's
   curve are:

  (u, v) = (x^2/y^2, (2 - x^2 - y^2)*x/y^3)
  (x, y) = (4*v*(u^2 - 1)/(u^4 - 2*u^2 + 4*v^2 + 1), (u^5 - 2*u^3 - 4*u*v^2 + u)/(u^5 - 2*u^2*v^2 - 2*u^3 - 2*v^2 + u))

7.  The curve25519 and curve448 functions

   The "curve25519" and "curve448" functions performs scalar
   multiplication on the Montgomery form of the above curves.  (This is
   used when implementing Diffie-Hellman.)  The functions take a scalar
   and a u-coordinate as inputs and produce a u-coordinate as output.
   Although the functions work internally with integers, the inputs and
   outputs are 32-byte strings and this specification defines their
   encoding.

   U-coordinates are elements of the underlying field GF(2^255-19) or
   GF(2^448-2^224-1) and are encoded as an array of bytes, u, in little-
   endian order such that u[0] + 256 * u[1] + 256^2 * u[2] + ... + 256^n
   * u[n] is congruent to the value modulo p and u[n] is minimal.  When
   receiving such an array, implementations of curve25519 (but not

Langley, et al.        Expires September 25, 2015               [Page 8]
Internet-Draft                  cfrgcurve                     March 2015

   curve448) MUST mask the most-significant bit 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.

   For example, the following functions implement this in Python,
   although the Python code is not intended to be performant nor side-
   channel free.  Here the "bits" parameter should be set to 255 for
   curve25519 and 448 for curve448:

 def decodeLittleEndian(b,bits):
     return sum([b[i] << 8*i for i in range((bits+7)/8)])

 def decodeUCoordinate(u,bits):
     u_list = [ord(b) for b in u]
     if bits % 8:
         u_list[-1] &= (1<<(bits%8))-1
     return decodeLittleEndian(u_list,bits)

 def encodeUCoordinate(u,bits):
     u = u % p
     return ''.join([chr((u >> 8*i) & 0xff) for i in range((bits+7)/8)])

   (EDITORS 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.)

   Scalars are assumed to be randomly generated bytes.  For curve25519,
   in order to decode 32 bytes into an integer scalar, set the three
   least significant bits of the first byte and the most significant bit
   of the last to zero, set the second most significant bit of the last
   byte to 1 and, finally, decode as little-endian.  This means that
   resulting integer is of the form 2^254 + 8 * {0, 1, ..., 2^(251) -
   1}.  Likewise, for curve448, set the two least significant bits of
   the first byte to 0, and the most significant bit of the last byte to
   1.  This means that the resulting integer is of the form 2^447 + 4 *
   {0, 1, ..., 2^(445) - 1}.

Langley, et al.        Expires September 25, 2015               [Page 9]
Internet-Draft                  cfrgcurve                     March 2015

   def decodeScalar25519(k):
       k_list = [ord(b) for b in k]
       k_list[0] &= 248
       k_list[31] &= 127
       k_list[31] |= 64
       return decodeLittleEndian(k_list,255)

   def decodeScalar448(k):
       k_list = [ord(b) for b in k]
       k_list[0] &= 252
       k_list[55] |= 128
       return decodeLittleEndian(k_list,448)

   To implement the "curve25519(k, u)" and "curve448(k, u)" functions
   (where "k" is the scalar and "u" is the u-coordinate) first decode
   "k" and "u" and then perform the following procedure, taken from
   [curve25519] and based on formulas from [montgomery].  All
   calculations are performed in GF(p), i.e., they are performed modulo
   p.  The constant a24 is (486662 - 2) / 4 = 121665 for curve25519, and
   (156326 - 2) / 4 = 39081 for curve448.

Langley, et al.        Expires September 25, 2015              [Page 10]
Internet-Draft                  cfrgcurve                     March 2015

   x_1 = u
   x_2 = 1
   z_2 = 0
   x_3 = u
   z_3 = 1
   swap = 0

   For t = bits-1 down to 0:
       k_t = (k >> t) & 1
       swap ^= k_t
       // Conditional swap; see text below.
       (x_2, x_3) = cswap(swap, x_2, x_3)
       (z_2, z_3) = cswap(swap, z_2, z_3)
       swap = k_t

       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(swap, x_2, x_3)
   (z_2, z_3) = cswap(swap, z_2, z_3)
   Return x_2 * (z_2^(p - 2))

   (TODO: Note the difference in the formula from Montgomery's original
   paper.  See https://www.ietf.org/mail-archive/web/cfrg/current/
   msg05872.html.)

   Finally, encode the resulting value as 32 or 56 bytes in little-
   endian order.

   When 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 "k".  It is also important that the arithmetic used not leak
   information about the integers modulo p (such as having b*c be
   distinguishable from c*c).

Langley, et al.        Expires September 25, 2015              [Page 11]
Internet-Draft                  cfrgcurve                     March 2015

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

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

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

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

7.1.  Test vectors

curve25519:

Input scalar:
  a546e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449ac4
Input scalar as a number (base 10):
  31029842492115040904895560451863089656472772604678260265531221036453811406496
Input U-coordinate:
  e6db6867583030db3594c1a424b15f7c726624ec26b3353b10a903a6d0ab1c4c
Input U-coordinate as a number:
  34426434033919594451155107781188821651316167215306631574996226621102155684838
Output U-coordinate:
  c3da55379de9c6908e94ea4df28d084f32eccf03491c71f754b4075577a28552

Input scalar:
  4b66e9d4d1b4673c5ad22691957d6af5c11b6421e0ea01d42ca4169e7918ba0d
Input scalar as a number (base 10):
  35156891815674817266734212754503633747128614016119564763269015315466259359304
Input U-coordinate:
  e5210f12786811d3f4b7959d0538ae2c31dbe7106fc03c3efc4cd549c715a493
Input U-coordinate as a number:
  8883857351183929894090759386610649319417338800022198945255395922347792736741
Output U-coordinate:
  95cbde9476e8907d7aade45cb4b873f88b595a68799fa152e6f8f7647aac7957

Langley, et al.        Expires September 25, 2015              [Page 12]
Internet-Draft                  cfrgcurve                     March 2015

curve448:

Input scalar:
  3d262fddf9ec8e88495266fea19a34d28882acef045104d0d1aae121
  700a779c984c24f8cdd78fbff44943eba368f54b29259a4f1c600ad3
Input scalar as a number (base 10):
  5991891753738964027837560161452132561572308560850261299268914594686 \
  22403380588640249457727683869421921443004045221642549886377526240828
Input U-coordinate:
  06fce640fa3487bfda5f6cf2d5263f8aad88334cbd07437f020f08f9
  814dc031ddbdc38c19c6da2583fa5429db94ada18aa7a7fb4ef8a086
Input U-coordinate as a number:
  3822399108141073301162299612348993770314163652405713251483465559224 \
  38025162094455820962429142971339584360034337310079791515452463053830
Output U-coordinate:
  ce3e4ff95a60dc6697da1db1d85e6afbdf79b50a2412d7546d5f239f
  e14fbaadeb445fc66a01b0779d98223961111e21766282f73dd96b6f

Input scalar:
  203d494428b8399352665ddca42f9de8fef600908e0d461cb021f8c5
  38345dd77c3e4806e25f46d3315c44e0a5b4371282dd2c8d5be3095f
Input scalar as a number (base 10):
  6332543359069705927792594815348623723825251552520289610564040013321 \
  22152890562527156973881968934311400345568203929409663925541994577184
Input U-coordinate:
  0fbcc2f993cd56d3305b0b7d9e55d4c1a8fb5dbb52f8e9a1e9b6201b
  165d015894e56c4d3570bee52fe205e28a78b91cdfbde71ce8d157db
Input U-coordinate as a number:
  6227617977583254444629220684312341806495903900248112997616251537672 \
  28042600197997696167956134770744996690267634159427999832340166786063
Output U-coordinate:
  884a02576239ff7a2f2f63b2db6a9ff37047ac13568e1e30fe63c4a7
  ad1b3ee3a5700df34321d62077e63633c575c1c954514e99da7c179d

8.  Diffie-Hellman

   The "curve25519" function can be used in an ECDH protocol as follows:

   Alice generates 32 random bytes in f[0] to f[31] and transmits K_A =
   curve25519(f, 9) to Bob, where 9 is the u-coordinate of the base
   point and is encoded as a byte with value 9, followed by 31 zero
   bytes.

   Bob similarly generates 32 random bytes in g[0] to g[31] and computes
   K_B = curve25519(g, 9) and transmits it to Alice.

   Alice computes curve25519(f, K_B); Bob computes curve25519(g, K_A)
   using their generated values and the received input.

Langley, et al.        Expires September 25, 2015              [Page 13]
Internet-Draft                  cfrgcurve                     March 2015

   Both now share K = curve25519(f, curve25519(g, 9)) = curve25519(g,
   curve25519(f, 9)) as a shared secret.  Both MUST check, without
   leaking extra information about the value of K, whether K is the all-
   zero value and abort if so (see below).  Alice and Bob can then use a
   key-derivation function, such as hashing K, to compute a key.

   The check for the all-zero value results from the fact that the
   curve25519 function produces that value if it operates on an input
   corresponding to a point with order dividing the co-factor, h, of the
   curve.  This check is cheap and so MUST always be carried out.  The
   check may be performed by ORing all the bytes together and checking
   whether the result is zero as this eliminates standard side-channels
   in software implementations.

8.1.  Test vectors

   curve25519:

   Alice's private key, f:
     77076d0a7318a57d3c16c17251b26645df4c2f87ebc0992ab177fba51db92c2a
   Alice's public key, curve25519(f, 9):
     8520f0098930a754748b7ddcb43ef75a0dbf3a0d26381af4eba4a98eaa9b4e6a
   Bob's private key, g:
     5dab087e624a8a4b79e17f8b83800ee66f3bb1292618b6fd1c2f8b27ff88e0eb
   Bob's public key, curve25519(g, 9):
     de9edb7d7b7dc1b4d35b61c2ece435373f8343c85b78674dadfc7e146f882b4f
   Their shared secret, K:
     4a5d9d5ba4ce2de1728e3bf480350f25e07e21c947d19e3376f09b3c1e161742

   curve448:

   Alice's private key, f:
     9a8f4925d1519f5775cf46b04b5800d4ee9ee8bae8bc5565d498c28d
     d9c9baf574a9419744897391006382a6f127ab1d9ac2d8c0a598726b
   Alice's public key, curve448(f, 5):
     9b08f7cc31b7e3e67d22d5aea121074a273bd2b83de09c63faa73d2c
     22c5d9bbc836647241d953d40c5b12da88120d53177f80e532c41fa0
   Bob's private key, g:
     1c306a7ac2a0e2e0990b294470cba339e6453772b075811d8fad0d1d
     6927c120bb5ee8972b0d3e21374c9c921b09d1b0366f10b65173992d
   Bob's public key, curve448(g, 5):
     3eb7a829b0cd20f5bcfc0b599b6feccf6da4627107bdb0d4f345b430
     27d8b972fc3e34fb4232a13ca706dcb57aec3dae07bdc1c67bf33609
   Their shared secret, K:
     fe2d52f1ca113e5441538037dc4a9d4cb381035fb4a990ac50ac4333
     63dc072301d1d4f2e82883b35103be96068c11e7c84b8fff74bb6ab0

Langley, et al.        Expires September 25, 2015              [Page 14]
Internet-Draft                  cfrgcurve                     March 2015

9.  Acknowledgements

   This document merges "draft-black-rpgecc-01" and "draft-turner-
   thecurve25519function-01".  The following authors of those documents
   wrote much of the text and figures but are not listed as authors on
   this document: Benjamin Black, Joppe W.  Bos, Craig Costello, Mike
   Hamburg, Patrick Longa, Michael Naehrig and Watson Ladd.

   The authors would also like to thank Tanja Lange and Rene Struik for
   their reviews.

10.  References

10.1.  Normative References

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

10.2.  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>.

   [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.

Langley, et al.        Expires September 25, 2015              [Page 15]
Internet-Draft                  cfrgcurve                     March 2015

   [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>.

Authors' Addresses

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

   Email: agl@google.com

   Rich Salz
   Akamai Technologies
   8 Cambridge Center
   Cambridge, MA  02142
   US

   Email: rsalz@akamai.com

   Sean Turner
   IECA, Inc.
   3057 Nutley Street
   Suite 106
   Fairfax, VA  22031
   US

   Email: turners@ieca.com

Langley, et al.        Expires September 25, 2015              [Page 16]