Elliptic Curves for Security
draft-irtf-cfrg-curves-05

CFRG                                                          A. Langley
Internet-Draft                                                    Google
Intended status: Informational                                M. Hamburg
Expires: February 21, 2016                  Rambus Cryptography Research
                                                               S. Turner
                                                              IECA, Inc.
                                                         August 20, 2015

                      Elliptic Curves for Security
                       draft-irtf-cfrg-curves-05

Abstract

   This memo specifies two elliptic curves over prime fields that offer
   high practical security in cryptographic applications, including
   Transport Layer Security (TLS).  These curves are intended to operate
   at the ~128-bit and ~224-bit security level, respectively, and are
   generated deterministically based on a list of required properties.

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
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   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 February 21, 2016.

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

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   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.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   3
   4.  Recommended Curves  . . . . . . . . . . . . . . . . . . . . .   3
     4.1.  Curve25519  . . . . . . . . . . . . . . . . . . . . . . .   3
     4.2.  Curve448  . . . . . . . . . . . . . . . . . . . . . . . .   4
   5.  The X25519 and X448 functions . . . . . . . . . . . . . . . .   6
     5.1.  Test vectors  . . . . . . . . . . . . . . . . . . . . . .   9
   6.  Diffie-Hellman  . . . . . . . . . . . . . . . . . . . . . . .  11
     6.1.  Curve25519  . . . . . . . . . . . . . . . . . . . . . . .  11
     6.2.  Curve448  . . . . . . . . . . . . . . . . . . . . . . . .  12
   7.  Deterministic Generation  . . . . . . . . . . . . . . . . . .  13
     7.1.  p = 1 mod 4 . . . . . . . . . . . . . . . . . . . . . . .  14
     7.2.  p = 3 mod 4 . . . . . . . . . . . . . . . . . . . . . . .  14
     7.3.  Base points . . . . . . . . . . . . . . . . . . . . . . .  15
   8.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  15
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  16
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  16
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  16
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  17

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 that 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 by NIST [NIST].

   This memo specifies two elliptic curves (curve25519 and curve448)
   that support constant-time, exception-free scalar multiplication that
   is resistant to a wide range of side-channel attacks, including
   timing and cache attacks.  They are Montgomery curves (where y^2 =
   x^3 + Ax^2 + x) and thus have birationally equivalent Edwards
   versions.  Edwards curves support the fastest (currently known)
   complete formulas for the elliptic-curve group operations,
   specifically the Edwards curve x^2 + y^2 = 1 + dx^2y^2 for primes p
   when p = 3 mod 4, and the twisted Edwards curve -x^2 + y^2 = 1 +

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   dx^2y^2 when p = 1 mod 4.  The maps to/from the Montgomery curves to
   their (twisted) Edwards equivalents are also given.

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.  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 -2 or 2.

   d  An element in the finite field GF(p), not equal to 0 or 1.

   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.

   u, v  Coordinates on a Montgomery curve.

   x, y  Coordinates on a (twisted) Edwards curve.

4.  Recommended Curves

4.1.  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 derivation procedure in Section 7
   results in the following Montgomery curve v^2 = u^3 + A*u^2 + u,
   called "curve25519":

   p  2^255-19

   A  486662

   order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

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   cofactor  8

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

   This curve is birationally equivalent to a twisted Edwards curve -x^2
   + y^2 = 1 + d*x^2*y^2, called "edwards25519", where:

   p  2^255-19

   d  370957059346694393431380835087545651895421138798432190163887855330
      85940283555

   order  2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed

   cofactor  8

   X(P)  151122213495354007725011514095885315114540126930418572060461132
      83949847762202

   Y(P)  463168356949264781694283940034751631413079938662562256157830336
      03165251855960

   The birational maps are:

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

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

4.2.  Curve448

   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 derivation procedure in
   Section 7 results in the following Montgomery curve, called
   "curve448":

   p  2^448-2^224-1

   A  156326

   order  2^446 -
      0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

   cofactor  4

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   The base point is u = 5, v = 3552939267855681752641275020637833348089
   763993877142718318808984351690887869674100029326737658645509101427741
   47268105838985595290606362.

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

   p  2^448-2^224-1

   d  611975850744529176160423220965553317543219696871016626328968936415
      087860042636474891785599283666020414768678979989378147065462815545
      017

   order  2^446 -
      0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

   cofactor  4

   X(P)  345397493039729516374008604150537410266655260075183290216406970
      281645695073672344430481787759340633221708391583424041788924124567
      700732

   Y(P)  363419362147803445274661903944002267176820680343659030140745099
      590306164083365386343198191849338272965044442230921818680526749009
      182718

   The birational maps are:

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

   Both of those curves are also 4-isogenous to the following Edwards
   curve x^2 + y^2 = 1 + d*x^2*y^2, called "edwards448", where:

   p  2^448-2^224-1

   d  -39081

   order  2^446 -
      0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d

   cofactor  4

   X(P)  224580040295924300187604334099896036246789641632564134246125461
      686950415467406032909029192869357953282578032075146446173674602635
      247710

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   Y(P)  298819210078481492676017930443930673437544040154080242095928241
      372331506189835876003536878655418784733982303233503462500531545062
      832660

   The 4-isogeny maps between the Montgomery curve this the Edwards
   curve are:

     (u, v) = (y^2/x^2, -(2 - x^2 - y^2)*y/x^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))

   The curve "edwards448" defined here is also called "Goldilocks" and
   is equal to the one defined in [goldilocks].

5.  The X25519 and X448 functions

   The "X25519" and "X448" functions perform 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 or 56-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 X25519 (but not
   X448) 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.

   Implementations MUST accept non-canonical values and process them as
   if they had been reduced modulo the field prime.  The non-canonical
   values are 2^255-19 through 2^255-1 for X25519 and 2^448-2^224-1
   through 2^448-1 for X448.

   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 X25519 and 448 for
   X448:

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 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]
     # Ignore any unused bits.
     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)])

   Scalars are assumed to be randomly generated bytes.  For X25519, in
   order to decode 32 random bytes as 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 X448, 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}.

   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 "X25519(k, u)" and "X448(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, which is 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/
   X25519 and (156326 - 2) / 4 = 39081 for curve448/X448.

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   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))

   (Note that these formulas are slightly different from Montgomery's
   original paper.  Implementations are free to use any correct
   formulas.)

   Finally, encode the resulting value as 32 or 56 bytes in little-
   endian order.  For X25519, the unused, most-significant bit MUST be
   zero.

   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).

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   The cswap function SHOULD be implemented in constant time (i.e.
   independent of the "swap" argument).  For example, this can be done
   as follows:

   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) = 0 - swap.

5.1.  Test vectors

   Two types of tests are provided.  The first is a pair of test vectors
   for each function that consist of expected outputs for the given
   inputs:

 X25519:

 Input scalar:
   a546e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449ac4
 Input scalar as a number (base 10):
   31029842492115040904895560451863089656
   472772604678260265531221036453811406496
 Input U-coordinate:
   e6db6867583030db3594c1a424b15f7c726624ec26b3353b10a903a6d0ab1c4c
 Input U-coordinate as a number:
   34426434033919594451155107781188821651
   316167215306631574996226621102155684838
 Output U-coordinate:
   c3da55379de9c6908e94ea4df28d084f32eccf03491c71f754b4075577a28552

 Input scalar:
   4b66e9d4d1b4673c5ad22691957d6af5c11b6421e0ea01d42ca4169e7918ba0d
 Input scalar as a number (base 10):
   35156891815674817266734212754503633747
   128614016119564763269015315466259359304
 Input U-coordinate:
   e5210f12786811d3f4b7959d0538ae2c31dbe7106fc03c3efc4cd549c715a493
 Input U-coordinate as a number:
   88838573511839298940907593866106493194
   17338800022198945255395922347792736741
 Output U-coordinate:
   95cbde9476e8907d7aade45cb4b873f88b595a68799fa152e6f8f7647aac7957

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 X448:

 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

   The second type of test vector consists of the result of calling the
   function in question a specified number of times.  Initially, set "k"
   and "u" to be the following values:

   For X25519:
     0900000000000000000000000000000000000000000000000000000000000000
   For X448:
     05000000000000000000000000000000000000000000000000000000
     00000000000000000000000000000000000000000000000000000000

   For each iteration, set "k" to be the result of calling the function
   and "u" to be the old value of "k".  The final result is the value
   left in "k".

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   X25519:

   After one iteration:
       422c8e7a6227d7bca1350b3e2bb7279f7897b87bb6854b783c60e80311ae3079
   After 1,000 iterations:
       684cf59ba83309552800ef566f2f4d3c1c3887c49360e3875f2eb94d99532c51
   After 1,000,000 iterations:
       7c3911e0ab2586fd864497297e575e6f3bc601c0883c30df5f4dd2d24f665424

   X448:

   After one iteration:
       3f482c8a9f19b01e6c46ee9711d9dc14fd4bf67af30765c2ae2b846a
       4d23a8cd0db897086239492caf350b51f833868b9bc2b3bca9cf4113
   After 1,000 iterations:
       aa3b4749d55b9daf1e5b00288826c467274ce3ebbdd5c17b975e09d4
       af6c67cf10d087202db88286e2b79fceea3ec353ef54faa26e219f38
   After 1,000,000 iterations:
       077f453681caca3693198420bbe515cae0002472519b3e67661a7e89
       cab94695c8f4bcd66e61b9b9c946da8d524de3d69bd9d9d66b997e37

6.  Diffie-Hellman

6.1.  Curve25519

   The "X25519" function can be used in an elliptic-curve Diffie-Hellman
   (ECDH) protocol as follows:

   Alice generates 32 random bytes in f[0] to f[31] and transmits K_A =
   X25519(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 = X25519(g, 9) and transmits it to Alice.

   Using their generated values and the received input, Alice computes
   X25519(f, K_B) and Bob computes X25519(g, K_A).

   Both now share K = X25519(f, X25519(g, 9)) = X25519(g, X25519(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 that includes K, K_A and K_B to derive a key.

   The check for the all-zero value results from the fact that the
   X25519 function produces that value if it operates on an input
   corresponding to a point with order dividing the co-factor, h, of the

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   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.

   Test vector:

   Alice's private key, f:
     77076d0a7318a57d3c16c17251b26645df4c2f87ebc0992ab177fba51db92c2a
   Alice's public key, X25519(f, 9):
     8520f0098930a754748b7ddcb43ef75a0dbf3a0d26381af4eba4a98eaa9b4e6a
   Bob's private key, g:
     5dab087e624a8a4b79e17f8b83800ee66f3bb1292618b6fd1c2f8b27ff88e0eb
   Bob's public key, X25519(g, 9):
     de9edb7d7b7dc1b4d35b61c2ece435373f8343c85b78674dadfc7e146f882b4f
   Their shared secret, K:
     4a5d9d5ba4ce2de1728e3bf480350f25e07e21c947d19e3376f09b3c1e161742

6.2.  Curve448

   The "X448" function can be used in an ECDH protocol very much like
   the "X22519" function.

   If "X448" is to be used, the only differences are that Alice and Bob
   generate 56 random bytes (not 32) and calculate K_A = X448(f, 5) or
   K_B = X448(g, 5) where 5 is the u-coordinate of the base point and is
   encoded as a byte with value 5, followed by 55 zero bytes.

   As with "X25519", both sides MUST check, without leaking extra
   information about the value of K, whether the resulting shared K is
   the all-zero value and abort if so.

   Test vector:

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   Alice's private key, f:
     9a8f4925d1519f5775cf46b04b5800d4ee9ee8bae8bc5565d498c28d
     d9c9baf574a9419744897391006382a6f127ab1d9ac2d8c0a598726b
   Alice's public key, X448(f, 5):
     9b08f7cc31b7e3e67d22d5aea121074a273bd2b83de09c63faa73d2c
     22c5d9bbc836647241d953d40c5b12da88120d53177f80e532c41fa0
   Bob's private key, g:
     1c306a7ac2a0e2e0990b294470cba339e6453772b075811d8fad0d1d
     6927c120bb5ee8972b0d3e21374c9c921b09d1b0366f10b65173992d
   Bob's public key, X448(g, 5):
     3eb7a829b0cd20f5bcfc0b599b6feccf6da4627107bdb0d4f345b430
     27d8b972fc3e34fb4232a13ca706dcb57aec3dae07bdc1c67bf33609
   Their shared secret, K:
     fe2d52f1ca113e5441538037dc4a9d4cb381035fb4a990ac50ac4333
     63dc072301d1d4f2e82883b35103be96068c11e7c84b8fff74bb6ab0

7.  Deterministic Generation

   This section specifies the procedure that was used to generate the
   above curves; specifically it defines how to generate the parameter A
   of the Montgomery curve y^2 = x^3 + Ax^2 + x.  This procedure is
   intended to be as objective as can reasonably be achieved so that
   it's clear that no untoward considerations influenced the choice of
   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) so
   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.

   For each curve at a specific security level:

   1.  The trace of Frobenius MUST NOT be in {0, 1} in order to rule out
       the attacks described in [smart], [satoh], and [semaev], as in
       [brainpool] and [safecurves].

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

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

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7.1.  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 will be the smaller of the two.

   To generate the Montgomery curve we find the minimal, positive A
   value, such that A > 2 and (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:

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

       for A in xrange(3, 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

7.2.  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 and (A-2) is
   divisible by four.  The "find3Mod4" function in the following Sage
   script returns this value given p:

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

                   Generating a curve where p = 3 mod 4

7.3.  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:

   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

8.  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, Patrick
   Longa, Michael Naehrig and Watson Ladd.

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

   The curve25519 function was developed by Daniel J.  Bernstein in
   [curve25519].

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9.  References

9.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

9.2.  Informative References

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

   [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>.

   [goldilocks]
              Hamburg, M., "Ed448-Goldilocks, a new elliptic curve",
              2015, <http://eprint.iacr.org/2015/625.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>.

   [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>.

   [safecurves]
              Bernstein, D. and T. Lange, "SafeCurves: choosing safe
              curves for elliptic-curve cryptography", Oct 2013,
              <http://safecurves.cr.yp.to/>.

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

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

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

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

Authors' Addresses

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

   Email: agl@google.com

   Mike Hamburg
   Rambus Cryptography Research
   425 Market Street, 11th Floor
   San Francisco, CA  94105
   US

   Email: mike@shiftleft.org

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

   Email: turners@ieca.com

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