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