Various stuff.

[mLib] / unihash.h

/* -*-c-*-
 *
 * $Id: unihash.h,v 1.1 2003/10/12 14:43:24 mdw Exp $
 *
 * Simple and efficient universal hashing for hashtables
 *
 * (c) 2003 Straylight/Edgeware
 */

/*----- Licensing notice --------------------------------------------------* 
 *
 * This file is part of the mLib utilities library.
 *
 * mLib 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.
 * 
 * mLib 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 mLib; if not, write to the Free
 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
 * MA 02111-1307, USA.
 */

/*----- Revision history --------------------------------------------------* 
 *
 * $Log: unihash.h,v $
 * Revision 1.1  2003/10/12 14:43:24  mdw
 * Universal hashing.
 *
 */

#ifndef MLIB_UNIHASH_H
#define MLIB_UNIHASH_H

#ifdef __cplusplus
  extern "C" {
#endif


/*----- Concept -----------------------------------------------------------*
 *
 * Let %$\gf{q}$% be a finite field.  Choose an arbitrary %$k \inr \gf{q}$%.
 * Let %$M$% be a message.  Injectively pad %$M$% and split it into blocks
 * $m_{n-1}, m_{n-2}, \ldots, m_2, m_1, m_0$% in %$\gf{q}%.
 * Then we compute
 *
 *   %$H_k(M) = k^{n+1} \sum_{0\le i<n} m_i k^{i+1}.$%
 *
 * Note that %$H_0(M) = 0$% for all messages %$M$%.
 *
 * If we deal with messages at most %$\ell$% blocks long then %$H_k(\cdot)$%
 * is %$(\ell + 1)/q$%-almost universal.  Moreover, if %$q = 2^f$% then
 * %$H_k(\cdot)$% is %$(\ell + 1)/q$%-almost XOR-universal.
 *
 * Proof.  Let %$A$% and %$B$% be two messages, represented by
 * %$a_{n-1}, \ldots, a_0$% and %$b_{m-1}, \ldots, b_0$% respectively; and
 * choose any %$\delta \in \gf{q}$%.  We must bound the probability that
 *
 * %$k^{n+1} + a_{n-1} k^{n} + \cdots + a_1 k^2 + a_0 k - {}$%
 *   %$k^{m+1} - b_{m-1} k^{m} - \cdots - b_1 k^2 - b_0 k = \delta$%.
 *
 * Firstly, we claim that if %$A$% and %$B$% are distinct, there is some
 * nonzero coefficient of %$k$%.  For if %$n \ne m$% then, without loss of
 * generality, let %$n > m$%, and hence the coefficient of %$k_n$% is
 * nonzero.  Alternatively, if %$n = m$% then there must be some
 * %$i \in \{ 0, \ldots, n - 1 \}$% with %$a_i \ne b_i$%, for otherwise the
 * messages would be identical; but then the coefficient of %$k^{i+1}$% is
 * %$a_i - b_i \ne 0$%.
 *
 * Hence we have a polynomial equation with degree at most %$\ell + 1$%;
 * there must be at most %$\ell + 1$% solutions for %$k$%; but we choose
 * %$k$% at random from a set of %$q$%; so the equation is true with
 * probability at most %$(\ell + 1)/q$%.
 *
 * This function can be used as a simple MAC with provable security against
 * computationally unbounded adversaries.  Simply XOR the hash with a random
 * string indexed from a large random pad by some nonce sent with the
 * message.  The probability of a forgery attempt being successful is then
 * %$(\ell + 1)/2^t$%, where %$t$% is the tag length and %$n$% is the longest
 * message permitted.
 */

/*----- Practicalities ----------------------------------------------------*
 *
 * We work in %$\gf{2^32}$%, represented as a field of polynomials modulo
 * %$\{104c11db7}_x$% (this is the standard CRC-32 polynomial).  Our blocks
 * are bytes.  We append a big-endian byte length.
 *
 * The choice of a 32-bit hash is made for pragmatic reasons: we're never
 * likely to actually want all 32 bits for a real hashtable anyway.  The
 * truncation result is needed to keep us afloat with smaller tables.
 *
 * We compute hashes using a slightly unrolled version of Horner's rule,
 * using the recurrence:
 *
 *   %$a_{i+b} = (a_i + m_i) k^b + m_{i+1} k^{b-1} + \cdots + m_{i+b-1} k$%
 *
 * which involves one full-width multiply and %$b - 1$% one-byte multiplies;
 * the latter may be efficiently computed using a table lookup.  Start with
 * %$a_0 = k$%.
 *
 * We precompute tables %$S[\cdot][\cdot][\cdot]$%, where
 *
 *   %$S[u][v][w] = k^{u+1} x^{8v} w$%
 *     for %$0 \le u < b$%, %$0 \le v < 4$%, %$0 \le w < 256)$%.
 *
 * A one-byte multiply is one lookup; a full-width multiply is four lookups
 * and three XORs.  The processing required is then %$b + 3$% lookups and
 * %$b + 3$% XORs per batch, or %$(b + 3)/b$% lookups and XORs per byte, at
 * the expense of %$4 b$% kilobytes of tables.  This compares relatively
 * favorably with CRC32.  Indeed, in tests, this implementation with $b = 4$%
 * is faster than a 32-bit CRC.
 */

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

#include <stddef.h>

#ifndef MLIB_BITS_H
#  include "bits.h"
#endif

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

#define UNIHASH_NBATCH 4
#define UNIHASH_POLY 0x04c11db7		/* From CRC32 */

typedef struct unihash_info {
  uint32 s[UNIHASH_NBATCH][4][256];	/* S-tables as described */
} unihash_info;

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

/* --- @unihash_setkey@ --- *
 *
 * Arguments:	@unihash_info *i@ = where to store the precomputed tables
 *		@uint32 k@ = the key to set, randomly chosen
 *
 * Returns:	---
 *
 * Use:		Calculates the tables required for efficient hashing.
 */

extern void unihash_setkey(unihash_info */*i*/, uint32 /*k*/);

/* --- @unihash_hash@ --- *
 *
 * Arguments:	@const unihash_info *i@ = pointer to precomputed table
 *		@uint32 a@ = @UNIHASH_INIT(i)@ or value from previous call
 *		@const void *p@ = pointer to data to hash
 *		@size_t sz@ = size of the data
 *
 * Returns:	---
 *
 * Use:		Hashes data.  Call this as many times as needed.  
 */

#define UNIHASH_INIT(i) ((i)->s[0][0][1]) /* %$k$% */

extern uint32 unihash_hash(const unihash_info */*i*/, uint32 /*a*/,
			   const void */*p*/, size_t /*sz*/);

/* --- @unihash@ --- *
 *
 * Arguments:	@const unihash_info *i@ = precomputed tables
 *		@const void *p@ = pointer to data to hash
 *		@size_t sz@ = size of the data
 *
 * Returns:	The hash value computed.
 *
 * Use:		All-in-one hashing function.  No faster than using the
 * 		separate calls, but more convenient.
 */

#define UNIHASH(i, p, sz) (unihash_hash((i), UNIHASH_INIT((i)), (p), (sz)))

extern uint32 unihash(const unihash_info */*i*/,
		      const void */*p*/, size_t /*sz*/);

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

#ifdef __cplusplus
  }
#endif

#endif