| 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 |