symm/poly1305.c: Implement `flushzero' to zero-pad to a block boundary.

[catacomb] / math / mpbarrett.h

/* -*-c-*-
 *
 * Barrett modular reduction
 *
 * (c) 1999 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.
 */

/*----- Notes on Barrett reduction ----------------------------------------*
 *
 * Barrett reduction is a technique for computing modular residues.  Unlike
 * Montgomery reduction, it doesn't have restrictions on the modulus (except
 * that it be positive) and doesn't confuse matters by putting an extra
 * factor all the way through your computation.
 *
 * It's useful for slightly less heavy-duty work than Montgomery reduction
 * because the precomputation phase is rather simpler, involving a single
 * division operation.
 *
 * Sometimes it's useful to exponentiate modulo an even number, so there's a
 * modexp routine provided which uses Barrett reduction rather than
 * Montgomery reduction.  This is handy when you're working on indices in an
 * even-order cyclic group or something.
 *
 * In more detail: suppose that %$b^{k-1} \le m < b^k$%.  Let %$\mu = {}$%
 * %$\lfloor b^{2k}/m \rfloor$%; %$\mu$% is a scaled approximation to the
 * reciprocal %$1/m$%.  Now, suppose we're given some %$a$% with
 * %$0 \le a < b^{2k}$%.  The first step is to calculate an approximation
 * %$q = \lfloor \mu \lfloor a/b^{k-1} \rfloor/b^{k+1} \rfloor$% to the
 * quotient %$a/m$%.  Then we have:
 *
 *	%$\lfloor a/m - a/b^{2k} - b^{k-1}/m + 1/b^{k+1} \rfloor \le {}$%
 *		%$q \le \lfloor a/m \rfloor
 *
 * But by assumption %$a < b^{2k}$% and %$2^{k-1} \le m$% so
 *
 *	%$\lfloor a/m \rfloor - 2 \le q \le \lfloor a/m \rfloor$%
 *
 * Now we approximate the remainder by calculating %$r = a - q m$%.
 * Certainly %$r \equiv a \pmod{m}$%; and
 *
 *	%$0 \le r \le (a - m \lfloor a/m \rfloor) + 2 m < 3 m$%.
 *
 * So the remainder can be fixed up with at most two conditional
 * subtractions.
 */

#ifndef CATACOMB_MPBARRETT_H
#define CATACOMB_MPBARRETT_H

#ifdef __cplusplus
  extern "C" {
#endif

/*----- Header files ------------------------------------------------------*/

#ifndef CATACOMB_MP_H
#  include "mp.h"
#endif

/*----- Data structures ---------------------------------------------------*/

typedef struct mpbarrett {
  mp *m;
  mp *mu;
  size_t k;
} mpbarrett;

/*----- Functions provided ------------------------------------------------*/

/* --- @mpbarrett_create@ --- *
 *
 * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
 *		@mp *m@ = modulus to work to
 *
 * Returns:	Zero on success, nonzero on error.
 *
 * Use:		Initializes a Barrett reduction context ready for use.
 */

extern int mpbarrett_create(mpbarrett */*mb*/, mp */*m*/);

/* --- @mpbarrett_destroy@ --- *
 *
 * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
 *
 * Returns:	---
 *
 * Use:		Destroys a Barrett reduction context releasing any resources
 *		claimed.
 */

extern void mpbarrett_destroy(mpbarrett */*mb*/);

/* --- @mpbarrett_reduce@ --- *
 *
 * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
 *		@mp *d@ = destination for result
 *		@mp *m@ = number to reduce
 *
 * Returns:	The residue of @m@ modulo the number in the reduction
 *		context.
 *
 * Use:		Performs an efficient modular reduction.
 */

extern mp *mpbarrett_reduce(mpbarrett */*mb*/, mp */*d*/, mp */*m*/);

/* --- @mpbarrett_exp@ --- *
 *
 * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
 *		@mp *d@ = fake destination
 *		@mp *a@ = base
 *		@mp *e@ = exponent
 *
 * Returns:	Result, %$a^e \bmod m$%.
 */

extern mp *mpbarrett_exp(mpbarrett */*mb*/, mp */*d*/, mp */*a*/, mp */*e*/);

/* --- @mpbarrett_mexp@ --- *
 *
 * Arguments:	@mpbarrett *mb@ = pointer to Barrett reduction context
 *		@mp *d@ = fake destination
 *		@const mp_expfactor *f@ = pointer to array of factors
 *		@size_t n@ = number of factors supplied
 *
 * Returns:	If the bases are %$g_0, g_1, \ldots, g_{n-1}$% and the
 *		exponents are %$e_0, e_1, \ldots, e_{n-1}$% then the result
 *		is:
 *
 *		%$g_0^{e_0} g_1^{e_1} \ldots g_{n-1}^{e_{n-1}} \bmod m$%
 */

extern mp *mpbarrett_mexp(mpbarrett */*mb*/, mp */*d*/,
			  const mp_expfactor */*f*/, size_t /*n*/);

/*----- That's all, folks -------------------------------------------------*/

#ifdef __cplusplus
  }
#endif

#endif