| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * Barrett modular reduction |
| 4 | * |
| 5 | * (c) 1999 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 | /*----- Notes on Barrett reduction ----------------------------------------* |
| 29 | * |
| 30 | * Barrett reduction is a technique for computing modular residues. Unlike |
| 31 | * Montgomery reduction, it doesn't have restrictions on the modulus (except |
| 32 | * that it be positive) and doesn't confuse matters by putting an extra |
| 33 | * factor all the way through your computation. |
| 34 | * |
| 35 | * It's useful for slightly less heavy-duty work than Montgomery reduction |
| 36 | * because the precomputation phase is rather simpler, involving a single |
| 37 | * division operation. |
| 38 | * |
| 39 | * Sometimes it's useful to exponentiate modulo an even number, so there's a |
| 40 | * modexp routine provided which uses Barrett reduction rather than |
| 41 | * Montgomery reduction. This is handy when you're working on indices in an |
| 42 | * even-order cyclic group or something. |
| 43 | * |
| 44 | * In more detail: suppose that %$b^{k-1} \le m < b^k$%. Let %$\mu = {}$% |
| 45 | * %$\lfloor b^{2k}/m \rfloor$%; %$\mu$% is a scaled approximation to the |
| 46 | * reciprocal %$1/m$%. Now, suppose we're given some %$a$% with |
| 47 | * %$0 \le a < b^{2k}$%. The first step is to calculate an approximation |
| 48 | * %$q = \lfloor \mu \lfloor a/b^{k-1} \rfloor/b^{k+1} \rfloor$% to the |
| 49 | * quotient %$a/m$%. Then we have: |
| 50 | * |
| 51 | * %$\lfloor a/m - a/b^{2k} - b^{k-1}/m + 1/b^{k+1} \rfloor \le {}$% |
| 52 | * %$q \le \lfloor a/m \rfloor |
| 53 | * |
| 54 | * But by assumption %$a < b^{2k}$% and %$2^{k-1} \le m$% so |
| 55 | * |
| 56 | * %$\lfloor a/m \rfloor - 2 \le q \le \lfloor a/m \rfloor$% |
| 57 | * |
| 58 | * Now we approximate the remainder by calculating %$r = a - q m$%. |
| 59 | * Certainly %$r \equiv a \pmod{m}$%; and |
| 60 | * |
| 61 | * %$0 \le r \le (a - m \lfloor a/m \rfloor) + 2 m < 3 m$%. |
| 62 | * |
| 63 | * So the remainder can be fixed up with at most two conditional |
| 64 | * subtractions. |
| 65 | */ |
| 66 | |
| 67 | #ifndef CATACOMB_MPBARRETT_H |
| 68 | #define CATACOMB_MPBARRETT_H |
| 69 | |
| 70 | #ifdef __cplusplus |
| 71 | extern "C" { |
| 72 | #endif |
| 73 | |
| 74 | /*----- Header files ------------------------------------------------------*/ |
| 75 | |
| 76 | #ifndef CATACOMB_MP_H |
| 77 | # include "mp.h" |
| 78 | #endif |
| 79 | |
| 80 | /*----- Data structures ---------------------------------------------------*/ |
| 81 | |
| 82 | typedef struct mpbarrett { |
| 83 | mp *m; |
| 84 | mp *mu; |
| 85 | size_t k; |
| 86 | } mpbarrett; |
| 87 | |
| 88 | /*----- Functions provided ------------------------------------------------*/ |
| 89 | |
| 90 | /* --- @mpbarrett_create@ --- * |
| 91 | * |
| 92 | * Arguments: @mpbarrett *mb@ = pointer to Barrett reduction context |
| 93 | * @mp *m@ = modulus to work to |
| 94 | * |
| 95 | * Returns: Zero on success, nonzero on error. |
| 96 | * |
| 97 | * Use: Initializes a Barrett reduction context ready for use. |
| 98 | */ |
| 99 | |
| 100 | extern int mpbarrett_create(mpbarrett */*mb*/, mp */*m*/); |
| 101 | |
| 102 | /* --- @mpbarrett_destroy@ --- * |
| 103 | * |
| 104 | * Arguments: @mpbarrett *mb@ = pointer to Barrett reduction context |
| 105 | * |
| 106 | * Returns: --- |
| 107 | * |
| 108 | * Use: Destroys a Barrett reduction context releasing any resources |
| 109 | * claimed. |
| 110 | */ |
| 111 | |
| 112 | extern void mpbarrett_destroy(mpbarrett */*mb*/); |
| 113 | |
| 114 | /* --- @mpbarrett_reduce@ --- * |
| 115 | * |
| 116 | * Arguments: @const mpbarrett *mb@ = pointer to Barrett reduction context |
| 117 | * @mp *d@ = destination for result |
| 118 | * @mp *m@ = number to reduce |
| 119 | * |
| 120 | * Returns: The residue of @m@ modulo the number in the reduction |
| 121 | * context. |
| 122 | * |
| 123 | * Use: Performs an efficient modular reduction. |
| 124 | */ |
| 125 | |
| 126 | extern mp *mpbarrett_reduce(const mpbarrett */*mb*/, mp */*d*/, mp */*m*/); |
| 127 | |
| 128 | /* --- @mpbarrett_exp@ --- * |
| 129 | * |
| 130 | * Arguments: @const mpbarrett *mb@ = pointer to Barrett reduction context |
| 131 | * @mp *d@ = fake destination |
| 132 | * @mp *a@ = base |
| 133 | * @mp *e@ = exponent |
| 134 | * |
| 135 | * Returns: Result, %$a^e \bmod m$%. |
| 136 | */ |
| 137 | |
| 138 | extern mp *mpbarrett_exp(const mpbarrett */*mb*/, mp */*d*/, |
| 139 | mp */*a*/, mp */*e*/); |
| 140 | |
| 141 | /* --- @mpbarrett_mexp@ --- * |
| 142 | * |
| 143 | * Arguments: @const mpbarrett *mb@ = pointer to Barrett reduction context |
| 144 | * @mp *d@ = fake destination |
| 145 | * @const mp_expfactor *f@ = pointer to array of factors |
| 146 | * @size_t n@ = number of factors supplied |
| 147 | * |
| 148 | * Returns: If the bases are %$g_0, g_1, \ldots, g_{n-1}$% and the |
| 149 | * exponents are %$e_0, e_1, \ldots, e_{n-1}$% then the result |
| 150 | * is: |
| 151 | * |
| 152 | * %$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} \bmod m$% |
| 153 | */ |
| 154 | |
| 155 | extern mp *mpbarrett_mexp(const mpbarrett */*mb*/, mp */*d*/, |
| 156 | const mp_expfactor */*f*/, size_t /*n*/); |
| 157 | |
| 158 | /*----- That's all, folks -------------------------------------------------*/ |
| 159 | |
| 160 | #ifdef __cplusplus |
| 161 | } |
| 162 | #endif |
| 163 | |
| 164 | #endif |