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pqcrypto/crypto_sign/falcon-1024/clean/fft.c

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/*
* FFT code.
*
* ==========================(LICENSE BEGIN)============================
*
* Copyright (c) 2017-2019 Falcon Project
*
* Permission is hereby granted, free of charge, to any person obtaining
* a copy of this software and associated documentation files (the
* "Software"), to deal in the Software without restriction, including
* without limitation the rights to use, copy, modify, merge, publish,
* distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to
* the following conditions:
*
* The above copyright notice and this permission notice shall be
* included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*
* ===========================(LICENSE END)=============================
*
* @author Thomas Pornin <thomas.pornin@nccgroup.com>
*/
#include "inner.h"
/*
* Rules for complex number macros:
* --------------------------------
*
* Operand order is: destination, source1, source2...
*
* Each operand is a real and an imaginary part.
*
* All overlaps are allowed.
*/
/*
* Addition of two complex numbers (d = a + b).
*/
#define FPC_ADD(d_re, d_im, a_re, a_im, b_re, b_im) do { \
fpr fpct_re, fpct_im; \
fpct_re = fpr_add(a_re, b_re); \
fpct_im = fpr_add(a_im, b_im); \
(d_re) = fpct_re; \
(d_im) = fpct_im; \
} while (0)
/*
* Subtraction of two complex numbers (d = a - b).
*/
#define FPC_SUB(d_re, d_im, a_re, a_im, b_re, b_im) do { \
fpr fpct_re, fpct_im; \
fpct_re = fpr_sub(a_re, b_re); \
fpct_im = fpr_sub(a_im, b_im); \
(d_re) = fpct_re; \
(d_im) = fpct_im; \
} while (0)
/*
* Multplication of two complex numbers (d = a * b).
*/
#define FPC_MUL(d_re, d_im, a_re, a_im, b_re, b_im) do { \
fpr fpct_a_re, fpct_a_im; \
fpr fpct_b_re, fpct_b_im; \
fpr fpct_d_re, fpct_d_im; \
fpct_a_re = (a_re); \
fpct_a_im = (a_im); \
fpct_b_re = (b_re); \
fpct_b_im = (b_im); \
fpct_d_re = fpr_sub( \
fpr_mul(fpct_a_re, fpct_b_re), \
fpr_mul(fpct_a_im, fpct_b_im)); \
fpct_d_im = fpr_add( \
fpr_mul(fpct_a_re, fpct_b_im), \
fpr_mul(fpct_a_im, fpct_b_re)); \
(d_re) = fpct_d_re; \
(d_im) = fpct_d_im; \
} while (0)
/*
* Squaring of a complex number (d = a * a).
*/
#define FPC_SQR(d_re, d_im, a_re, a_im) do { \
fpr fpct_a_re, fpct_a_im; \
fpr fpct_d_re, fpct_d_im; \
fpct_a_re = (a_re); \
fpct_a_im = (a_im); \
fpct_d_re = fpr_sub(fpr_sqr(fpct_a_re), fpr_sqr(fpct_a_im)); \
fpct_d_im = fpr_double(fpr_mul(fpct_a_re, fpct_a_im)); \
(d_re) = fpct_d_re; \
(d_im) = fpct_d_im; \
} while (0)
/*
* Inversion of a complex number (d = 1 / a).
*/
#define FPC_INV(d_re, d_im, a_re, a_im) do { \
fpr fpct_a_re, fpct_a_im; \
fpr fpct_d_re, fpct_d_im; \
fpr fpct_m; \
fpct_a_re = (a_re); \
fpct_a_im = (a_im); \
fpct_m = fpr_add(fpr_sqr(fpct_a_re), fpr_sqr(fpct_a_im)); \
fpct_m = fpr_inv(fpct_m); \
fpct_d_re = fpr_mul(fpct_a_re, fpct_m); \
fpct_d_im = fpr_mul(fpr_neg(fpct_a_im), fpct_m); \
(d_re) = fpct_d_re; \
(d_im) = fpct_d_im; \
} while (0)
/*
* Division of complex numbers (d = a / b).
*/
#define FPC_DIV(d_re, d_im, a_re, a_im, b_re, b_im) do { \
fpr fpct_a_re, fpct_a_im; \
fpr fpct_b_re, fpct_b_im; \
fpr fpct_d_re, fpct_d_im; \
fpr fpct_m; \
fpct_a_re = (a_re); \
fpct_a_im = (a_im); \
fpct_b_re = (b_re); \
fpct_b_im = (b_im); \
fpct_m = fpr_add(fpr_sqr(fpct_b_re), fpr_sqr(fpct_b_im)); \
fpct_m = fpr_inv(fpct_m); \
fpct_b_re = fpr_mul(fpct_b_re, fpct_m); \
fpct_b_im = fpr_mul(fpr_neg(fpct_b_im), fpct_m); \
fpct_d_re = fpr_sub( \
fpr_mul(fpct_a_re, fpct_b_re), \
fpr_mul(fpct_a_im, fpct_b_im)); \
fpct_d_im = fpr_add( \
fpr_mul(fpct_a_re, fpct_b_im), \
fpr_mul(fpct_a_im, fpct_b_re)); \
(d_re) = fpct_d_re; \
(d_im) = fpct_d_im; \
} while (0)
/*
* Let w = exp(i*pi/N); w is a primitive 2N-th root of 1. We define the
* values w_j = w^(2j+1) for all j from 0 to N-1: these are the roots
* of X^N+1 in the field of complex numbers. A crucial property is that
* w_{N-1-j} = conj(w_j) = 1/w_j for all j.
*
* FFT representation of a polynomial f (taken modulo X^N+1) is the
* set of values f(w_j). Since f is real, conj(f(w_j)) = f(conj(w_j)),
* thus f(w_{N-1-j}) = conj(f(w_j)). We thus store only half the values,
* for j = 0 to N/2-1; the other half can be recomputed easily when (if)
* needed. A consequence is that FFT representation has the same size
* as normal representation: N/2 complex numbers use N real numbers (each
* complex number is the combination of a real and an imaginary part).
*
* We use a specific ordering which makes computations easier. Let rev()
* be the bit-reversal function over log(N) bits. For j in 0..N/2-1, we
* store the real and imaginary parts of f(w_j) in slots:
*
* Re(f(w_j)) -> slot rev(j)/2
* Im(f(w_j)) -> slot rev(j)/2+N/2
*
* (Note that rev(j) is even for j < N/2.)
*/
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_FFT(fpr *f, unsigned logn) {
/*
* FFT algorithm in bit-reversal order uses the following
* iterative algorithm:
*
* t = N
* for m = 1; m < N; m *= 2:
* ht = t/2
* for i1 = 0; i1 < m; i1 ++:
* j1 = i1 * t
* s = GM[m + i1]
* for j = j1; j < (j1 + ht); j ++:
* x = f[j]
* y = s * f[j + ht]
* f[j] = x + y
* f[j + ht] = x - y
* t = ht
*
* GM[k] contains w^rev(k) for primitive root w = exp(i*pi/N).
*
* In the description above, f[] is supposed to contain complex
* numbers. In our in-memory representation, the real and
* imaginary parts of f[k] are in array slots k and k+N/2.
*
* We only keep the first half of the complex numbers. We can
* see that after the first iteration, the first and second halves
* of the array of complex numbers have separate lives, so we
* simply ignore the second part.
*/
unsigned u;
size_t t, n, hn, m;
/*
* First iteration: compute f[j] + i * f[j+N/2] for all j < N/2
* (because GM[1] = w^rev(1) = w^(N/2) = i).
* In our chosen representation, this is a no-op: everything is
* already where it should be.
*/
/*
* Subsequent iterations are truncated to use only the first
* half of values.
*/
n = (size_t)1 << logn;
hn = n >> 1;
t = hn;
for (u = 1, m = 2; u < logn; u ++, m <<= 1) {
size_t ht, hm, i1, j1;
ht = t >> 1;
hm = m >> 1;
for (i1 = 0, j1 = 0; i1 < hm; i1 ++, j1 += t) {
size_t j, j2;
j2 = j1 + ht;
fpr s_re, s_im;
s_re = fpr_gm_tab[((m + i1) << 1) + 0];
s_im = fpr_gm_tab[((m + i1) << 1) + 1];
for (j = j1; j < j2; j ++) {
fpr x_re, x_im, y_re, y_im;
x_re = f[j];
x_im = f[j + hn];
y_re = f[j + ht];
y_im = f[j + ht + hn];
FPC_MUL(y_re, y_im, y_re, y_im, s_re, s_im);
FPC_ADD(f[j], f[j + hn],
x_re, x_im, y_re, y_im);
FPC_SUB(f[j + ht], f[j + ht + hn],
x_re, x_im, y_re, y_im);
}
}
t = ht;
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_iFFT(fpr *f, unsigned logn) {
/*
* Inverse FFT algorithm in bit-reversal order uses the following
* iterative algorithm:
*
* t = 1
* for m = N; m > 1; m /= 2:
* hm = m/2
* dt = t*2
* for i1 = 0; i1 < hm; i1 ++:
* j1 = i1 * dt
* s = iGM[hm + i1]
* for j = j1; j < (j1 + t); j ++:
* x = f[j]
* y = f[j + t]
* f[j] = x + y
* f[j + t] = s * (x - y)
* t = dt
* for i1 = 0; i1 < N; i1 ++:
* f[i1] = f[i1] / N
*
* iGM[k] contains (1/w)^rev(k) for primitive root w = exp(i*pi/N)
* (actually, iGM[k] = 1/GM[k] = conj(GM[k])).
*
* In the main loop (not counting the final division loop), in
* all iterations except the last, the first and second half of f[]
* (as an array of complex numbers) are separate. In our chosen
* representation, we do not keep the second half.
*
* The last iteration recombines the recomputed half with the
* implicit half, and should yield only real numbers since the
* target polynomial is real; moreover, s = i at that step.
* Thus, when considering x and y:
* y = conj(x) since the final f[j] must be real
* Therefore, f[j] is filled with 2*Re(x), and f[j + t] is
* filled with 2*Im(x).
* But we already have Re(x) and Im(x) in array slots j and j+t
* in our chosen representation. That last iteration is thus a
* simple doubling of the values in all the array.
*
* We make the last iteration a no-op by tweaking the final
* division into a division by N/2, not N.
*/
size_t u, n, hn, t, m;
n = (size_t)1 << logn;
t = 1;
m = n;
hn = n >> 1;
for (u = logn; u > 1; u --) {
size_t hm, dt, i1, j1;
hm = m >> 1;
dt = t << 1;
for (i1 = 0, j1 = 0; j1 < hn; i1 ++, j1 += dt) {
size_t j, j2;
j2 = j1 + t;
fpr s_re, s_im;
s_re = fpr_gm_tab[((hm + i1) << 1) + 0];
s_im = fpr_neg(fpr_gm_tab[((hm + i1) << 1) + 1]);
for (j = j1; j < j2; j ++) {
fpr x_re, x_im, y_re, y_im;
x_re = f[j];
x_im = f[j + hn];
y_re = f[j + t];
y_im = f[j + t + hn];
FPC_ADD(f[j], f[j + hn],
x_re, x_im, y_re, y_im);
FPC_SUB(x_re, x_im, x_re, x_im, y_re, y_im);
FPC_MUL(f[j + t], f[j + t + hn],
x_re, x_im, s_re, s_im);
}
}
t = dt;
m = hm;
}
/*
* Last iteration is a no-op, provided that we divide by N/2
* instead of N. We need to make a special case for logn = 0.
*/
if (logn > 0) {
fpr ni;
ni = fpr_p2_tab[logn];
for (u = 0; u < n; u ++) {
f[u] = fpr_mul(f[u], ni);
}
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_add(
fpr *a, const fpr *b, unsigned logn) {
size_t n, u;
n = (size_t)1 << logn;
for (u = 0; u < n; u ++) {
a[u] = fpr_add(a[u], b[u]);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_sub(
fpr *a, const fpr *b, unsigned logn) {
size_t n, u;
n = (size_t)1 << logn;
for (u = 0; u < n; u ++) {
a[u] = fpr_sub(a[u], b[u]);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_neg(fpr *a, unsigned logn) {
size_t n, u;
n = (size_t)1 << logn;
for (u = 0; u < n; u ++) {
a[u] = fpr_neg(a[u]);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_adj_fft(fpr *a, unsigned logn) {
size_t n, u;
n = (size_t)1 << logn;
for (u = (n >> 1); u < n; u ++) {
a[u] = fpr_neg(a[u]);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(
fpr *a, const fpr *b, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr a_re, a_im, b_re, b_im;
a_re = a[u];
a_im = a[u + hn];
b_re = b[u];
b_im = b[u + hn];
FPC_MUL(a[u], a[u + hn], a_re, a_im, b_re, b_im);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_muladj_fft(
fpr *a, const fpr *b, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr a_re, a_im, b_re, b_im;
a_re = a[u];
a_im = a[u + hn];
b_re = b[u];
b_im = fpr_neg(b[u + hn]);
FPC_MUL(a[u], a[u + hn], a_re, a_im, b_re, b_im);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(fpr *a, unsigned logn) {
/*
* Since each coefficient is multiplied with its own conjugate,
* the result contains only real values.
*/
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr a_re, a_im;
a_re = a[u];
a_im = a[u + hn];
a[u] = fpr_add(fpr_sqr(a_re), fpr_sqr(a_im));
a[u + hn] = fpr_zero;
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_mulconst(fpr *a, fpr x, unsigned logn) {
size_t n, u;
n = (size_t)1 << logn;
for (u = 0; u < n; u ++) {
a[u] = fpr_mul(a[u], x);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_div_fft(
fpr *a, const fpr *b, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr a_re, a_im, b_re, b_im;
a_re = a[u];
a_im = a[u + hn];
b_re = b[u];
b_im = b[u + hn];
FPC_DIV(a[u], a[u + hn], a_re, a_im, b_re, b_im);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_invnorm2_fft(fpr *d,
const fpr *a, const fpr *b, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr a_re, a_im;
fpr b_re, b_im;
a_re = a[u];
a_im = a[u + hn];
b_re = b[u];
b_im = b[u + hn];
d[u] = fpr_inv(fpr_add(
fpr_add(fpr_sqr(a_re), fpr_sqr(a_im)),
fpr_add(fpr_sqr(b_re), fpr_sqr(b_im))));
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_add_muladj_fft(fpr *d,
const fpr *F, const fpr *G,
const fpr *f, const fpr *g, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr F_re, F_im, G_re, G_im;
fpr f_re, f_im, g_re, g_im;
fpr a_re, a_im, b_re, b_im;
F_re = F[u];
F_im = F[u + hn];
G_re = G[u];
G_im = G[u + hn];
f_re = f[u];
f_im = f[u + hn];
g_re = g[u];
g_im = g[u + hn];
FPC_MUL(a_re, a_im, F_re, F_im, f_re, fpr_neg(f_im));
FPC_MUL(b_re, b_im, G_re, G_im, g_re, fpr_neg(g_im));
d[u] = fpr_add(a_re, b_re);
d[u + hn] = fpr_add(a_im, b_im);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_mul_autoadj_fft(
fpr *a, const fpr *b, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
a[u] = fpr_mul(a[u], b[u]);
a[u + hn] = fpr_mul(a[u + hn], b[u]);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_div_autoadj_fft(
fpr *a, const fpr *b, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr ib;
ib = fpr_inv(b[u]);
a[u] = fpr_mul(a[u], ib);
a[u + hn] = fpr_mul(a[u + hn], ib);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_LDL_fft(
const fpr *g00,
fpr *g01, fpr *g11, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr g00_re, g00_im, g01_re, g01_im, g11_re, g11_im;
fpr mu_re, mu_im;
g00_re = g00[u];
g00_im = g00[u + hn];
g01_re = g01[u];
g01_im = g01[u + hn];
g11_re = g11[u];
g11_im = g11[u + hn];
FPC_DIV(mu_re, mu_im, g01_re, g01_im, g00_re, g00_im);
FPC_MUL(g01_re, g01_im, mu_re, mu_im, g01_re, fpr_neg(g01_im));
FPC_SUB(g11[u], g11[u + hn], g11_re, g11_im, g01_re, g01_im);
g01[u] = mu_re;
g01[u + hn] = fpr_neg(mu_im);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_LDLmv_fft(
fpr *d11, fpr *l10,
const fpr *g00, const fpr *g01,
const fpr *g11, unsigned logn) {
size_t n, hn, u;
n = (size_t)1 << logn;
hn = n >> 1;
for (u = 0; u < hn; u ++) {
fpr g00_re, g00_im, g01_re, g01_im, g11_re, g11_im;
fpr mu_re, mu_im;
g00_re = g00[u];
g00_im = g00[u + hn];
g01_re = g01[u];
g01_im = g01[u + hn];
g11_re = g11[u];
g11_im = g11[u + hn];
FPC_DIV(mu_re, mu_im, g01_re, g01_im, g00_re, g00_im);
FPC_MUL(g01_re, g01_im, mu_re, mu_im, g01_re, fpr_neg(g01_im));
FPC_SUB(d11[u], d11[u + hn], g11_re, g11_im, g01_re, g01_im);
l10[u] = mu_re;
l10[u + hn] = fpr_neg(mu_im);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(
fpr *f0, fpr *f1,
const fpr *f, unsigned logn) {
/*
* The FFT representation we use is in bit-reversed order
* (element i contains f(w^(rev(i))), where rev() is the
* bit-reversal function over the ring degree. This changes
* indexes with regards to the Falcon specification.
*/
size_t n, hn, qn, u;
n = (size_t)1 << logn;
hn = n >> 1;
qn = hn >> 1;
/*
* We process complex values by pairs. For logn = 1, there is only
* one complex value (the other one is the implicit conjugate),
* so we add the two lines below because the loop will be
* skipped.
*/
f0[0] = f[0];
f1[0] = f[hn];
for (u = 0; u < qn; u ++) {
fpr a_re, a_im, b_re, b_im;
fpr t_re, t_im;
a_re = f[(u << 1) + 0];
a_im = f[(u << 1) + 0 + hn];
b_re = f[(u << 1) + 1];
b_im = f[(u << 1) + 1 + hn];
FPC_ADD(t_re, t_im, a_re, a_im, b_re, b_im);
f0[u] = fpr_half(t_re);
f0[u + qn] = fpr_half(t_im);
FPC_SUB(t_re, t_im, a_re, a_im, b_re, b_im);
FPC_MUL(t_re, t_im, t_re, t_im,
fpr_gm_tab[((u + hn) << 1) + 0],
fpr_neg(fpr_gm_tab[((u + hn) << 1) + 1]));
f1[u] = fpr_half(t_re);
f1[u + qn] = fpr_half(t_im);
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_poly_merge_fft(
fpr *f,
const fpr *f0, const fpr *f1, unsigned logn) {
size_t n, hn, qn, u;
n = (size_t)1 << logn;
hn = n >> 1;
qn = hn >> 1;
/*
* An extra copy to handle the special case logn = 1.
*/
f[0] = f0[0];
f[hn] = f1[0];
for (u = 0; u < qn; u ++) {
fpr a_re, a_im, b_re, b_im;
fpr t_re, t_im;
a_re = f0[u];
a_im = f0[u + qn];
FPC_MUL(b_re, b_im, f1[u], f1[u + qn],
fpr_gm_tab[((u + hn) << 1) + 0],
fpr_gm_tab[((u + hn) << 1) + 1]);
FPC_ADD(t_re, t_im, a_re, a_im, b_re, b_im);
f[(u << 1) + 0] = t_re;
f[(u << 1) + 0 + hn] = t_im;
FPC_SUB(t_re, t_im, a_re, a_im, b_re, b_im);
f[(u << 1) + 1] = t_re;
f[(u << 1) + 1 + hn] = t_im;
}
}
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