+/* -*-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