/* gf128mul.h - GF(2^128) multiplication functions * * Copyright (c) 2003, Dr Brian Gladman, Worcester, UK. * Copyright (c) 2006 Rik Snel * * Based on Dr Brian Gladman's (GPL'd) work published at * http://fp.gladman.plus.com/cryptography_technology/index.htm * See the original copyright notice below. * * This program is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License as published by the Free * Software Foundation; either version 2 of the License, or (at your option) * any later version. */ /* --------------------------------------------------------------------------- Copyright (c) 2003, Dr Brian Gladman, Worcester, UK. All rights reserved. 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DISCLAIMER This software is provided 'as is' with no explicit or implied warranties in respect of its properties, including, but not limited to, correctness and/or fitness for purpose. --------------------------------------------------------------------------- Issue Date: 31/01/2006 An implementation of field multiplication in Galois Field GF(2^128) */ #ifndef _CRYPTO_GF128MUL_H #define _CRYPTO_GF128MUL_H #include #include #include /* Comment by Rik: * * For some background on GF(2^128) see for example: * http://csrc.nist.gov/groups/ST/toolkit/BCM/documents/proposedmodes/gcm/gcm-revised-spec.pdf * * The elements of GF(2^128) := GF(2)[X]/(X^128-X^7-X^2-X^1-1) can * be mapped to computer memory in a variety of ways. Let's examine * three common cases. * * Take a look at the 16 binary octets below in memory order. The msb's * are left and the lsb's are right. char b[16] is an array and b[0] is * the first octet. * * 10000000 00000000 00000000 00000000 .... 00000000 00000000 00000000 * b[0] b[1] b[2] b[3] b[13] b[14] b[15] * * Every bit is a coefficient of some power of X. We can store the bits * in every byte in little-endian order and the bytes themselves also in * little endian order. I will call this lle (little-little-endian). * The above buffer represents the polynomial 1, and X^7+X^2+X^1+1 looks * like 11100001 00000000 .... 00000000 = { 0xE1, 0x00, }. * This format was originally implemented in gf128mul and is used * in GCM (Galois/Counter mode) and in ABL (Arbitrary Block Length). * * Another convention says: store the bits in bigendian order and the * bytes also. This is bbe (big-big-endian). Now the buffer above * represents X^127. X^7+X^2+X^1+1 looks like 00000000 .... 10000111, * b[15] = 0x87 and the rest is 0. LRW uses this convention and bbe * is partly implemented. * * Both of the above formats are easy to implement on big-endian * machines. * * XTS and EME (the latter of which is patent encumbered) use the ble * format (bits are stored in big endian order and the bytes in little * endian). The above buffer represents X^7 in this case and the * primitive polynomial is b[0] = 0x87. * * The common machine word-size is smaller than 128 bits, so to make * an efficient implementation we must split into machine word sizes. * This implementation uses 64-bit words for the moment. Machine * endianness comes into play. The lle format in relation to machine * endianness is discussed below by the original author of gf128mul Dr * Brian Gladman. * * Let's look at the bbe and ble format on a little endian machine. * * bbe on a little endian machine u32 x[4]: * * MS x[0] LS MS x[1] LS * ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls * 103..96 111.104 119.112 127.120 71...64 79...72 87...80 95...88 * * MS x[2] LS MS x[3] LS * ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls * 39...32 47...40 55...48 63...56 07...00 15...08 23...16 31...24 * * ble on a little endian machine * * MS x[0] LS MS x[1] LS * ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls * 31...24 23...16 15...08 07...00 63...56 55...48 47...40 39...32 * * MS x[2] LS MS x[3] LS * ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls * 95...88 87...80 79...72 71...64 127.120 199.112 111.104 103..96 * * Multiplications in GF(2^128) are mostly bit-shifts, so you see why * ble (and lbe also) are easier to implement on a little-endian * machine than on a big-endian machine. The converse holds for bbe * and lle. * * Note: to have good alignment, it seems to me that it is sufficient * to keep elements of GF(2^128) in type u64[2]. On 32-bit wordsize * machines this will automatically aligned to wordsize and on a 64-bit * machine also. */ /* Multiply a GF(2^128) field element by x. Field elements are held in arrays of bytes in which field bits 8n..8n + 7 are held in byte[n], with lower indexed bits placed in the more numerically significant bit positions within bytes. On little endian machines the bit indexes translate into the bit positions within four 32-bit words in the following way MS x[0] LS MS x[1] LS ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls 24...31 16...23 08...15 00...07 56...63 48...55 40...47 32...39 MS x[2] LS MS x[3] LS ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls 88...95 80...87 72...79 64...71 120.127 112.119 104.111 96..103 On big endian machines the bit indexes translate into the bit positions within four 32-bit words in the following way MS x[0] LS MS x[1] LS ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls 00...07 08...15 16...23 24...31 32...39 40...47 48...55 56...63 MS x[2] LS MS x[3] LS ms ls ms ls ms ls ms ls ms ls ms ls ms ls ms ls 64...71 72...79 80...87 88...95 96..103 104.111 112.119 120.127 */ /* A slow generic version of gf_mul, implemented for lle * It multiplies a and b and puts the result in a */ void gf128mul_lle(be128 *a, const be128 *b); /* * The following functions multiply a field element by x in * the polynomial field representation. They use 64-bit word operations * to gain speed but compensate for machine endianness and hence work * correctly on both styles of machine. * * They are defined here for performance. */ static inline u64 gf128mul_mask_from_bit(u64 x, int which) { /* a constant-time version of 'x & ((u64)1 << which) ? (u64)-1 : 0' */ return ((s64)(x << (63 - which)) >> 63); } static inline void gf128mul_x_lle(be128 *r, const be128 *x) { u64 a = be64_to_cpu(x->a); u64 b = be64_to_cpu(x->b); /* equivalent to gf128mul_table_le[(b << 7) & 0xff] << 48 * (see crypto/gf128mul.c): */ u64 _tt = gf128mul_mask_from_bit(b, 0) & ((u64)0xe1 << 56); r->b = cpu_to_be64((b >> 1) | (a << 63)); r->a = cpu_to_be64((a >> 1) ^ _tt); } static inline void gf128mul_x_bbe(be128 *r, const be128 *x) { u64 a = be64_to_cpu(x->a); u64 b = be64_to_cpu(x->b); /* equivalent to gf128mul_table_be[a >> 63] (see crypto/gf128mul.c): */ u64 _tt = gf128mul_mask_from_bit(a, 63) & 0x87; r->a = cpu_to_be64((a << 1) | (b >> 63)); r->b = cpu_to_be64((b << 1) ^ _tt); } /* needed by XTS */ static inline void gf128mul_x_ble(le128 *r, const le128 *x) { u64 a = le64_to_cpu(x->a); u64 b = le64_to_cpu(x->b); /* equivalent to gf128mul_table_be[b >> 63] (see crypto/gf128mul.c): */ u64 _tt = gf128mul_mask_from_bit(a, 63) & 0x87; r->a = cpu_to_le64((a << 1) | (b >> 63)); r->b = cpu_to_le64((b << 1) ^ _tt); } /* 4k table optimization */ struct gf128mul_4k { be128 t[256]; }; struct gf128mul_4k *gf128mul_init_4k_lle(const be128 *g); void gf128mul_4k_lle(be128 *a, const struct gf128mul_4k *t); void gf128mul_x8_ble(le128 *r, const le128 *x); static inline void gf128mul_free_4k(struct gf128mul_4k *t) { kfree_sensitive(t); } /* 64k table optimization, implemented for bbe */ struct gf128mul_64k { struct gf128mul_4k *t[16]; }; /* First initialize with the constant factor with which you * want to multiply and then call gf128mul_64k_bbe with the other * factor in the first argument, and the table in the second. * Afterwards, the result is stored in *a. */ struct gf128mul_64k *gf128mul_init_64k_bbe(const be128 *g); void gf128mul_free_64k(struct gf128mul_64k *t); void gf128mul_64k_bbe(be128 *a, const struct gf128mul_64k *t); #endif /* _CRYPTO_GF128MUL_H */