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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: rho.h,v 1.3 2004/04/08 01:36:15 mdw Exp $ |
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4 | * |
5 | * Pollard's rho algorithm for discrete logs |
6 | * |
7 | * (c) 2000 Straylight/Edgeware |
8 | */ |
9 | |
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10 | /*----- Licensing notice --------------------------------------------------* |
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11 | * |
12 | * This file is part of Catacomb. |
13 | * |
14 | * Catacomb is free software; you can redistribute it and/or modify |
15 | * it under the terms of the GNU Library General Public License as |
16 | * published by the Free Software Foundation; either version 2 of the |
17 | * License, or (at your option) any later version. |
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18 | * |
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19 | * Catacomb is distributed in the hope that it will be useful, |
20 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
21 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
22 | * GNU Library General Public License for more details. |
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23 | * |
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24 | * You should have received a copy of the GNU Library General Public |
25 | * License along with Catacomb; if not, write to the Free |
26 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
27 | * MA 02111-1307, USA. |
28 | */ |
29 | |
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30 | #ifndef CATACOMB_RHO_H |
31 | #define CATACOMB_RHO_H |
32 | |
33 | #ifdef __cplusplus |
34 | extern "C" { |
35 | #endif |
36 | |
37 | /*----- Header files ------------------------------------------------------*/ |
38 | |
39 | #ifndef CATACOMB_MP_H |
40 | # include "mp.h" |
41 | #endif |
42 | |
43 | /*----- Data structures ---------------------------------------------------*/ |
44 | |
45 | /* --- The group operations table --- */ |
46 | |
47 | typedef struct rho_ops { |
48 | void (*sqr)(void *x, void *c); |
49 | void (*mul)(void *x, void *y, void *c); |
50 | int (*eq)(void *x, void *y); |
51 | int (*split)(void *x); |
52 | void (*drop)(void *x); |
53 | } rho_ops; |
54 | |
55 | /* --- The Pollard's rho context structure --- */ |
56 | |
57 | typedef struct rho_ctx { |
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58 | const rho_ops *ops; /* Group operations table */ |
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59 | void *c; /* Context for group operations */ |
60 | void *g, *a; /* Generator and argument for log */ |
61 | mp *n; /* Cyclic group order */ |
62 | } rho_ctx; |
63 | |
64 | /*----- Functions provided ------------------------------------------------*/ |
65 | |
66 | /* --- @rho@ --- * |
67 | * |
68 | * Arguments: @rho_ctx *cc@ = pointer to the context structure |
69 | * @void *x, *y@ = two (equal) base values (try 1) |
70 | * @mp *a, *b@ = logs of %$x$% (see below) |
71 | * |
72 | * Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm |
73 | * failed. (This is unlikely, though possible.) |
74 | * |
75 | * Use: Uses Pollard's rho algorithm to compute discrete logs in the |
76 | * group %$G$% generated by %$g$%. |
77 | * |
78 | * The algorithm works by finding a cycle in a pseudo-random |
79 | * walk. The function @ops->split@ should return an element |
80 | * from %$\{\,0, 1, 2\,\}$% according to its argument, in order |
81 | * to determine the walk. At each step in the walk, we know a |
82 | * group element %$x \in G$% together with its representation as |
83 | * a product of powers of %$g$% and $%a$% (i.e., we know that |
84 | * %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%). |
85 | * |
86 | * Locating a cycle gives us a collision |
87 | * |
88 | * %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$% |
89 | * |
90 | * Taking logs of both sides (to base %$g$%) gives us that |
91 | * |
92 | * %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$% |
93 | * |
94 | * Good initial values are %$x = y = 1$% (the multiplicative |
95 | * identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%. |
96 | * If that doesn't work then start choosing more `interesting' |
97 | * values. |
98 | * |
99 | * Note that the algorithm requires minimal space but |
100 | * %$O(\sqrt{n})$% time. Don't do this on large groups, |
101 | * particularly if you can find a decent factor base. |
102 | * |
103 | * Finally, note that this function will free the input values |
104 | * when it's finished with them. This probably isn't a great |
105 | * problem. |
106 | */ |
107 | |
108 | extern mp *rho(rho_ctx */*cc*/, void */*x*/, void */*y*/, |
109 | mp */*a*/, mp */*b*/); |
110 | |
111 | /* --- @rho_prime@ --- * |
112 | * |
113 | * Arguments: @mp *g@ = generator for the group |
114 | * @mp *a@ = value to find the logarithm of |
115 | * @mp *n@ = order of the group |
116 | * @mp *p@ = prime size of the underlying prime field |
117 | * |
118 | * Returns: The discrete logarithm %$\log_g a$%. |
119 | * |
120 | * Use: Computes discrete logarithms in a subgroup of a prime field. |
121 | */ |
122 | |
123 | extern mp *rho_prime(mp */*g*/, mp */*a*/, mp */*n*/, mp */*p*/); |
124 | |
125 | /*----- That's all, folks -------------------------------------------------*/ |
126 | |
127 | #ifdef __cplusplus |
128 | } |
129 | #endif |
130 | |
131 | #endif |