mdw@git.distorted.org.uk Git - secnet/blob - montladder.h

   1 /* -*-c-*-
   2  *
   3  * Definitions for Montgomery's ladder
   4  *
   5  * (c) 2017 Straylight/Edgeware
   6  */
   7
   8 /*----- Licensing notice --------------------------------------------------*
   9  *
  10  * This file is part of secnet.
  11  * See README for full list of copyright holders.
  12  *
  13  * secnet is free software; you can redistribute it and/or modify it
  14  * under the terms of the GNU General Public License as published by
  15  * the Free Software Foundation; either version d of the License, or
  16  * (at your option) any later version.
  17  *
  18  * secnet is distributed in the hope that it will be useful, but
  19  * WITHOUT ANY WARRANTY; without even the implied warranty of
  20  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  21  * General Public License for more details.
  22  *
  23  * You should have received a copy of the GNU General Public License
  24  * version 3 along with secnet; if not, see
  25  * https://www.gnu.org/licenses/gpl.html.
  26  *
  27  * This file was originally part of Catacomb, but has been automatically
  28  * modified for incorporation into secnet: see `import-catacomb-crypto'
  29  * for details.
  30  *
  31  * Catacomb is free software; you can redistribute it and/or modify
  32  * it under the terms of the GNU Library General Public License as
  33  * published by the Free Software Foundation; either version 2 of the
  34  * License, or (at your option) any later version.
  35  *
  36  * Catacomb is distributed in the hope that it will be useful,
  37  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  38  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  39  * GNU Library General Public License for more details.
  40  *
  41  * You should have received a copy of the GNU Library General Public
  42  * License along with Catacomb; if not, write to the Free
  43  * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
  44  * MA 02111-1307, USA.
  45  */
  46
  47 #ifndef CATACOMB_MONTLADDER_H
  48 #define CATACOMB_MONTLADDER_H
  49
  50 #ifdef __cplusplus
  51   extern "C" {
  52 #endif
  53
  54 /*----- Notes on the Montgomery ladder ------------------------------------*
  55  *
  56  * The algorithm here is Montgomery's famous binary ladder for calculating
  57  * x-coordinates of scalar products on a particular shape of elliptic curve,
  58  * as elucidated by Daniel Bernstein.
  59  *
  60  * Let Q = (x_1, y_1) be the base point, for some unknown y_1 (which will
  61  * turn out to be unimportant).  Define x_n, z_n by x(n Q) = (x_n : z_n).
  62  * Given x_n, z_n, x_{n+1}, z_{n+1}, Montgomery's differential addition
  63  * formulae calculate x_{2i}, z_{2i}, x_{2i+1}, z_{2i+1}.  Furthermore,
  64  * calculating x_{2i}, z_{2i} requires only x_n, z_n, and the calculation of
  65  * x_{2i+1}, z_{2i+1} is symmetrical.
  66  */
  67
  68 /*----- Functions provided ------------------------------------------------*/
  69
  70 /* F designates a field, both naming the type of its elements and acting as a
  71  * prefix for the standard field operations `F_add', `F_sub', `F_mul',
  72  * `F_sqr', and `F_inv' (the last of which should return zero its own
  73  * inverse); and the constant-time utility `F_condswap'.
  74  *
  75  * The macro calculates the x-coordinate of the product k Q, where Q is a
  76  * point on the elliptic curve B y^2 = x^3 + A x^2 + x or its quadratic
  77  * twist, for some irrelevant B.  The x-coordinate of Q is given as X1 (a
  78  * pointer to a field element).  The scalar k is given as a vector of NK
  79  * unsigned integers KW, each containing NBITS significant bits, with the
  80  * least-significant element first.  The result is written to the field
  81  * element pointed to by Z.
  82  *
  83  * The curve coefficient A is given indirectly, as the name of a macro MULA0
  84  * such that
  85  *
  86  *      MULA0(z, x)
  87  *
  88  * will store in z the value (A - 2)/4 x.
  89  */
  90 #define MONT_LADDER(f, mula0, kw, nk, nbits, z, x1) do {                \
  91   f _x, _z, _u, _w;                                                     \
  92   f _t0, _t1, _t2, _t3, _t4;                                            \
  93   uint32 _m = 0, _mm = 0, _k;                                           \
  94   unsigned _i, _j;                                                      \
  95                                                                         \
  96   /* Initialize the main variables.  We'll have, (x, z) and (u, w)      \
  97    * holding (x_n, z_n) and (x_{n+1}, z_{n+1}) in some order, but       \
  98    * there's some weirdness: if m = 0 then (x, z) = (x_n, z_n) and      \
  99    * (u, v) = (x_{n+1}, z_{n+1}); if m /= 0, then the pairs are         \
 100    * swapped over.                                                      \
 101    *                                                                    \
 102    * Initially, we have (x_0, z_0) = (1, 0), representing the identity  \
 103    * at projective-infinity, which works fine; and we have z_1 = 1.     \
 104    */                                                                   \
 105   _u = *(x1); f##_set(&_w, 1); f##_set(&_x, 1); f##_set(&_z, 0);        \
 106                                                                         \
 107   /* The main ladder loop.  Work through each bit of the clamped key. */ \
 108   for (_i = (nk); _i--; ) {                                             \
 109     _k = (kw)[_i];                                                      \
 110     for (_j = 0; _j < (nbits); _j++) {                                  \
 111       /* We're at bit i of the scalar key (represented by 32 (7 - i) +  \
 112        * (31 - j) in our loop variables -- don't worry about that).     \
 113        * Let k = 2^i k_i + k'_i, with 0 <= k'_i < 2^i.  In particular,  \
 114        * then, k_0 = k.  Write Q(i) = (x_i, z_i).                       \
 115        *                                                                \
 116        * We currently have, in (x, z) and (u, w), Q(k_i) and Q(k_i +    \
 117        * 1), in some order.  The ladder step will double the point in   \
 118        * (x, z), and leave the sum of (x : z) and (u : w) in (u, w).    \
 119        */                                                               \
 120                                                                         \
 121       _mm = -((_k >> ((nbits) - 1))&1u); _k <<= 1;                      \
 122       f##_condswap(&_x, &_u, _m ^ _mm);                                 \
 123       f##_condswap(&_z, &_w, _m ^ _mm);                                 \
 124       _m = _mm;                                                         \
 125                                                                         \
 126       f##_add(&_t0, &_x, &_z);          /* x + z */                     \
 127       f##_sub(&_t1, &_x, &_z);          /* x - z */                     \
 128       f##_add(&_t2, &_u, &_w);          /* u + w */                     \
 129       f##_sub(&_t3, &_u, &_w);          /* u - w */                     \
 130       f##_mul(&_t2, &_t2, &_t1);        /* (x - z) (u + w) */           \
 131       f##_mul(&_t3, &_t3, &_t0);        /* (x + z) (u - w) */           \
 132       f##_sqr(&_t0, &_t0);              /* (x + z)^2 */                 \
 133       f##_sqr(&_t1, &_t1);              /* (x - z)^2 */                 \
 134       f##_mul(&_x, &_t0, &_t1);         /* (x + z)^2 (x - z)^2 */       \
 135       f##_sub(&_t1, &_t0, &_t1);        /* (x + z)^2 - (x - z)^2 */     \
 136       mula0(&_t4, &_t1);             /* A_0 ((x + z)^2 - (x - z)^2) */  \
 137       f##_add(&_t0, &_t0, &_t4);        /* A_0 ... + (x + z)^2 */       \
 138       f##_mul(&_z, &_t0, &_t1);  /* (...^2 - ...^2) (A_0 ... + ...) */  \
 139       f##_add(&_t0, &_t2, &_t3); /* (x - z) (u + w) + (x + z) (u - w) */ \
 140       f##_sub(&_t1, &_t2, &_t3); /* (x - z) (u + w) - (x + z) (u - w) */ \
 141       f##_sqr(&_u, &_t0);               /* (... + ...)^2 */             \
 142       f##_sqr(&_t1, &_t1);              /* (... - ...)^2 */             \
 143       f##_mul(&_w, &_t1, (x1));         /* x_1 (... - ...)^2 */         \
 144     }                                                                   \
 145   }                                                                     \
 146                                                                         \
 147   /* Almost done.  Undo the swap, if any. */                            \
 148   f##_condswap(&_x, &_u, _m);                                           \
 149   f##_condswap(&_z, &_w, _m);                                           \
 150                                                                         \
 151   /* And convert to affine. */                                          \
 152   f##_inv(&_t0, &_z);                                                   \
 153   f##_mul((z), &_x, &_t0);                                              \
 154 } while (0)
 155
 156 /*----- That's all, folks -------------------------------------------------*/
 157
 158 #ifdef __cplusplus
 159   }
 160 #endif
 161
 162 #endif