350 lines
12 KiB
C
350 lines
12 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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/**
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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_CLEAN_reed_solomon_encode(uint8_t *cdw, const uint8_t *msg) {
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size_t i, j, k;
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uint8_t gate_value = 0;
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uint16_t tmp[PARAM_G] = {0};
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uint16_t PARAM_RS_POLY [] = {RS_POLY_COEFS};
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uint8_t prev, x;
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for (i = 0; i < PARAM_N1; ++i) {
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cdw[i] = 0;
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}
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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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for (j = 0; j < PARAM_G; ++j) {
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tmp[j] = PQCLEAN_HQCRMRS128_CLEAN_gf_mul(gate_value, PARAM_RS_POLY[j]);
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}
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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[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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for (size_t i = 0; i < 2 * PARAM_DELTA; ++i) {
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for (size_t j = 1; j < PARAM_N1; ++j) {
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syndromes[i] ^= PQCLEAN_HQCRMRS128_CLEAN_gf_mul(cdw[j], alpha_ij_pow[i][j - 1]);
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}
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syndromes[i] ^= cdw[0];
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}
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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_CLEAN_gf_mul(d, PQCLEAN_HQCRMRS128_CLEAN_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_CLEAN_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_CLEAN_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_CLEAN_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_CLEAN_fft(w, sigma, PARAM_DELTA + 1);
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PQCLEAN_HQCRMRS128_CLEAN_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_CLEAN_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_CLEAN_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_CLEAN_gf_mul(inverse_power_j, inverse);
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tmp1 ^= PQCLEAN_HQCRMRS128_CLEAN_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_CLEAN_gf_mul(tmp2, (1 ^ PQCLEAN_HQCRMRS128_CLEAN_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_CLEAN_gf_mul(tmp1, PQCLEAN_HQCRMRS128_CLEAN_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_CLEAN_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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