Pollard's rho algorithm for computing discrete logs.

[u/mdw/catacomb] / rho.h

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+/* -*-c-*-
+ *
+ * $Id: rho.h,v 1.1 2000/07/09 21:32:30 mdw Exp $
+ *
+ * Pollard's rho algorithm for discrete logs
+ *
+ * (c) 2000 Straylight/Edgeware
+ */
+
+/*----- Licensing notice --------------------------------------------------* 
+ *
+ * This file is part of Catacomb.
+ *
+ * Catacomb is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU Library General Public License as
+ * published by the Free Software Foundation; either version 2 of the
+ * License, or (at your option) any later version.
+ * 
+ * Catacomb is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ * GNU Library General Public License for more details.
+ * 
+ * You should have received a copy of the GNU Library General Public
+ * License along with Catacomb; if not, write to the Free
+ * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
+ * MA 02111-1307, USA.
+ */
+
+/*----- Revision history --------------------------------------------------* 
+ *
+ * $Log: rho.h,v $
+ * Revision 1.1  2000/07/09 21:32:30  mdw
+ * Pollard's rho algorithm for computing discrete logs.
+ *
+ */
+
+#ifndef CATACOMB_RHO_H
+#define CATACOMB_RHO_H
+
+#ifdef __cplusplus
+  extern "C" {
+#endif
+
+/*----- Header files ------------------------------------------------------*/
+
+#ifndef CATACOMB_MP_H
+#  include "mp.h"
+#endif
+
+/*----- Data structures ---------------------------------------------------*/
+
+/* --- The group operations table --- */
+
+typedef struct rho_ops {
+  void (*sqr)(void *x, void *c);
+  void (*mul)(void *x, void *y, void *c);
+  int (*eq)(void *x, void *y);
+  int (*split)(void *x);
+  void (*drop)(void *x);
+} rho_ops;
+
+/* --- The Pollard's rho context structure --- */
+
+typedef struct rho_ctx {
+  rho_ops *ops;                                /* Group operations table */
+  void *c;                             /* Context for group operations */
+  void *g, *a;                         /* Generator and argument for log */
+  mp *n;                               /* Cyclic group order */
+} rho_ctx;
+
+/*----- Functions provided ------------------------------------------------*/
+
+/* --- @rho@ --- *
+ *
+ * Arguments:  @rho_ctx *cc@ = pointer to the context structure
+ *             @void *x, *y@ = two (equal) base values (try 1)
+ *             @mp *a, *b@ = logs of %$x$% (see below)
+ *
+ * Returns:    The discrete logarithm %$\log_g a$%, or null if the algorithm
+ *             failed.  (This is unlikely, though possible.)
+ *
+ * Use:                Uses Pollard's rho algorithm to compute discrete logs in the
+ *             group %$G$% generated by %$g$%.
+ *
+ *             The algorithm works by finding a cycle in a pseudo-random
+ *             walk.  The function @ops->split@ should return an element
+ *             from %$\{\,0, 1, 2\,\}$% according to its argument, in order
+ *             to determine the walk.  At each step in the walk, we know a
+ *             group element %$x \in G$% together with its representation as
+ *             a product of powers of %$g$% and $%a$% (i.e., we know that
+ *             %$x = g^\alpha a^\beta$% for some %$\alpha$%, %$\beta$%).
+ *
+ *             Locating a cycle gives us a collision
+ *
+ *               %$g^{\alpha} a^{\beta} = g^{\alpha'} a^{\beta'}$%
+ *
+ *             Taking logs of both sides (to base %$g$%) gives us that
+ *
+ *               %$\log a\equiv\frac{\alpha-\alpha'}{\beta'-\beta}\bmod{n}$%
+ *
+ *             Good initial values are %$x = y = 1$% (the multiplicative
+ *             identity of %$G$%) and %$\alpha\equiv\beta\equiv0\bmod{n}$%.
+ *             If that doesn't work then start choosing more `interesting'
+ *             values.
+ *
+ *             Note that the algorithm requires minimal space but
+ *             %$O(\sqrt{n})$% time.  Don't do this on large groups,
+ *             particularly if you can find a decent factor base.
+ *
+ *             Finally, note that this function will free the input values
+ *             when it's finished with them.  This probably isn't a great
+ *             problem.
+ */
+
+extern mp *rho(rho_ctx */*cc*/, void */*x*/, void */*y*/,
+              mp */*a*/, mp */*b*/);
+
+/* --- @rho_prime@ --- *
+ *
+ * Arguments:  @mp *g@ = generator for the group
+ *             @mp *a@ = value to find the logarithm of
+ *             @mp *n@ = order of the group
+ *             @mp *p@ = prime size of the underlying prime field
+ *
+ * Returns:    The discrete logarithm %$\log_g a$%.
+ *
+ * Use:                Computes discrete logarithms in a subgroup of a prime field.
+ */
+
+extern mp *rho_prime(mp */*g*/, mp */*a*/, mp */*n*/, mp */*p*/);
+
+/*----- That's all, folks -------------------------------------------------*/
+
+#ifdef __cplusplus
+  }
+#endif
+
+#endif