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gf2_128.c
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#include <stdint.h>
#include "gf2_16.h"
#include "gf2_128.h"
// In this file, we construct the Galois field GF(2^128)
// as a finite field extension of GF(2^16). The
// field is constructed using the following irreducible
// polynomial, whose coefficents should be understood
// as the binary representation of elements of GF(2^16):
//
// x^8 + x^3 + x + 8
//
uint128_t gf2_128_mult( uint128_t a, uint128_t b ) {
uint16_t a0, a1, a2, a3, a4, a5, a6, a7;
uint16_t b0, b1, b2, b3, b4, b5, b6, b7;
uint16_t za0, za1, za2, za3, za4, za5, za6, za7;
uint16_t zb0, zb1, zb2, zb3, zb4, zb5, zb6, zb7;
uint32_t alog0, alog1, alog2, alog3;
uint32_t alog4, alog5, alog6, alog7;
uint32_t blog0, blog1, blog2, blog3;
uint32_t blog4, blog5, blog6, blog7;
uint16_t c0, c1, c2, c3, c4, c5, c6, c7;
uint16_t c8, c9, c10, c11, c12, c13, c14;
uint16_t d0, d1, d2, d3, d4, d5, d6, d7;
a0 = (uint16_t) a;
a1 = (uint16_t) (a >> 16);
a2 = (uint16_t) (a >> 32);
a3 = (uint16_t) (a >> 48);
a4 = (uint16_t) (a >> 64);
a5 = (uint16_t) (a >> 80);
a6 = (uint16_t) (a >> 96);
a7 = (uint16_t) (a >> 112);
b0 = (uint16_t) b;
b1 = (uint16_t) (b >> 16);
b2 = (uint16_t) (b >> 32);
b3 = (uint16_t) (b >> 48);
b4 = (uint16_t) (b >> 64);
b5 = (uint16_t) (b >> 80);
b6 = (uint16_t) (b >> 96);
b7 = (uint16_t) (b >> 112);
alog0 = gf2_16_log_table[a0];
alog1 = gf2_16_log_table[a1];
alog2 = gf2_16_log_table[a2];
alog3 = gf2_16_log_table[a3];
alog4 = gf2_16_log_table[a4];
alog5 = gf2_16_log_table[a5];
alog6 = gf2_16_log_table[a6];
alog7 = gf2_16_log_table[a7];
blog0 = gf2_16_log_table[b0];
blog1 = gf2_16_log_table[b1];
blog2 = gf2_16_log_table[b2];
blog3 = gf2_16_log_table[b3];
blog4 = gf2_16_log_table[b4];
blog5 = gf2_16_log_table[b5];
blog6 = gf2_16_log_table[b6];
blog7 = gf2_16_log_table[b7];
za0 = zeroMask( a0 );
za1 = zeroMask( a1 );
za2 = zeroMask( a2 );
za3 = zeroMask( a3 );
za4 = zeroMask( a4 );
za5 = zeroMask( a5 );
za6 = zeroMask( a6 );
za7 = zeroMask( a7 );
zb0 = zeroMask( b0 );
zb1 = zeroMask( b1 );
zb2 = zeroMask( b2 );
zb3 = zeroMask( b3 );
zb4 = zeroMask( b4 );
zb5 = zeroMask( b5 );
zb6 = zeroMask( b6 );
zb7 = zeroMask( b7 );
c0 = gf2_16_expadd( za0|zb0, alog0, blog0 );
c1 = gf2_16_expadd( za0|zb1, alog0, blog1 );
c1 ^= gf2_16_expadd( za1|zb0, alog1, blog0 );
c2 = gf2_16_expadd( za0|zb2, alog0, blog2 );
c2 ^= gf2_16_expadd( za1|zb1, alog1, blog1 );
c2 ^= gf2_16_expadd( za2|zb0, alog2, blog0 );
c3 = gf2_16_expadd( za0|zb3, alog0, blog3 );
c3 ^= gf2_16_expadd( za1|zb2, alog1, blog2 );
c3 ^= gf2_16_expadd( za2|zb1, alog2, blog1 );
c3 ^= gf2_16_expadd( za3|zb0, alog3, blog0 );
c4 = gf2_16_expadd( za0|zb4, alog0, blog4 );
c4 ^= gf2_16_expadd( za1|zb3, alog1, blog3 );
c4 ^= gf2_16_expadd( za2|zb2, alog2, blog2 );
c4 ^= gf2_16_expadd( za3|zb1, alog3, blog1 );
c4 ^= gf2_16_expadd( za4|zb0, alog4, blog0 );
c5 = gf2_16_expadd( za0|zb5, alog0, blog5 );
c5 ^= gf2_16_expadd( za1|zb4, alog1, blog4 );
c5 ^= gf2_16_expadd( za2|zb3, alog2, blog3 );
c5 ^= gf2_16_expadd( za3|zb2, alog3, blog2 );
c5 ^= gf2_16_expadd( za4|zb1, alog4, blog1 );
c5 ^= gf2_16_expadd( za5|zb0, alog5, blog0 );
c6 = gf2_16_expadd( za0|zb6, alog0, blog6 );
c6 ^= gf2_16_expadd( za1|zb5, alog1, blog5 );
c6 ^= gf2_16_expadd( za2|zb4, alog2, blog4 );
c6 ^= gf2_16_expadd( za3|zb3, alog3, blog3 );
c6 ^= gf2_16_expadd( za4|zb2, alog4, blog2 );
c6 ^= gf2_16_expadd( za5|zb1, alog5, blog1 );
c6 ^= gf2_16_expadd( za6|zb0, alog6, blog0 );
c7 = gf2_16_expadd( za0|zb7, alog0, blog7 );
c7 ^= gf2_16_expadd( za1|zb6, alog1, blog6 );
c7 ^= gf2_16_expadd( za2|zb5, alog2, blog5 );
c7 ^= gf2_16_expadd( za3|zb4, alog3, blog4 );
c7 ^= gf2_16_expadd( za4|zb3, alog4, blog3 );
c7 ^= gf2_16_expadd( za5|zb2, alog5, blog2 );
c7 ^= gf2_16_expadd( za6|zb1, alog6, blog1 );
c7 ^= gf2_16_expadd( za7|zb0, alog7, blog0 );
c8 = gf2_16_expadd( za1|zb7, alog1, blog7 );
c8 ^= gf2_16_expadd( za2|zb6, alog2, blog6 );
c8 ^= gf2_16_expadd( za3|zb5, alog3, blog5 );
c8 ^= gf2_16_expadd( za4|zb4, alog4, blog4 );
c8 ^= gf2_16_expadd( za5|zb3, alog5, blog3 );
c8 ^= gf2_16_expadd( za6|zb2, alog6, blog2 );
c8 ^= gf2_16_expadd( za7|zb1, alog7, blog1 );
c9 = gf2_16_expadd( za2|zb7, alog2, blog7 );
c9 ^= gf2_16_expadd( za3|zb6, alog3, blog6 );
c9 ^= gf2_16_expadd( za4|zb5, alog4, blog5 );
c9 ^= gf2_16_expadd( za5|zb4, alog5, blog4 );
c9 ^= gf2_16_expadd( za6|zb3, alog6, blog3 );
c9 ^= gf2_16_expadd( za7|zb2, alog7, blog2 );
c10 = gf2_16_expadd( za3|zb7, alog3, blog7 );
c10^= gf2_16_expadd( za4|zb6, alog4, blog6 );
c10^= gf2_16_expadd( za5|zb5, alog5, blog5 );
c10^= gf2_16_expadd( za6|zb4, alog6, blog4 );
c10^= gf2_16_expadd( za7|zb3, alog7, blog3 );
c11 = gf2_16_expadd( za4|zb7, alog4, blog7 );
c11^= gf2_16_expadd( za5|zb6, alog5, blog6 );
c11^= gf2_16_expadd( za6|zb5, alog6, blog5 );
c11^= gf2_16_expadd( za7|zb4, alog7, blog4 );
c12 = gf2_16_expadd( za5|zb7, alog5, blog7 );
c12^= gf2_16_expadd( za6|zb6, alog6, blog6 );
c12^= gf2_16_expadd( za7|zb5, alog7, blog5 );
c13 = gf2_16_expadd( za6|zb7, alog6, blog7 );
c13^= gf2_16_expadd( za7|zb6, alog7, blog6 );
c14 = gf2_16_expadd( za7|zb7, alog7, blog7 );
// Now, modular reduction
uint16_t log8 = 3; // gf2_16_log_table[8];
uint16_t c14x8 = gf2_16_expadd( zeroMask(c14), gf2_16_log_table[c14], log8 );
uint16_t c13x8 = gf2_16_expadd( zeroMask(c13), gf2_16_log_table[c13], log8 );
uint16_t c12x8 = gf2_16_expadd( zeroMask(c12), gf2_16_log_table[c12], log8 );
uint16_t c11x8 = gf2_16_expadd( zeroMask(c11), gf2_16_log_table[c11], log8 );
uint16_t c10x8 = gf2_16_expadd( zeroMask(c10), gf2_16_log_table[c10], log8 );
uint16_t c9x8 = gf2_16_expadd( zeroMask( c9), gf2_16_log_table[ c9], log8 );
uint16_t c8x8 = gf2_16_expadd( zeroMask( c8), gf2_16_log_table[ c8], log8 );
d7 = c14 ^ c12 ^ c7;
d6 = c14x8 ^ c13 ^ c11 ^ c6;
d5 = c13x8 ^ c12 ^ c10 ^ c5;
d4 = c14 ^ c12x8 ^ c11 ^ c9 ^ c4;
d3 = c13 ^ c11x8 ^ c10 ^ c8 ^ c3;
d2 = c14 ^ c10x8 ^ c9 ^ c2;
d1 = c14x8 ^ c13 ^ c9x8 ^ c8 ^ c1;
d0 = c13x8 ^ c8x8 ^ c0;
uint128_t d =
(((uint128_t) d7) << 112) |
(((uint128_t) d6) << 96) |
(((uint128_t) d5) << 80) |
(((uint128_t) d4) << 64) |
(((uint128_t) d3) << 48) |
(((uint128_t) d2) << 32) |
(((uint128_t) d1) << 16) |
((uint128_t) d0);
return d;
}
uint16_t gf2_128_mult_low_coeff( uint128_t a, uint128_t b ) {
uint16_t a0, a1, a2, a3, a4, a5, a6, a7;
uint16_t b0, b1, b2, b3, b4, b5, b6, b7;
uint16_t za0, za1, za2, za3, za4, za5, za6, za7;
uint16_t zb0, zb1, zb2, zb3, zb4, zb5, zb6, zb7;
uint32_t alog0, alog1, alog2, alog3;
uint32_t alog4, alog5, alog6, alog7;
uint32_t blog0, blog1, blog2, blog3;
uint32_t blog4, blog5, blog6, blog7;
uint16_t c0, c1, c2, c3, c4, c5, c6, c7;
uint16_t c8, c9, c10, c11, c12, c13, c14;
uint16_t d0;
a0 = (uint16_t) a;
a1 = (uint16_t) (a >> 16);
a2 = (uint16_t) (a >> 32);
a3 = (uint16_t) (a >> 48);
a4 = (uint16_t) (a >> 64);
a5 = (uint16_t) (a >> 80);
a6 = (uint16_t) (a >> 96);
a7 = (uint16_t) (a >> 112);
b0 = (uint16_t) b;
b1 = (uint16_t) (b >> 16);
b2 = (uint16_t) (b >> 32);
b3 = (uint16_t) (b >> 48);
b4 = (uint16_t) (b >> 64);
b5 = (uint16_t) (b >> 80);
b6 = (uint16_t) (b >> 96);
b7 = (uint16_t) (b >> 112);
alog0 = gf2_16_log_table[a0];
alog1 = gf2_16_log_table[a1];
alog2 = gf2_16_log_table[a2];
alog3 = gf2_16_log_table[a3];
alog4 = gf2_16_log_table[a4];
alog5 = gf2_16_log_table[a5];
alog6 = gf2_16_log_table[a6];
alog7 = gf2_16_log_table[a7];
blog0 = gf2_16_log_table[b0];
blog1 = gf2_16_log_table[b1];
blog2 = gf2_16_log_table[b2];
blog3 = gf2_16_log_table[b3];
blog4 = gf2_16_log_table[b4];
blog5 = gf2_16_log_table[b5];
blog6 = gf2_16_log_table[b6];
blog7 = gf2_16_log_table[b7];
za0 = zeroMask( a0 );
za1 = zeroMask( a1 );
za2 = zeroMask( a2 );
za3 = zeroMask( a3 );
za4 = zeroMask( a4 );
za5 = zeroMask( a5 );
za6 = zeroMask( a6 );
za7 = zeroMask( a7 );
zb0 = zeroMask( b0 );
zb1 = zeroMask( b1 );
zb2 = zeroMask( b2 );
zb3 = zeroMask( b3 );
zb4 = zeroMask( b4 );
zb5 = zeroMask( b5 );
zb6 = zeroMask( b6 );
zb7 = zeroMask( b7 );
c0 = gf2_16_expadd( za0|zb0, alog0, blog0 );
c8 = gf2_16_expadd( za1|zb7, alog1, blog7 );
c8 ^= gf2_16_expadd( za2|zb6, alog2, blog6 );
c8 ^= gf2_16_expadd( za3|zb5, alog3, blog5 );
c8 ^= gf2_16_expadd( za4|zb4, alog4, blog4 );
c8 ^= gf2_16_expadd( za5|zb3, alog5, blog3 );
c8 ^= gf2_16_expadd( za6|zb2, alog6, blog2 );
c8 ^= gf2_16_expadd( za7|zb1, alog7, blog1 );
c13 = gf2_16_expadd( za6|zb7, alog6, blog7 );
c13^= gf2_16_expadd( za7|zb6, alog7, blog6 );
// Now, modular reduction
uint16_t log8 = 3; // gf2_16_log_table[8];
uint16_t c13x8 = gf2_16_expadd( zeroMask(c13), gf2_16_log_table[c13], log8 );
uint16_t c8x8 = gf2_16_expadd( zeroMask( c8), gf2_16_log_table[ c8], log8 );
d0 = c13x8 ^ c8x8 ^ c0;
return d0;
}
uint128_t gf2_128_pointwise_mult( uint16_t xlog, uint128_t a ) {
uint16_t a0, a1, a2, a3, a4, a5, a6, a7;
uint16_t za0, za1, za2, za3, za4, za5, za6, za7;
uint32_t alog0, alog1, alog2, alog3;
uint32_t alog4, alog5, alog6, alog7;
uint16_t d0, d1, d2, d3, d4, d5, d6, d7;
a0 = (uint16_t) a;
a1 = (uint16_t) (a >> 16);
a2 = (uint16_t) (a >> 32);
a3 = (uint16_t) (a >> 48);
a4 = (uint16_t) (a >> 64);
a5 = (uint16_t) (a >> 80);
a6 = (uint16_t) (a >> 96);
a7 = (uint16_t) (a >> 112);
alog0 = gf2_16_log_table[a0];
alog1 = gf2_16_log_table[a1];
alog2 = gf2_16_log_table[a2];
alog3 = gf2_16_log_table[a3];
alog4 = gf2_16_log_table[a4];
alog5 = gf2_16_log_table[a5];
alog6 = gf2_16_log_table[a6];
alog7 = gf2_16_log_table[a7];
za0 = zeroMask( a0 );
za1 = zeroMask( a1 );
za2 = zeroMask( a2 );
za3 = zeroMask( a3 );
za4 = zeroMask( a4 );
za5 = zeroMask( a5 );
za6 = zeroMask( a6 );
za7 = zeroMask( a7 );
d7 = gf2_16_expadd( za7, alog7, xlog );
d6 = gf2_16_expadd( za6, alog6, xlog );
d5 = gf2_16_expadd( za5, alog5, xlog );
d4 = gf2_16_expadd( za4, alog4, xlog );
d3 = gf2_16_expadd( za3, alog3, xlog );
d2 = gf2_16_expadd( za2, alog2, xlog );
d1 = gf2_16_expadd( za1, alog1, xlog );
d0 = gf2_16_expadd( za0, alog0, xlog );
uint128_t d =
(((uint128_t) d7) << 112) |
(((uint128_t) d6) << 96) |
(((uint128_t) d5) << 80) |
(((uint128_t) d4) << 64) |
(((uint128_t) d3) << 48) |
(((uint128_t) d2) << 32) |
(((uint128_t) d1) << 16) |
((uint128_t) d0);
return d;
}
uint128_t gf2_128_square( uint128_t a ) {
uint16_t a0, a1, a2, a3, a4, a5, a6, a7;
uint16_t za0, za1, za2, za3, za4, za5, za6, za7;
uint32_t alog0, alog1, alog2, alog3;
uint32_t alog4, alog5, alog6, alog7;
uint16_t c0, c2, c4, c6;
uint16_t c8, c10, c12, c14;
uint16_t d0, d1, d2, d3, d4, d5, d6, d7;
a0 = (uint16_t) a;
a1 = (uint16_t) (a >> 16);
a2 = (uint16_t) (a >> 32);
a3 = (uint16_t) (a >> 48);
a4 = (uint16_t) (a >> 64);
a5 = (uint16_t) (a >> 80);
a6 = (uint16_t) (a >> 96);
a7 = (uint16_t) (a >> 112);
alog0 = gf2_16_log_table[a0];
alog1 = gf2_16_log_table[a1];
alog2 = gf2_16_log_table[a2];
alog3 = gf2_16_log_table[a3];
alog4 = gf2_16_log_table[a4];
alog5 = gf2_16_log_table[a5];
alog6 = gf2_16_log_table[a6];
alog7 = gf2_16_log_table[a7];
za0 = zeroMask( a0 );
za1 = zeroMask( a1 );
za2 = zeroMask( a2 );
za3 = zeroMask( a3 );
za4 = zeroMask( a4 );
za5 = zeroMask( a5 );
za6 = zeroMask( a6 );
za7 = zeroMask( a7 );
c0 = gf2_16_expadd( za0, alog0, alog0 );
c2 = gf2_16_expadd( za1, alog1, alog1 );
c4 = gf2_16_expadd( za2, alog2, alog2 );
c6 = gf2_16_expadd( za3, alog3, alog3 );
c8 = gf2_16_expadd( za4, alog4, alog4 );
c10 = gf2_16_expadd( za5, alog5, alog5 );
c12 = gf2_16_expadd( za6, alog6, alog6 );
c14 = gf2_16_expadd( za7, alog7, alog7 );
// Now, modular reduction
uint16_t log8 = 3; // gf2_16_log_table[8];
uint16_t c14x8 = gf2_16_expadd( zeroMask(c14), gf2_16_log_table[c14], log8 );
uint16_t c12x8 = gf2_16_expadd( zeroMask(c12), gf2_16_log_table[c12], log8 );
uint16_t c10x8 = gf2_16_expadd( zeroMask(c10), gf2_16_log_table[c10], log8 );
uint16_t c8x8 = gf2_16_expadd( zeroMask( c8), gf2_16_log_table[ c8], log8 );
d7 = c14 ^ c12;
d6 = c14x8 ^ c6;
d5 = c12 ^ c10;
d4 = c14 ^ c12x8 ^ c4;
d3 = c10 ^ c8;
d2 = c14 ^ c10x8 ^ c2;
d1 = c14x8 ^ c8;
d0 = c8x8 ^ c0;
uint128_t d =
(((uint128_t) d7) << 112) |
(((uint128_t) d6) << 96) |
(((uint128_t) d5) << 80) |
(((uint128_t) d4) << 64) |
(((uint128_t) d3) << 48) |
(((uint128_t) d2) << 32) |
(((uint128_t) d1) << 16) |
((uint128_t) d0);
return d;
}
/*
2d3c 6cfa b56b a301 6049 5dd2 71eb 0000
fa8d fecd a42d 5693 4c80 f18a 9fc9 0000
2d3d 6cfa b56a a301 6049 5dd2 71eb 0000
297c fdfd 41f5 ed81 758c 1587 6664 0000
21ab 4b1a 16f5 f846 cc9d 1b72 355d 0000
a04c 3c06 57ce b6ef 58b8 e8bc 67de 0000
ce94 5686 745d 11f9 376d 14ec af3a 0000
f2c4 81c3 d3a4 d2ec 372a d568 7232 0001
*/
uint128_t gf2_128_square16( uint128_t a ) {
uint16_t a0, a1, a2, a3, a4, a5, a6, a7;
uint16_t za1, za2, za3, za4, za5, za6, za7;
uint32_t alog1, alog2, alog3;
uint32_t alog4, alog5, alog6, alog7;
uint16_t d0, d1, d2, d3, d4, d5, d6, d7;
a0 = (uint16_t) a;
a1 = (uint16_t) (a >> 16);
a2 = (uint16_t) (a >> 32);
a3 = (uint16_t) (a >> 48);
a4 = (uint16_t) (a >> 64);
a5 = (uint16_t) (a >> 80);
a6 = (uint16_t) (a >> 96);
a7 = (uint16_t) (a >> 112);
alog1 = gf2_16_log_table[a1];
alog2 = gf2_16_log_table[a2];
alog3 = gf2_16_log_table[a3];
alog4 = gf2_16_log_table[a4];
alog5 = gf2_16_log_table[a5];
alog6 = gf2_16_log_table[a6];
alog7 = gf2_16_log_table[a7];
za1 = zeroMask( a1 );
za2 = zeroMask( a2 );
za3 = zeroMask( a3 );
za4 = zeroMask( a4 );
za5 = zeroMask( a5 );
za6 = zeroMask( a6 );
za7 = zeroMask( a7 );
d7 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0x2d3c ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0x6cfa ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0xb56b ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0xa301 ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0x6049 ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0x5dd2 ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0x71eb ] );
d6 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0xfa8d ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0xfecd ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0xa42d ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0x5693 ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0x4c80 ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0xf18a ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0x9fc9 ] );
d5 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0x2d3d ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0x6cfa ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0xb56a ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0xa301 ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0x6049 ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0x5dd2 ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0x71eb ] );
d4 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0x297c ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0xfdfd ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0x41f5 ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0xed81 ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0x758c ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0x1587 ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0x6664 ] );
d3 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0x21ab ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0x4b1a ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0x16f5 ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0xf846 ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0xcc9d ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0x1b72 ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0x355d ] );
d2 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0xa04c ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0x3c06 ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0x57ce ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0xb6ef ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0x58b8 ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0xe8bc ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0x67de ] );
d1 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0xce94 ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0x5686 ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0x745d ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0x11f9 ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0x376d ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0x14ec ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0xaf3a ] );
d0 = gf2_16_expadd( za7, alog7, gf2_16_log_table[ 0xf2c4 ] ) ^
gf2_16_expadd( za6, alog6, gf2_16_log_table[ 0x81c3 ] ) ^
gf2_16_expadd( za5, alog5, gf2_16_log_table[ 0xd3a4 ] ) ^
gf2_16_expadd( za4, alog4, gf2_16_log_table[ 0xd2ec ] ) ^
gf2_16_expadd( za3, alog3, gf2_16_log_table[ 0x372a ] ) ^
gf2_16_expadd( za2, alog2, gf2_16_log_table[ 0xd568 ] ) ^
gf2_16_expadd( za1, alog1, gf2_16_log_table[ 0x7232 ] ) ^
a0;
uint128_t d =
(((uint128_t) d7) << 112) |
(((uint128_t) d6) << 96) |
(((uint128_t) d5) << 80) |
(((uint128_t) d4) << 64) |
(((uint128_t) d3) << 48) |
(((uint128_t) d2) << 32) |
(((uint128_t) d1) << 16) |
((uint128_t) d0);
return d;
}
uint128_t gf2_128_inv( uint128_t a ) {
// Let q = 2^16. Let r = (q^8 - 1)/(q - 1) = 2^112 + 2^96 + 2^80 + 2^64 + 2^48 + 2^32 + 2^16 + 1
// This rather special number has to do with caluclating finite field norms.
// For all x in GF((2^16)^8), x^r yields a value in GF(2^16); that is, for which
// the high coefficents are 0. We exploit this fact to perform fast inversions
// in GF(2^128) by reducing them to inversion in GF(2^16).
//
// We are going to calculate a^(-1) = a^(-r) * a^(r-1). The algorithm below goes
// roughly as follows:
//
// s = a^(r-1)
// = a^(q^7 + q^6 + q^5 + q^4 + q^3 + q^2 + q)
// = ((((((a^q * a)^q *a)^q * a)^q * a)^q * a)^q * a)^q
// t = a * s = a^r
// b = t^(-1) * s = a^(-1)
// Compute s = a^(r-1)
uint128_t s = a;
for(int i=0;;i++) {
s = gf2_128_square16( s );
if( i>=6 ) break;
s = gf2_128_mult( s, a );
}
// t = s * a = a^r
uint16_t t0 = gf2_128_mult_low_coeff( s, a );
uint16_t t0_inv_log = Q - gf2_16_log_table[ t0 ];
// b = t^(-1) * s = a^(-r) * a^(r-1) = a(-1)
uint128_t b = gf2_128_pointwise_mult( t0_inv_log, s );
return b;
}