f41f820e |
1 | /* -*-c-*- |
2 | * |
3 | * $Id: rho.h,v 1.1 2000/07/09 21:32:30 mdw Exp $ |
4 | * |
5 | * Pollard's rho algorithm for discrete logs |
6 | * |
7 | * (c) 2000 Straylight/Edgeware |
8 | */ |
9 | |
10 | /*----- Licensing notice --------------------------------------------------* |
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. |
18 | * |
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. |
23 | * |
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 | |
30 | /*----- Revision history --------------------------------------------------* |
31 | * |
32 | * $Log: rho.h,v $ |
33 | * Revision 1.1 2000/07/09 21:32:30 mdw |
34 | * Pollard's rho algorithm for computing discrete logs. |
35 | * |
36 | */ |
37 | |
38 | #ifndef CATACOMB_RHO_H |
39 | #define CATACOMB_RHO_H |
40 | |
41 | #ifdef __cplusplus |
42 | extern "C" { |
43 | #endif |
44 | |
45 | /*----- Header files ------------------------------------------------------*/ |
46 | |
47 | #ifndef CATACOMB_MP_H |
48 | # include "mp.h" |
49 | #endif |
50 | |
51 | /*----- Data structures ---------------------------------------------------*/ |
52 | |
53 | /* --- The group operations table --- */ |
54 | |
55 | typedef struct rho_ops { |
56 | void (*sqr)(void *x, void *c); |
57 | void (*mul)(void *x, void *y, void *c); |
58 | int (*eq)(void *x, void *y); |
59 | int (*split)(void *x); |
60 | void (*drop)(void *x); |
61 | } rho_ops; |
62 | |
63 | /* --- The Pollard's rho context structure --- */ |
64 | |
65 | typedef struct rho_ctx { |
66 | rho_ops *ops; /* Group operations table */ |
67 | void *c; /* Context for group operations */ |
68 | void *g, *a; /* Generator and argument for log */ |
69 | mp *n; /* Cyclic group order */ |
70 | } rho_ctx; |
71 | |
72 | /*----- Functions provided ------------------------------------------------*/ |
73 | |
74 | /* --- @rho@ --- * |
75 | * |
76 | * Arguments: @rho_ctx *cc@ = pointer to the context structure |
77 | * @void *x, *y@ = two (equal) base values (try 1) |
78 | * @mp *a, *b@ = logs of %$x$% (see below) |
79 | * |
80 | * Returns: The discrete logarithm %$\log_g a$%, or null if the algorithm |
81 | * failed. (This is unlikely, though possible.) |
82 | * |
83 | * Use: Uses Pollard's rho algorithm to compute discrete logs in the |
84 | * group %$G$% generated by %$g$%. |
85 | * |
86 | * The algorithm works by finding a cycle in a pseudo-random |
87 | * walk. The function @ops->split@ should return an element |
88 | * from %$\{\,0, 1, 2\,\}$% according to its argument, in order |
89 | * to determine the walk. At each step in the walk, we know a |
90 | * group element %$x \in G$% together with its representation as |
91 | * a product of powers of %$g$% and $%a$% (i.e., we know that |
92 | * %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%). |
93 | * |
94 | * Locating a cycle gives us a collision |
95 | * |
96 | * %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$% |
97 | * |
98 | * Taking logs of both sides (to base %$g$%) gives us that |
99 | * |
100 | * %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$% |
101 | * |
102 | * Good initial values are %$x = y = 1$% (the multiplicative |
103 | * identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%. |
104 | * If that doesn't work then start choosing more `interesting' |
105 | * values. |
106 | * |
107 | * Note that the algorithm requires minimal space but |
108 | * %$O(\sqrt{n})$% time. Don't do this on large groups, |
109 | * particularly if you can find a decent factor base. |
110 | * |
111 | * Finally, note that this function will free the input values |
112 | * when it's finished with them. This probably isn't a great |
113 | * problem. |
114 | */ |
115 | |
116 | extern mp *rho(rho_ctx */*cc*/, void */*x*/, void */*y*/, |
117 | mp */*a*/, mp */*b*/); |
118 | |
119 | /* --- @rho_prime@ --- * |
120 | * |
121 | * Arguments: @mp *g@ = generator for the group |
122 | * @mp *a@ = value to find the logarithm of |
123 | * @mp *n@ = order of the group |
124 | * @mp *p@ = prime size of the underlying prime field |
125 | * |
126 | * Returns: The discrete logarithm %$\log_g a$%. |
127 | * |
128 | * Use: Computes discrete logarithms in a subgroup of a prime field. |
129 | */ |
130 | |
131 | extern mp *rho_prime(mp */*g*/, mp */*a*/, mp */*n*/, mp */*p*/); |
132 | |
133 | /*----- That's all, folks -------------------------------------------------*/ |
134 | |
135 | #ifdef __cplusplus |
136 | } |
137 | #endif |
138 | |
139 | #endif |