467 lines
21 KiB
C
467 lines
21 KiB
C
#include "fft.h"
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#include "gf.h"
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#include "parameters.h"
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#include "parsing.h"
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#include "reed_solomon.h"
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#include <stdint.h>
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#include <stdio.h>
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#include <string.h>
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/**
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* @file reed_solomon.c
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* Constant time implementation of Reed-Solomon codes
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*/
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static void compute_syndromes(uint16_t *syndromes, uint8_t *cdw);
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static uint16_t compute_elp(uint16_t *sigma, const uint16_t *syndromes);
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static void compute_roots(uint8_t *error, uint16_t *sigma);
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static void compute_z_poly(uint16_t *z, const uint16_t *sigma, uint16_t degree, const uint16_t *syndromes);
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static void compute_error_values(uint16_t *error_values, const uint16_t *z, const uint8_t *error);
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static void correct_errors(uint8_t *cdw, const uint16_t *error_values);
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static const __m256i alpha_ij256_1[45] = {
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{0x0010000800040002, 0x001d008000400020, 0x00cd00e80074003a, 0x004c002600130087},
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{0x001d004000100004, 0x004c001300cd0074, 0x008f00ea00b4002d, 0x009d006000180006},
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{0x00cd003a00400008, 0x008f0075002d0026, 0x002500270060000c, 0x004600c100b50035},
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{0x004c00cd001d0010, 0x009d0018008f00b4, 0x004600ee006a0025, 0x005f00b9005d0014},
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{0x00b4002600740020, 0x006a009c00600003, 0x00b900a0000500c1, 0x00fd000f005e00be},
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{0x008f002d00cd0040, 0x004600b500250060, 0x0065006100b90050, 0x00d900df006b0078},
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{0x0018007500130080, 0x005d008c00b5009c, 0x006b003c005e00a1, 0x0081001a004300a3},
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{0x009d008f004c001d, 0x005f005d0046006a, 0x00d900fe00fd0065, 0x0085003b0081000d},
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{0x0025000c002d003a, 0x006500a1005000c1, 0x00d0008600df00e7, 0x00a800a9006600ed},
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{0x006a006000b40074, 0x00fd005e00b90005, 0x003b0067001100df, 0x00e600550084002e},
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{0x00ee002700ea00e8, 0x00fe003c006100a0, 0x00b8007600670086, 0x00e3009100390054},
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{0x00460025008f00cd, 0x00d9006b006500b9, 0x00a800b8003b00d0, 0x0082009600fc00e4},
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{0x0014003500060087, 0x000d00a3007800be, 0x00e40054002e00ed, 0x00510064006200e5},
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{0x005d00b500180013, 0x00810043006b005e, 0x00fc003900840066, 0x0012005900c80062},
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{0x00b900c100600026, 0x003b001a00df000f, 0x00960091005500a9, 0x002c002400590064},
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{0x005f0046009d004c, 0x0085008100d900fd, 0x008200e300e600a8, 0x0002002c00120051},
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{0x0099000a004e0098, 0x004f0093004400d6, 0x00dd00dc00d70092, 0x00980001000b0045},
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{0x006500500025002d, 0x00a8006600d000df, 0x00c30007009600bf, 0x0027002600ad00fb},
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{0x001e00ba0094005a, 0x0049006d003e00e2, 0x003d00a200ae00b3, 0x008c006000e80083},
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{0x00fd00b9006a00b4, 0x00e60084003b0011, 0x002c00ac001c0096, 0x00be00c100030020},
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{0x006b00a100b50075, 0x00fc00290066001a, 0x00ad00f500590057, 0x00e700b90035002d},
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{0x00fe006100ee00ea, 0x00e3003900b80067, 0x003a00b000ac0007, 0x00af000f002800c0},
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{0x005b002f009f00c9, 0x009500d10021007c, 0x0075004700f400a6, 0x001f00df00c200ee},
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{0x00d900650046008f, 0x008200fc00a8003b, 0x0027003a002c00c3, 0x0017001a00e700ba},
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{0x0011000f00050003, 0x001c00ff00550033, 0x00c100b4006c0024, 0x004d003b00e2005e},
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{0x000d007800140006, 0x0051006200e4002e, 0x00ba00c0002000fb, 0x00d100a900bd00bb},
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{0x00d000e70050000c, 0x00c3005700bf00a9, 0x002f00b50026007d, 0x00db005500c500d9},
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{0x0081006b005d0018, 0x001200c800fc0084, 0x00e70028000300ad, 0x00190091009e00bd},
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{0x00f8007f00690030, 0x00f700e000f1004d, 0x00b6005f009c0040, 0x00a2009600aa00ec},
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{0x003b00df00b90060, 0x002c005900960055, 0x001a000f00c10026, 0x00240064009100a9},
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{0x009700b600de00c0, 0x001b009b006e0072, 0x00ed00b100a0008f, 0x00580059004b0052},
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{0x008500d9005f009d, 0x00020012008200e6, 0x001700af00be0027, 0x00040024001900d1},
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{0x00b8008600610027, 0x003a00f500070091, 0x001500d0000f00b5, 0x002d002c00a600f1},
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{0x004f00440099004e, 0x0098000b00dd00d7, 0x0092009300d6000a, 0x004e0001004500dc},
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{0x0084001a005e009c, 0x000300e9005900ff, 0x0091002e00e200b9, 0x0005002600eb001c},
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{0x00a800d000650025, 0x002700ad00c30096, 0x00db0015001a002f, 0x00610060003600f2},
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{0x005200ce0089004a, 0x00d40010008a0037, 0x00570049007c0078, 0x00d300c1001d0048},
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{0x0049003e001e0094, 0x008c00e8003d00ae, 0x003800630033007f, 0x004300b900ea0016},
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{0x00e400ed00780035, 0x00ba002d00fb0064, 0x00f200f100a900d9, 0x003e000f002500ad},
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{0x00e6003b00fd006a, 0x00be0003002c001c, 0x00240037004d001a, 0x002e00df00050074},
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{0x00c600c500d300d4, 0x00ca009d00cf00a7, 0x008b00c80072003e, 0x009a001a005f00c9},
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{0x00fc0066006b00b5, 0x00e7003500ad0059, 0x003600a6009100c5, 0x00bf003b00780025},
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{0x007b001700b10077, 0x00e1009f000800ef, 0x0040002b00ff00b8, 0x00ab00a9005b008c},
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{0x00e300b800fe00ee, 0x00af0028003a00ac, 0x002d007a00370015, 0x00320055003400de},
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{0x009600a900df00c1, 0x001a00b900260024, 0x0060002c00640055, 0x00590091003b000f}
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};
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static const __m256i alpha_ij256_2[45] = {
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{0x00b4005a002d0098, 0x008f00c900ea0075, 0x0018000c00060003, 0x0000000000600030},
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{0x006a00940025004e, 0x0046009f00ee00b5, 0x005d005000140005, 0x0000000000b90069},
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{0x00b900ba0050000a, 0x0065002f006100a1, 0x006b00e70078000f, 0x0000000000df007f},
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{0x00fd001e00650099, 0x00d9005b00fe006b, 0x008100d0000d0011, 0x00000000003b00f8},
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{0x001100e200df00d6, 0x003b007c0067001a, 0x008400a9002e0033, 0x000000000055004d},
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{0x003b003e00d00044, 0x00a8002100b80066, 0x00fc00bf00e40055, 0x00000000009600f1},
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{0x0084006d00660093, 0x00fc00d100390029, 0x00c80057006200ff, 0x00000000005900e0},
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{0x00e6004900a8004f, 0x0082009500e300fc, 0x001200c30051001c, 0x00000000002c00f7},
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{0x009600b300bf0092, 0x00c300a600070057, 0x00ad007d00fb0024, 0x0000000000260040},
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{0x001c00ae009600d7, 0x002c00f400ac0059, 0x000300260020006c, 0x0000000000c1009c},
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{0x00ac00a2000700dc, 0x003a004700b000f5, 0x002800b500c000b4, 0x00000000000f005f},
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{0x002c003d00c300dd, 0x00270075003a00ad, 0x00e7002f00ba00c1, 0x00000000001a00b6},
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{0x0020008300fb0045, 0x00ba00ee00c0002d, 0x00bd00d900bb005e, 0x0000000000a900ec},
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{0x000300e800ad000b, 0x00e700c200280035, 0x009e00c500bd00e2, 0x00000000009100aa},
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{0x00c1006000260001, 0x001a00df000f00b9, 0x0091005500a9003b, 0x0000000000640096},
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{0x00be008c00270098, 0x0017001f00af00e7, 0x001900db00d1004d, 0x00000000002400a2},
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{0x00d60099000a004e, 0x0092004f00930044, 0x004500dd00dc00d7, 0x000000000001000b},
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{0x001a007f002f000a, 0x00db0073001500c5, 0x003600f500f20064, 0x00000000006000cd},
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{0x00330034007f0099, 0x00380062006300a8, 0x00ea0008001600ac, 0x0000000000b900d4},
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{0x004d0033001a00d6, 0x002400a700370091, 0x00050060007400e9, 0x0000000000df005e},
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{0x009100a800c50044, 0x0036003d00a6006e, 0x007800ba00250026, 0x00000000003b0086},
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{0x0037006300150093, 0x002d00d8007a00a6, 0x0034006b00de006a, 0x0000000000550085},
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{0x00a700620073004f, 0x00b5005a00d8003d, 0x00da00ce00fe00be, 0x00000000009600d5},
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{0x0024003800db0092, 0x006100b5002d0036, 0x00bf0021003e00df, 0x000000000059006e},
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{0x00e900ac006400d7, 0x00df00be006a0026, 0x00ae00910084007c, 0x00000000002c00ef},
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{0x0074001600f200dc, 0x003e00fe00de0025, 0x002b0082003f0084, 0x00000000002600fa},
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{0x0060000800f500dd, 0x002100ce006b00ba, 0x00cf005600820091, 0x0000000000c1002d},
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{0x000500ea00360045, 0x00bf00da00340078, 0x005a00cf002b00ae, 0x00000000000f0023},
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{0x005e00d400cd000b, 0x006e00d500850086, 0x0023002d00fa00ef, 0x00000000001a001e},
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{0x00df00b900600001, 0x005900960055003b, 0x000f00c10026002c, 0x0000000000a9001a},
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{0x006700f000460098, 0x00fb00e0007b0015, 0x0088006500d40074, 0x00000000009100da},
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{0x002e00430061004e, 0x00080048003200bf, 0x005c008600c2009c, 0x0000000000640063},
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{0x005500ed006b000a, 0x000c003600c300c4, 0x0073006600b600b9, 0x0000000000240082},
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{0x00d7004f00440099, 0x000a0098000b00dd, 0x00dc0092009300d6, 0x0000000000010045},
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{0x00ae0072003b00d6, 0x000f006a00200024, 0x00ef0096004d0067, 0x000000000060006c},
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{0x005900f100210044, 0x008600a1000c00cf, 0x007d00a600b300a9, 0x0000000000b9008f},
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{0x00f4001900e40093, 0x00c500b1008c00cd, 0x004c00fb008d00e6, 0x0000000000df0028},
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{0x006c007900f1004f, 0x002900bd00bc0027, 0x00ee004000090037, 0x00000000003b00d3},
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{0x002600f500820092, 0x00b300b800b60050, 0x0065002700360059, 0x00000000005500ce},
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{0x009c006c005900d7, 0x00640072007c000f, 0x001100b900b400eb, 0x0000000000960084},
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{0x00a00013003d00dc, 0x005600ab009e00d9, 0x0085007f009f0020, 0x00000000005900e5},
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{0x000f002700cf00dd, 0x007d0038007300ed, 0x00e4003e00650060, 0x00000000002c0007},
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{0x00e20014003a0045, 0x00cd001200310021, 0x00950015004300a0, 0x0000000000260090},
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{0x007c00bc000c000b, 0x0025008300e00073, 0x007900fc009700fd, 0x0000000000c10002},
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{0x00a900df00c10001, 0x00b9002600240096, 0x002c00640055001a, 0x00000000000f0060}
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};
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/**
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* @brief Encodes a message message of PARAM_K bits to a Reed-Solomon codeword codeword of PARAM_N1 bytes
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*
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* Following @cite lin1983error (Chapter 4 - Cyclic Codes),
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* We perform a systematic encoding using a linear (PARAM_N1 - PARAM_K)-stage shift register
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* with feedback connections based on the generator polynomial PARAM_RS_POLY of the Reed-Solomon code.
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*
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* @param[out] cdw Array of size VEC_N1_SIZE_64 receiving the encoded message
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* @param[in] msg Array of size VEC_K_SIZE_64 storing the message
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*/
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void PQCLEAN_HQCRMRS128_AVX2_reed_solomon_encode(uint8_t *cdw, const uint8_t *msg) {
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size_t i, k;
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uint8_t gate_value = 0;
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uint8_t prev, x;
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union {
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uint16_t arr16[16 * CEIL_DIVIDE(PARAM_G, 16)];
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__m256i dummy;
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} tmp = {0};
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union {
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uint16_t arr16[16 * CEIL_DIVIDE(PARAM_G, 16)];
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__m256i dummy;
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} PARAM_RS_POLY = {{ RS_POLY_COEFS }};
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__m256i *tmp256 = (__m256i *)tmp.arr16;
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__m256i *param256 = (__m256i *)PARAM_RS_POLY.arr16;
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for (i = 0; i < PARAM_K; ++i) {
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gate_value = (uint8_t) (msg[PARAM_K - 1 - i] ^ cdw[PARAM_N1 - PARAM_K - 1]);
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_mm256_storeu_si256(&tmp256[0], PQCLEAN_HQCRMRS128_AVX2_gf_mul_vect(_mm256_set1_epi16(gate_value), param256[0]));
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_mm256_storeu_si256(&tmp256[1], PQCLEAN_HQCRMRS128_AVX2_gf_mul_vect(_mm256_set1_epi16(gate_value), param256[1]));
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prev = 0;
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for (k = 0; k < PARAM_N1 - PARAM_K; k++) {
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x = cdw[k];
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cdw[k] = (uint8_t) (prev ^ tmp.arr16[k]);
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prev = x;
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}
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}
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memcpy(cdw + PARAM_N1 - PARAM_K, msg, PARAM_K);
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}
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/**
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* @brief Computes 2 * PARAM_DELTA syndromes
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*
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* @param[out] syndromes Array of size 2 * PARAM_DELTA receiving the computed syndromes
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* @param[in] cdw Array of size PARAM_N1 storing the received vector
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*/
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void compute_syndromes(uint16_t *syndromes, uint8_t *cdw) {
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__m256i *syndromes256 = (__m256i *) syndromes;
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__m256i last_syndromes256;
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syndromes256[0] = _mm256_set1_epi16(cdw[0]);
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for (size_t i = 0; i < PARAM_N1 - 1; ++i) {
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syndromes256[0] ^= PQCLEAN_HQCRMRS128_AVX2_gf_mul_vect(_mm256_set1_epi16(cdw[i + 1]), alpha_ij256_1[i]);
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}
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last_syndromes256 = _mm256_set1_epi16(cdw[0]);
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for (size_t i = 0; i < PARAM_N1 - 1; ++i) {
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last_syndromes256 ^= PQCLEAN_HQCRMRS128_AVX2_gf_mul_vect(_mm256_set1_epi16(cdw[i + 1]), alpha_ij256_2[i]);
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}
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__m128i *s128 = (__m128i *) &last_syndromes256;
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_mm_store_si128((__m128i *) (syndromes + 16), *s128);
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uint64_t *s8 = (uint64_t *) (syndromes + 24);
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s8[0] = _mm_extract_epi64(s128[1], 0);
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uint32_t *s12 = (uint32_t *) (syndromes + 28);
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uint32_t *s32 = ((uint32_t *) &last_syndromes256) + 6;
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s12[0] = *s32;
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}
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/**
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* @brief Computes the error locator polynomial (ELP) sigma
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*
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* This is a constant time implementation of Berlekamp's simplified algorithm (see @cite lin1983error (Chapter 6 - BCH Codes). <br>
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* We use the letter p for rho which is initialized at -1. <br>
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* The array X_sigma_p represents the polynomial X^(mu-rho)*sigma_p(X). <br>
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* Instead of maintaining a list of sigmas, we update in place both sigma and X_sigma_p. <br>
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* sigma_copy serves as a temporary save of sigma in case X_sigma_p needs to be updated. <br>
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* We can properly correct only if the degree of sigma does not exceed PARAM_DELTA.
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* This means only the first PARAM_DELTA + 1 coefficients of sigma are of value
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* and we only need to save its first PARAM_DELTA - 1 coefficients.
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*
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* @returns the degree of the ELP sigma
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* @param[out] sigma Array of size (at least) PARAM_DELTA receiving the ELP
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* @param[in] syndromes Array of size (at least) 2*PARAM_DELTA storing the syndromes
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*/
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static uint16_t compute_elp(uint16_t *sigma, const uint16_t *syndromes) {
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uint16_t deg_sigma = 0;
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uint16_t deg_sigma_p = 0;
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uint16_t deg_sigma_copy = 0;
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uint16_t sigma_copy[PARAM_DELTA + 1] = {0};
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uint16_t X_sigma_p[PARAM_DELTA + 1] = {0, 1};
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uint16_t pp = (uint16_t) -1; // 2*rho
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uint16_t d_p = 1;
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uint16_t d = syndromes[0];
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uint16_t mask1, mask2, mask12;
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uint16_t deg_X, deg_X_sigma_p;
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uint16_t dd;
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uint16_t mu;
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uint16_t i;
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sigma[0] = 1;
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for (mu = 0; (mu < (2 * PARAM_DELTA)); ++mu) {
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// Save sigma in case we need it to update X_sigma_p
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memcpy(sigma_copy, sigma, 2 * (PARAM_DELTA));
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deg_sigma_copy = deg_sigma;
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dd = PQCLEAN_HQCRMRS128_AVX2_gf_mul(d, PQCLEAN_HQCRMRS128_AVX2_gf_inverse(d_p));
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for (i = 1; (i <= mu + 1) && (i <= PARAM_DELTA); ++i) {
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sigma[i] ^= PQCLEAN_HQCRMRS128_AVX2_gf_mul(dd, X_sigma_p[i]);
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}
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deg_X = mu - pp;
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deg_X_sigma_p = deg_X + deg_sigma_p;
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// mask1 = 0xffff if(d != 0) and 0 otherwise
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mask1 = -((uint16_t) - d >> 15);
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// mask2 = 0xffff if(deg_X_sigma_p > deg_sigma) and 0 otherwise
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mask2 = -((uint16_t) (deg_sigma - deg_X_sigma_p) >> 15);
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// mask12 = 0xffff if the deg_sigma increased and 0 otherwise
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mask12 = mask1 & mask2;
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deg_sigma ^= mask12 & (deg_X_sigma_p ^ deg_sigma);
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if (mu == (2 * PARAM_DELTA - 1)) {
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break;
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}
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pp ^= mask12 & (mu ^ pp);
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d_p ^= mask12 & (d ^ d_p);
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for (i = PARAM_DELTA; i; --i) {
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X_sigma_p[i] = (mask12 & sigma_copy[i - 1]) ^ (~mask12 & X_sigma_p[i - 1]);
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}
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deg_sigma_p ^= mask12 & (deg_sigma_copy ^ deg_sigma_p);
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d = syndromes[mu + 1];
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for (i = 1; (i <= mu + 1) && (i <= PARAM_DELTA); ++i) {
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d ^= PQCLEAN_HQCRMRS128_AVX2_gf_mul(sigma[i], syndromes[mu + 1 - i]);
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}
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}
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return deg_sigma;
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}
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/**
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* @brief Computes the error polynomial error from the error locator polynomial sigma
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*
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* See function PQCLEAN_HQCRMRS128_AVX2_fft for more details.
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*
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* @param[out] error Array of 2^PARAM_M elements receiving the error polynomial
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* @param[out] error_compact Array of PARAM_DELTA + PARAM_N1 elements receiving a compact representation of the vector error
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* @param[in] sigma Array of 2^PARAM_FFT elements storing the error locator polynomial
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*/
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static void compute_roots(uint8_t *error, uint16_t *sigma) {
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uint16_t w[1 << PARAM_M] = {0};
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PQCLEAN_HQCRMRS128_AVX2_fft(w, sigma, PARAM_DELTA + 1);
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PQCLEAN_HQCRMRS128_AVX2_fft_retrieve_error_poly(error, w);
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}
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/**
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* @brief Computes the polynomial z(x)
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*
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* See @cite lin1983error (Chapter 6 - BCH Codes) for more details.
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*
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* @param[out] z Array of PARAM_DELTA + 1 elements receiving the polynomial z(x)
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* @param[in] sigma Array of 2^PARAM_FFT elements storing the error locator polynomial
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* @param[in] degree Integer that is the degree of polynomial sigma
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* @param[in] syndromes Array of 2 * PARAM_DELTA storing the syndromes
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*/
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static void compute_z_poly(uint16_t *z, const uint16_t *sigma, uint16_t degree, const uint16_t *syndromes) {
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size_t i, j;
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uint16_t mask;
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z[0] = 1;
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for (i = 1; i < PARAM_DELTA + 1; ++i) {
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mask = -((uint16_t) (i - degree - 1) >> 15);
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z[i] = mask & sigma[i];
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}
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z[1] ^= syndromes[0];
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for (i = 2; i <= PARAM_DELTA; ++i) {
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mask = -((uint16_t) (i - degree - 1) >> 15);
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z[i] ^= mask & syndromes[i - 1];
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for (j = 1; j < i; ++j) {
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z[i] ^= mask & PQCLEAN_HQCRMRS128_AVX2_gf_mul(sigma[j], syndromes[i - j - 1]);
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}
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}
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}
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/**
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* @brief Computes the error values
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*
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* See @cite lin1983error (Chapter 6 - BCH Codes) for more details.
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*
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* @param[out] error_values Array of PARAM_DELTA elements receiving the error values
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* @param[in] z Array of PARAM_DELTA + 1 elements storing the polynomial z(x)
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* @param[in] z_degree Integer that is the degree of polynomial z(x)
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* @param[in] error_compact Array of PARAM_DELTA + PARAM_N1 storing compact representation of the error
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*/
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static void compute_error_values(uint16_t *error_values, const uint16_t *z, const uint8_t *error) {
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uint16_t beta_j[PARAM_DELTA] = {0};
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uint16_t e_j[PARAM_DELTA] = {0};
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uint16_t delta_counter;
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uint16_t delta_real_value;
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uint16_t found;
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uint16_t mask1;
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uint16_t mask2;
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uint16_t tmp1;
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uint16_t tmp2;
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uint16_t inverse;
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uint16_t inverse_power_j;
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// Compute the beta_{j_i} page 31 of the documentation
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delta_counter = 0;
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for (size_t i = 0; i < PARAM_N1; i++) {
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found = 0;
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mask1 = (uint16_t) (-((int32_t)error[i]) >> 31); // error[i] != 0
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for (size_t j = 0; j < PARAM_DELTA; j++) {
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mask2 = ~((uint16_t) (-((int32_t) j ^ delta_counter) >> 31)); // j == delta_counter
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beta_j[j] += mask1 & mask2 & gf_exp[i];
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found += mask1 & mask2 & 1;
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}
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delta_counter += found;
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}
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delta_real_value = delta_counter;
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// Compute the e_{j_i} page 31 of the documentation
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for (size_t i = 0; i < PARAM_DELTA; ++i) {
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tmp1 = 1;
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tmp2 = 1;
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inverse = PQCLEAN_HQCRMRS128_AVX2_gf_inverse(beta_j[i]);
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inverse_power_j = 1;
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for (size_t j = 1; j <= PARAM_DELTA; ++j) {
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inverse_power_j = PQCLEAN_HQCRMRS128_AVX2_gf_mul(inverse_power_j, inverse);
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tmp1 ^= PQCLEAN_HQCRMRS128_AVX2_gf_mul(inverse_power_j, z[j]);
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}
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for (size_t k = 1; k < PARAM_DELTA; ++k) {
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tmp2 = PQCLEAN_HQCRMRS128_AVX2_gf_mul(tmp2, (1 ^ PQCLEAN_HQCRMRS128_AVX2_gf_mul(inverse, beta_j[(i + k) % PARAM_DELTA])));
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}
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mask1 = (uint16_t) (((int16_t) i - delta_real_value) >> 15); // i < delta_real_value
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e_j[i] = mask1 & PQCLEAN_HQCRMRS128_AVX2_gf_mul(tmp1, PQCLEAN_HQCRMRS128_AVX2_gf_inverse(tmp2));
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}
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// Place the delta e_{j_i} values at the right coordinates of the output vector
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delta_counter = 0;
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for (size_t i = 0; i < PARAM_N1; ++i) {
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found = 0;
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mask1 = (uint16_t) (-((int32_t)error[i]) >> 31); // error[i] != 0
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for (size_t j = 0; j < PARAM_DELTA; j++) {
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mask2 = ~((uint16_t) (-((int32_t) j ^ delta_counter) >> 31)); // j == delta_counter
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error_values[i] += mask1 & mask2 & e_j[j];
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found += mask1 & mask2 & 1;
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}
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delta_counter += found;
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}
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}
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/**
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* @brief Correct the errors
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*
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* @param[out] cdw Array of PARAM_N1 elements receiving the corrected vector
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* @param[in] error Array of the error vector
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* @param[in] error_values Array of PARAM_DELTA elements storing the error values
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*/
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static void correct_errors(uint8_t *cdw, const uint16_t *error_values) {
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for (size_t i = 0; i < PARAM_N1; ++i) {
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cdw[i] ^= error_values[i];
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}
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}
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/**
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* @brief Decodes the received word
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*
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* This function relies on six steps:
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* <ol>
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* <li> The first step, is the computation of the 2*PARAM_DELTA syndromes.
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* <li> The second step is the computation of the error-locator polynomial sigma.
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* <li> The third step, done by additive FFT, is finding the error-locator numbers by calculating the roots of the polynomial sigma and takings their inverses.
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* <li> The fourth step, is the polynomial z(x).
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* <li> The fifth step, is the computation of the error values.
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* <li> The sixth step is the correction of the errors in the received polynomial.
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* </ol>
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* For a more complete picture on Reed-Solomon decoding, see Shu. Lin and Daniel J. Costello in Error Control Coding: Fundamentals and Applications @cite lin1983error
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*
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* @param[out] msg Array of size VEC_K_SIZE_64 receiving the decoded message
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* @param[in] cdw Array of size VEC_N1_SIZE_64 storing the received word
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*/
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void PQCLEAN_HQCRMRS128_AVX2_reed_solomon_decode(uint8_t *msg, uint8_t *cdw) {
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uint16_t syndromes[2 * PARAM_DELTA] = {0};
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uint16_t sigma[1 << PARAM_FFT] = {0};
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uint8_t error[1 << PARAM_M] = {0};
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uint16_t z[PARAM_N1] = {0};
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uint16_t error_values[PARAM_N1] = {0};
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uint16_t deg;
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// Calculate the 2*PARAM_DELTA syndromes
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compute_syndromes(syndromes, cdw);
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// Compute the error locator polynomial sigma
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// Sigma's degree is at most PARAM_DELTA but the FFT requires the extra room
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deg = compute_elp(sigma, syndromes);
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// Compute the error polynomial error
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compute_roots(error, sigma);
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// Compute the polynomial z(x)
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compute_z_poly(z, sigma, deg, syndromes);
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// Compute the error values
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compute_error_values(error_values, z, error);
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// Correct the errors
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correct_errors(cdw, error_values);
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// Retrieve the message from the decoded codeword
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memcpy(msg, cdw + (PARAM_G - 1), PARAM_K);
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}
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