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cauchy_prime_field_erasure_coding.hh
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/*
Cauchy Prime Field Erasure Coding
Copyright 2024 Ahmet Inan <[email protected]>
*/
#pragma once
namespace CODE {
template <typename PF, typename IO, int MAX_LEN>
struct CauchyPrimeFieldErasureCoding
{
static_assert(MAX_LEN < int(PF::P-1), "Block length must be smaller than largest field value");
PF temp[MAX_LEN];
uint8_t used[(PF::P+7)/8];
PF row_num, row_den;
// $a_{ij} = \frac{1}{x_i + y_j}$
PF cauchy_matrix(int i, int j)
{
PF row(i), col(j);
return rcp(row + col);
}
// $b_{ij} = \frac{\prod_{k=1}^{n}{(x_j + y_k)(x_k + y_i)}}{(x_j + y_i)\prod_{k \ne j}^{n}{(x_j - x_k)}\prod_{k \ne i}^{n}{(y_i - y_k)}}$
PF inverse_cauchy_matrix(const IO *rows, int i, int j, int n)
{
#if 0
PF row_j(rows[j]), col_i(i);
PF prod_xy(1), prod_x(1), prod_y(1);
for (int k = 0; k < n; k++) {
PF row_k(rows[k]), col_k(k);
prod_xy *= (row_j + col_k) * (row_k + col_i);
if (k != j)
prod_x *= (row_j - row_k);
if (k != i)
prod_y *= (col_i - col_k);
}
return prod_xy / ((row_j + col_i) * prod_x * prod_y);
#else
PF row_j(rows[j]), col_i(i);
if (j == 0) {
PF num(1), den(1);
for (int k = 0, r = 2; k < n; k++, --r) {
PF row_k(rows[k]), col_k(k);
num = mul(num, add(row_k, col_i));
if (k != i)
den = mul(den, sub(col_i, col_k));
if (!r) {
r = 3;
num = reduce(num);
den = reduce(den);
}
}
row_num = reduce(num);
row_den = reduce(den);
}
PF num(row_num), den(row_den);
for (int k = 0, r = 2; k < n; k++, --r) {
PF row_k(rows[k]), col_k(k);
num = mul(num, add(row_j, col_k));
if (k != j)
den = mul(den, sub(row_j, row_k));
if (!r) {
r = 3;
num = reduce(num);
den = reduce(den);
}
}
num = reduce(num);
den = reduce(den);
return num / (add(row_j, col_i) * den);
#endif
}
void mac(const IO *a, PF b, int len, bool first, bool last)
{
if (first && last) {
for (int i = 0; i < len; i++)
temp[i] = b * PF(a[i]);
} else if (first) {
for (int i = 0; i < len; i++)
temp[i] = mul(b, PF(a[i]));
} else if (last) {
for (int i = 0; i < len; i++)
temp[i] = reduce(add(temp[i], mul(b, PF(a[i]))));
} else {
for (int i = 0; i < len; i++)
temp[i] = add(temp[i], mul(b, PF(a[i])));
}
}
void mac_sub(IO *c, const IO *a, PF b, IO s, int len, bool first, bool last)
{
int v = PF::P-1;
if (first && last) {
for (int i = 0; i < len; i++)
c[i] = (b * PF(a[i] == s ? v : a[i]))();
} else if (first) {
for (int i = 0; i < len; i++)
temp[i] = mul(b, PF(a[i] == s ? v : a[i]));
} else if (last) {
for (int i = 0; i < len; i++)
c[i] = reduce(add(temp[i], mul(b, PF(a[i] == s ? v : a[i]))))();
} else {
for (int i = 0; i < len; i++)
temp[i] = add(temp[i], mul(b, PF(a[i] == s ? v : a[i])));
}
}
int find_unused(int block_len)
{
for (int i = 0; i < int(PF::P+7)/8; ++i)
used[i] = 0;
for (int i = 0; i < block_len; ++i)
used[temp[i]()/8] |= 1 << temp[i]()%8;
int s = 0;
while (used[s/8] & 1 << s%8)
++s;
return s;
}
int encode(const IO *data, IO *block, int block_id, int block_len, int block_cnt)
{
assert(block_id >= block_cnt && block_id < int(PF::P) / 2);
assert(block_len <= MAX_LEN);
for (int k = 0; k < block_cnt; k++) {
PF a_ik = cauchy_matrix(block_id, k);
mac(data + block_len * k, a_ik, block_len, !k, k == block_cnt - 1);
}
int sub = find_unused(block_len);
for (int i = 0; i < block_len; ++i)
block[i] = temp[i]() == PF::P-1 ? sub : temp[i]();
return sub;
}
void decode(IO *data, const IO *blocks, const IO *block_subs, const IO *block_ids, int block_idx, int block_len, int block_cnt)
{
assert(block_len <= MAX_LEN);
for (int k = 0; k < block_cnt; k++) {
PF b_ik = inverse_cauchy_matrix(block_ids, block_idx, k, block_cnt);
mac_sub(data, blocks + block_len * k, b_ik, block_subs[k], block_len, !k, k == block_cnt - 1);
}
}
};
}