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/*
D. Monniaux
CNRS / VERIMAG
Demonstration of number theoretic transform (Fast Fourier transform in a finite field) in Z/(2^16+1)Z for buffer of length 2^16.
FFT original code from Rosetta Code.
*/
#include <assert.h>
#include <stdint.h>
#include <inttypes.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "../clock.h"
typedef uint32_t modint;
typedef int32_t smodint;
static modint invm(modint a0, modint b0)
{
smodint a=a0, b=b0, q;
smodint x0 = 0, x1 = 1, t;
if (b == 1) return 1;
while (a > 1) {
q = a / b;
t = b; b = a % b; a = t;
t = x0; x0 = x1 - q * x0; x1 = t;
}
if (x1 < 0) x1 += b0;
assert((x1 * a0) % b0 == 1);
return x1;
}
static inline modint mulm(modint a, modint b, modint m) {
return (a * b) % m;
}
static inline modint addm(modint a, modint b, modint m) {
return (a + b) % m;
}
static inline modint subm(modint a, modint b, modint m) {
return (a + (m - b)) % m;
}
static inline modint powm_u(modint a, modint exponent, modint m) {
modint x = 1;
while (exponent > 0) {
if (exponent % 2) x = mulm(x, a, m);
exponent /= 2;
a = mulm(a, a, m);
}
return x;
}
static inline modint powm(modint a, smodint exponent, modint m) {
if (exponent == 0) return 1;
else if (exponent > 0) return powm_u(a, exponent, m);
else /* exponent < 0 */ return powm(invm(a, m), -exponent, m);
}
static void _fft(modint modulus,
modint root_of_unit,
modint buf[], modint out[],
unsigned n, unsigned step)
{
if (step < n) {
modint root_of_unit2 = mulm(root_of_unit, root_of_unit, modulus);
_fft(modulus, root_of_unit2, out, buf, n, step * 2);
_fft(modulus, root_of_unit2, out + step, buf + step, n, step * 2);
modint exp = 1;
for (unsigned i = 0; i < n; i += 2 * step) {
modint t = mulm(exp, out[i + step], modulus);
buf[i / 2] = addm(out[i], t, modulus);
buf[(i + n)/2] = subm(out[i], t, modulus);
exp = mulm(exp, root_of_unit, modulus);
}
}
}
void fft(modint modulus, modint root_of_unit, modint buf[], unsigned n)
{
modint *out = malloc(sizeof(modint) * n);
memcpy(out, buf, sizeof(modint) * n);
_fft(modulus, root_of_unit, buf, out, n, 1);
free(out);
}
static void negate(modint MODULUS, modint buf[restrict], unsigned n) {
for(unsigned i=0; i<n; i++) {
if (buf[i]) buf[i] = MODULUS-buf[i];
}
}
static void mulvecm(modint modulus, modint buf[restrict], unsigned n, modint coef) {
for(unsigned i=0; i<n; i++) {
buf[i] = mulm(buf[i], coef, modulus);
}
}
#define LOG_LENGTH 16
#define LENGTH (1 << LOG_LENGTH)
#define MUL_MODULUS LENGTH
#define MODULUS (MUL_MODULUS + 1)
modint randm(modint modulus) {
static modint state = 0x42;
state = state * UINT64_C(9223372036854775837) + 0x2017;
return state % modulus;
}
int main() {
#if 0
modint root_of_unit = 1;
for(modint i=1; i<MODULUS; i++) {
if (powm_u(i, MUL_MODULUS/2, MODULUS) != 1) {
root_of_unit = i;
break;
}
}
#else
modint root_of_unit = 3;
#endif
assert(root_of_unit != 1);
//printf("root of unit = %" PRIu64 "\n", root_of_unit);
modint *buf = malloc(LENGTH * sizeof(modint)),
*save = malloc(LENGTH * sizeof(modint));
for(unsigned i=0; i<LENGTH; i++) {
buf[i] = randm(MODULUS);
}
memcpy(save, buf, LENGTH * sizeof(modint));
clock_start();
fft(MODULUS, root_of_unit, buf, LENGTH);
fft(MODULUS, invm(root_of_unit, MODULUS), buf, LENGTH);
clock_stop();
print_total_clock();
#if 0
/* can be replaced by x -> -x */
mulvecm(MODULUS, buf, LENGTH, invm(LENGTH, MODULUS));
#else
negate(MODULUS, buf, LENGTH);
#endif
printf("compare = %d\n", memcmp(buf, save, LENGTH * sizeof(modint)));
/*
printf("buf[0] = %" PRIu64 "\n", buf[0]);
printf("buf[1] = %" PRIu64 "\n", buf[1]);
printf("buf[2] = %" PRIu64 "\n", buf[2]);
printf("buf[3] = %" PRIu64 "\n", buf[3]);
printf("buf[4] = %" PRIu64 "\n", buf[4]);
*/
free(buf);
free(save);
}
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