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

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/*
* Falcon signature generation.
*
* ==========================(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"
/* =================================================================== */
/*
* Compute degree N from logarithm 'logn'.
*/
#define MKN(logn) ((size_t)1 << (logn))
/* =================================================================== */
/*
* Binary case:
* N = 2^logn
* phi = X^N+1
*/
/*
* Get the size of the LDL tree for an input with polynomials of size
* 2^logn. The size is expressed in the number of elements.
*/
static inline unsigned
ffLDL_treesize(unsigned logn) {
/*
* For logn = 0 (polynomials are constant), the "tree" is a
* single element. Otherwise, the tree node has size 2^logn, and
* has two child trees for size logn-1 each. Thus, treesize s()
* must fulfill these two relations:
*
* s(0) = 1
* s(logn) = (2^logn) + 2*s(logn-1)
*/
return (logn + 1) << logn;
}
/*
* Inner function for ffLDL_fft(). It expects the matrix to be both
* auto-adjoint and quasicyclic; also, it uses the source operands
* as modifiable temporaries.
*
* tmp[] must have room for at least one polynomial.
*/
static void
ffLDL_fft_inner(fpr *tree,
fpr *g0, fpr *g1, unsigned logn, fpr *tmp) {
size_t n, hn;
n = MKN(logn);
if (n == 1) {
tree[0] = g0[0];
return;
}
hn = n >> 1;
/*
* The LDL decomposition yields L (which is written in the tree)
* and the diagonal of D. Since d00 = g0, we just write d11
* into tmp.
*/
PQCLEAN_FALCON1024_CLEAN_poly_LDLmv_fft(tmp, tree, g0, g1, g0, logn);
/*
* Split d00 (currently in g0) and d11 (currently in tmp). We
* reuse g0 and g1 as temporary storage spaces:
* d00 splits into g1, g1+hn
* d11 splits into g0, g0+hn
*/
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(g1, g1 + hn, g0, logn);
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(g0, g0 + hn, tmp, logn);
/*
* Each split result is the first row of a new auto-adjoint
* quasicyclic matrix for the next recursive step.
*/
ffLDL_fft_inner(tree + n,
g1, g1 + hn, logn - 1, tmp);
ffLDL_fft_inner(tree + n + ffLDL_treesize(logn - 1),
g0, g0 + hn, logn - 1, tmp);
}
/*
* Compute the ffLDL tree of an auto-adjoint matrix G. The matrix
* is provided as three polynomials (FFT representation).
*
* The "tree" array is filled with the computed tree, of size
* (logn+1)*(2^logn) elements (see ffLDL_treesize()).
*
* Input arrays MUST NOT overlap, except possibly the three unmodified
* arrays g00, g01 and g11. tmp[] should have room for at least three
* polynomials of 2^logn elements each.
*/
static void
ffLDL_fft(fpr *tree, const fpr *g00,
const fpr *g01, const fpr *g11,
unsigned logn, fpr *tmp) {
size_t n, hn;
fpr *d00, *d11;
n = MKN(logn);
if (n == 1) {
tree[0] = g00[0];
return;
}
hn = n >> 1;
d00 = tmp;
d11 = tmp + n;
tmp += n << 1;
memcpy(d00, g00, n * sizeof * g00);
PQCLEAN_FALCON1024_CLEAN_poly_LDLmv_fft(d11, tree, g00, g01, g11, logn);
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(tmp, tmp + hn, d00, logn);
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(d00, d00 + hn, d11, logn);
memcpy(d11, tmp, n * sizeof * tmp);
ffLDL_fft_inner(tree + n,
d11, d11 + hn, logn - 1, tmp);
ffLDL_fft_inner(tree + n + ffLDL_treesize(logn - 1),
d00, d00 + hn, logn - 1, tmp);
}
/*
* Normalize an ffLDL tree: each leaf of value x is replaced with
* sigma / sqrt(x).
*/
static void
ffLDL_binary_normalize(fpr *tree, unsigned logn) {
/*
* TODO: make an iterative version.
*/
size_t n;
n = MKN(logn);
if (n == 1) {
/*
* We actually store in the tree leaf the inverse of
* the value mandated by the specification: this
* saves a division both here and in the sampler.
*/
tree[0] = fpr_mul(fpr_sqrt(tree[0]), fpr_inv_sigma);
} else {
ffLDL_binary_normalize(tree + n, logn - 1);
ffLDL_binary_normalize(tree + n + ffLDL_treesize(logn - 1),
logn - 1);
}
}
/* =================================================================== */
/*
* Convert an integer polynomial (with small values) into the
* representation with complex numbers.
*/
static void
smallints_to_fpr(fpr *r, const int8_t *t, unsigned logn) {
size_t n, u;
n = MKN(logn);
for (u = 0; u < n; u ++) {
r[u] = fpr_of(t[u]);
}
}
/*
* The expanded private key contains:
* - The B0 matrix (four elements)
* - The ffLDL tree
*/
static inline size_t
skoff_b00(unsigned logn) {
(void)logn;
return 0;
}
static inline size_t
skoff_b01(unsigned logn) {
return MKN(logn);
}
static inline size_t
skoff_b10(unsigned logn) {
return 2 * MKN(logn);
}
static inline size_t
skoff_b11(unsigned logn) {
return 3 * MKN(logn);
}
static inline size_t
skoff_tree(unsigned logn) {
return 4 * MKN(logn);
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_expand_privkey(fpr *expanded_key,
const int8_t *f, const int8_t *g,
const int8_t *F, const int8_t *G,
unsigned logn, uint8_t *tmp) {
size_t n;
fpr *rf, *rg, *rF, *rG;
fpr *b00, *b01, *b10, *b11;
fpr *g00, *g01, *g11, *gxx;
fpr *tree;
n = MKN(logn);
b00 = expanded_key + skoff_b00(logn);
b01 = expanded_key + skoff_b01(logn);
b10 = expanded_key + skoff_b10(logn);
b11 = expanded_key + skoff_b11(logn);
tree = expanded_key + skoff_tree(logn);
/*
* We load the private key elements directly into the B0 matrix,
* since B0 = [[g, -f], [G, -F]].
*/
rf = b01;
rg = b00;
rF = b11;
rG = b10;
smallints_to_fpr(rf, f, logn);
smallints_to_fpr(rg, g, logn);
smallints_to_fpr(rF, F, logn);
smallints_to_fpr(rG, G, logn);
/*
* Compute the FFT for the key elements, and negate f and F.
*/
PQCLEAN_FALCON1024_CLEAN_FFT(rf, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(rg, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(rF, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(rG, logn);
PQCLEAN_FALCON1024_CLEAN_poly_neg(rf, logn);
PQCLEAN_FALCON1024_CLEAN_poly_neg(rF, logn);
/*
* The Gram matrix is G = B·B*. Formulas are:
* g00 = b00*adj(b00) + b01*adj(b01)
* g01 = b00*adj(b10) + b01*adj(b11)
* g10 = b10*adj(b00) + b11*adj(b01)
* g11 = b10*adj(b10) + b11*adj(b11)
*
* For historical reasons, this implementation uses
* g00, g01 and g11 (upper triangle).
*/
g00 = (fpr *)tmp;
g01 = g00 + n;
g11 = g01 + n;
gxx = g11 + n;
memcpy(g00, b00, n * sizeof * b00);
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(g00, logn);
memcpy(gxx, b01, n * sizeof * b01);
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(gxx, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(g00, gxx, logn);
memcpy(g01, b00, n * sizeof * b00);
PQCLEAN_FALCON1024_CLEAN_poly_muladj_fft(g01, b10, logn);
memcpy(gxx, b01, n * sizeof * b01);
PQCLEAN_FALCON1024_CLEAN_poly_muladj_fft(gxx, b11, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(g01, gxx, logn);
memcpy(g11, b10, n * sizeof * b10);
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(g11, logn);
memcpy(gxx, b11, n * sizeof * b11);
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(gxx, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(g11, gxx, logn);
/*
* Compute the Falcon tree.
*/
ffLDL_fft(tree, g00, g01, g11, logn, gxx);
/*
* Normalize tree.
*/
ffLDL_binary_normalize(tree, logn);
}
typedef int (*samplerZ)(void *ctx, fpr mu, fpr sigma);
/*
* Perform Fast Fourier Sampling for target vector t. The Gram matrix
* is provided (G = [[g00, g01], [adj(g01), g11]]). The sampled vector
* is written over (t0,t1). The Gram matrix is modified as well. The
* tmp[] buffer must have room for four polynomials.
*/
static void
ffSampling_fft_dyntree(samplerZ samp, void *samp_ctx,
fpr *t0, fpr *t1,
fpr *g00, fpr *g01, fpr *g11,
unsigned logn, fpr *tmp) {
size_t n, hn;
fpr *z0, *z1;
/*
* Deepest level: the LDL tree leaf value is just g00 (the
* array has length only 1 at this point); we normalize it
* with regards to sigma, then use it for sampling.
*/
if (logn == 0) {
fpr leaf;
leaf = g00[0];
leaf = fpr_mul(fpr_sqrt(leaf), fpr_inv_sigma);
t0[0] = fpr_of(samp(samp_ctx, t0[0], leaf));
t1[0] = fpr_of(samp(samp_ctx, t1[0], leaf));
return;
}
n = (size_t)1 << logn;
hn = n >> 1;
/*
* Decompose G into LDL. We only need d00 (identical to g00),
* d11, and l10; we do that in place.
*/
PQCLEAN_FALCON1024_CLEAN_poly_LDL_fft(g00, g01, g11, logn);
/*
* Split d00 and d11 and expand them into half-size quasi-cyclic
* Gram matrices. We also save l10 in tmp[].
*/
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(tmp, tmp + hn, g00, logn);
memcpy(g00, tmp, n * sizeof * tmp);
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(tmp, tmp + hn, g11, logn);
memcpy(g11, tmp, n * sizeof * tmp);
memcpy(tmp, g01, n * sizeof * g01);
memcpy(g01, g00, hn * sizeof * g00);
memcpy(g01 + hn, g11, hn * sizeof * g00);
/*
* The half-size Gram matrices for the recursive LDL tree
* building are now:
* - left sub-tree: g00, g00+hn, g01
* - right sub-tree: g11, g11+hn, g01+hn
* l10 is in tmp[].
*/
/*
* We split t1 and use the first recursive call on the two
* halves, using the right sub-tree. The result is merged
* back into tmp + 2*n.
*/
z1 = tmp + n;
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(z1, z1 + hn, t1, logn);
ffSampling_fft_dyntree(samp, samp_ctx, z1, z1 + hn,
g11, g11 + hn, g01 + hn, logn - 1, z1 + n);
PQCLEAN_FALCON1024_CLEAN_poly_merge_fft(tmp + (n << 1), z1, z1 + hn, logn);
/*
* Compute tb0 = t0 + (t1 - z1) * l10.
* At that point, l10 is in tmp, t1 is unmodified, and z1 is
* in tmp + (n << 1). The buffer in z1 is free.
*
* In the end, z1 is written over t1, and tb0 is in t0.
*/
memcpy(z1, t1, n * sizeof * t1);
PQCLEAN_FALCON1024_CLEAN_poly_sub(z1, tmp + (n << 1), logn);
memcpy(t1, tmp + (n << 1), n * sizeof * tmp);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(tmp, z1, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(t0, tmp, logn);
/*
* Second recursive invocation, on the split tb0 (currently in t0)
* and the left sub-tree.
*/
z0 = tmp;
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(z0, z0 + hn, t0, logn);
ffSampling_fft_dyntree(samp, samp_ctx, z0, z0 + hn,
g00, g00 + hn, g01, logn - 1, z0 + n);
PQCLEAN_FALCON1024_CLEAN_poly_merge_fft(t0, z0, z0 + hn, logn);
}
/*
* Perform Fast Fourier Sampling for target vector t and LDL tree T.
* tmp[] must have size for at least two polynomials of size 2^logn.
*/
static void
ffSampling_fft(samplerZ samp, void *samp_ctx,
fpr *z0, fpr *z1,
const fpr *tree,
const fpr *t0, const fpr *t1, unsigned logn,
fpr *tmp) {
size_t n, hn;
const fpr *tree0, *tree1;
/*
* When logn == 2, we inline the last two recursion levels.
*/
if (logn == 2) {
fpr x0, x1, y0, y1, w0, w1, w2, w3, sigma;
fpr a_re, a_im, b_re, b_im, c_re, c_im;
tree0 = tree + 4;
tree1 = tree + 8;
/*
* We split t1 into w*, then do the recursive invocation,
* with output in w*. We finally merge back into z1.
*/
a_re = t1[0];
a_im = t1[2];
b_re = t1[1];
b_im = t1[3];
c_re = fpr_add(a_re, b_re);
c_im = fpr_add(a_im, b_im);
w0 = fpr_half(c_re);
w1 = fpr_half(c_im);
c_re = fpr_sub(a_re, b_re);
c_im = fpr_sub(a_im, b_im);
w2 = fpr_mul(fpr_add(c_re, c_im), fpr_invsqrt8);
w3 = fpr_mul(fpr_sub(c_im, c_re), fpr_invsqrt8);
x0 = w2;
x1 = w3;
sigma = tree1[3];
w2 = fpr_of(samp(samp_ctx, x0, sigma));
w3 = fpr_of(samp(samp_ctx, x1, sigma));
a_re = fpr_sub(x0, w2);
a_im = fpr_sub(x1, w3);
b_re = tree1[0];
b_im = tree1[1];
c_re = fpr_sub(fpr_mul(a_re, b_re), fpr_mul(a_im, b_im));
c_im = fpr_add(fpr_mul(a_re, b_im), fpr_mul(a_im, b_re));
x0 = fpr_add(c_re, w0);
x1 = fpr_add(c_im, w1);
sigma = tree1[2];
w0 = fpr_of(samp(samp_ctx, x0, sigma));
w1 = fpr_of(samp(samp_ctx, x1, sigma));
a_re = w0;
a_im = w1;
b_re = w2;
b_im = w3;
c_re = fpr_mul(fpr_sub(b_re, b_im), fpr_invsqrt2);
c_im = fpr_mul(fpr_add(b_re, b_im), fpr_invsqrt2);
z1[0] = w0 = fpr_add(a_re, c_re);
z1[2] = w2 = fpr_add(a_im, c_im);
z1[1] = w1 = fpr_sub(a_re, c_re);
z1[3] = w3 = fpr_sub(a_im, c_im);
/*
* Compute tb0 = t0 + (t1 - z1) * L. Value tb0 ends up in w*.
*/
w0 = fpr_sub(t1[0], w0);
w1 = fpr_sub(t1[1], w1);
w2 = fpr_sub(t1[2], w2);
w3 = fpr_sub(t1[3], w3);
a_re = w0;
a_im = w2;
b_re = tree[0];
b_im = tree[2];
w0 = fpr_sub(fpr_mul(a_re, b_re), fpr_mul(a_im, b_im));
w2 = fpr_add(fpr_mul(a_re, b_im), fpr_mul(a_im, b_re));
a_re = w1;
a_im = w3;
b_re = tree[1];
b_im = tree[3];
w1 = fpr_sub(fpr_mul(a_re, b_re), fpr_mul(a_im, b_im));
w3 = fpr_add(fpr_mul(a_re, b_im), fpr_mul(a_im, b_re));
w0 = fpr_add(w0, t0[0]);
w1 = fpr_add(w1, t0[1]);
w2 = fpr_add(w2, t0[2]);
w3 = fpr_add(w3, t0[3]);
/*
* Second recursive invocation.
*/
a_re = w0;
a_im = w2;
b_re = w1;
b_im = w3;
c_re = fpr_add(a_re, b_re);
c_im = fpr_add(a_im, b_im);
w0 = fpr_half(c_re);
w1 = fpr_half(c_im);
c_re = fpr_sub(a_re, b_re);
c_im = fpr_sub(a_im, b_im);
w2 = fpr_mul(fpr_add(c_re, c_im), fpr_invsqrt8);
w3 = fpr_mul(fpr_sub(c_im, c_re), fpr_invsqrt8);
x0 = w2;
x1 = w3;
sigma = tree0[3];
w2 = y0 = fpr_of(samp(samp_ctx, x0, sigma));
w3 = y1 = fpr_of(samp(samp_ctx, x1, sigma));
a_re = fpr_sub(x0, y0);
a_im = fpr_sub(x1, y1);
b_re = tree0[0];
b_im = tree0[1];
c_re = fpr_sub(fpr_mul(a_re, b_re), fpr_mul(a_im, b_im));
c_im = fpr_add(fpr_mul(a_re, b_im), fpr_mul(a_im, b_re));
x0 = fpr_add(c_re, w0);
x1 = fpr_add(c_im, w1);
sigma = tree0[2];
w0 = fpr_of(samp(samp_ctx, x0, sigma));
w1 = fpr_of(samp(samp_ctx, x1, sigma));
a_re = w0;
a_im = w1;
b_re = w2;
b_im = w3;
c_re = fpr_mul(fpr_sub(b_re, b_im), fpr_invsqrt2);
c_im = fpr_mul(fpr_add(b_re, b_im), fpr_invsqrt2);
z0[0] = fpr_add(a_re, c_re);
z0[2] = fpr_add(a_im, c_im);
z0[1] = fpr_sub(a_re, c_re);
z0[3] = fpr_sub(a_im, c_im);
return;
}
/*
* Case logn == 1 is reachable only when using Falcon-2 (the
* smallest size for which Falcon is mathematically defined, but
* of course way too insecure to be of any use).
*/
if (logn == 1) {
fpr x0, x1, y0, y1, sigma;
fpr a_re, a_im, b_re, b_im, c_re, c_im;
x0 = t1[0];
x1 = t1[1];
sigma = tree[3];
z1[0] = y0 = fpr_of(samp(samp_ctx, x0, sigma));
z1[1] = y1 = fpr_of(samp(samp_ctx, x1, sigma));
a_re = fpr_sub(x0, y0);
a_im = fpr_sub(x1, y1);
b_re = tree[0];
b_im = tree[1];
c_re = fpr_sub(fpr_mul(a_re, b_re), fpr_mul(a_im, b_im));
c_im = fpr_add(fpr_mul(a_re, b_im), fpr_mul(a_im, b_re));
x0 = fpr_add(c_re, t0[0]);
x1 = fpr_add(c_im, t0[1]);
sigma = tree[2];
z0[0] = fpr_of(samp(samp_ctx, x0, sigma));
z0[1] = fpr_of(samp(samp_ctx, x1, sigma));
return;
}
/*
* Normal end of recursion is for logn == 0. Since the last
* steps of the recursions were inlined in the blocks above
* (when logn == 1 or 2), this case is not reachable, and is
* retained here only for documentation purposes.
if (logn == 0) {
fpr x0, x1, sigma;
x0 = t0[0];
x1 = t1[0];
sigma = tree[0];
z0[0] = fpr_of(samp(samp_ctx, x0, sigma));
z1[0] = fpr_of(samp(samp_ctx, x1, sigma));
return;
}
*/
/*
* General recursive case (logn >= 3).
*/
n = (size_t)1 << logn;
hn = n >> 1;
tree0 = tree + n;
tree1 = tree + n + ffLDL_treesize(logn - 1);
/*
* We split t1 into z1 (reused as temporary storage), then do
* the recursive invocation, with output in tmp. We finally
* merge back into z1.
*/
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(z1, z1 + hn, t1, logn);
ffSampling_fft(samp, samp_ctx, tmp, tmp + hn,
tree1, z1, z1 + hn, logn - 1, tmp + n);
PQCLEAN_FALCON1024_CLEAN_poly_merge_fft(z1, tmp, tmp + hn, logn);
/*
* Compute tb0 = t0 + (t1 - z1) * L. Value tb0 ends up in tmp[].
*/
memcpy(tmp, t1, n * sizeof * t1);
PQCLEAN_FALCON1024_CLEAN_poly_sub(tmp, z1, logn);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(tmp, tree, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(tmp, t0, logn);
/*
* Second recursive invocation.
*/
PQCLEAN_FALCON1024_CLEAN_poly_split_fft(z0, z0 + hn, tmp, logn);
ffSampling_fft(samp, samp_ctx, tmp, tmp + hn,
tree0, z0, z0 + hn, logn - 1, tmp + n);
PQCLEAN_FALCON1024_CLEAN_poly_merge_fft(z0, tmp, tmp + hn, logn);
}
/*
* Compute a signature: the signature contains two vectors, s1 and s2.
* The s1 vector is not returned. The squared norm of (s1,s2) is
* computed, and if it is short enough, then s2 is returned into the
* s2[] buffer, and 1 is returned; otherwise, s2[] is untouched and 0 is
* returned; the caller should then try again. This function uses an
* expanded key.
*
* tmp[] must have room for at least six polynomials.
*/
static int
do_sign_tree(samplerZ samp, void *samp_ctx, int16_t *s2,
const fpr *expanded_key,
const uint16_t *hm,
unsigned logn, fpr *tmp) {
size_t n, u;
fpr *t0, *t1, *tx, *ty;
const fpr *b00, *b01, *b10, *b11, *tree;
fpr ni;
uint32_t sqn, ng;
int16_t *s1tmp, *s2tmp;
n = MKN(logn);
t0 = tmp;
t1 = t0 + n;
b00 = expanded_key + skoff_b00(logn);
b01 = expanded_key + skoff_b01(logn);
b10 = expanded_key + skoff_b10(logn);
b11 = expanded_key + skoff_b11(logn);
tree = expanded_key + skoff_tree(logn);
/*
* Set the target vector to [hm, 0] (hm is the hashed message).
*/
for (u = 0; u < n; u ++) {
t0[u] = fpr_of(hm[u]);
/* This is implicit.
t1[u] = fpr_zero;
*/
}
/*
* Apply the lattice basis to obtain the real target
* vector (after normalization with regards to modulus).
*/
PQCLEAN_FALCON1024_CLEAN_FFT(t0, logn);
ni = fpr_inverse_of_q;
memcpy(t1, t0, n * sizeof * t0);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(t1, b01, logn);
PQCLEAN_FALCON1024_CLEAN_poly_mulconst(t1, fpr_neg(ni), logn);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(t0, b11, logn);
PQCLEAN_FALCON1024_CLEAN_poly_mulconst(t0, ni, logn);
tx = t1 + n;
ty = tx + n;
/*
* Apply sampling. Output is written back in [tx, ty].
*/
ffSampling_fft(samp, samp_ctx, tx, ty, tree, t0, t1, logn, ty + n);
/*
* Get the lattice point corresponding to that tiny vector.
*/
memcpy(t0, tx, n * sizeof * tx);
memcpy(t1, ty, n * sizeof * ty);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(tx, b00, logn);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(ty, b10, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(tx, ty, logn);
memcpy(ty, t0, n * sizeof * t0);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(ty, b01, logn);
memcpy(t0, tx, n * sizeof * tx);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(t1, b11, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(t1, ty, logn);
PQCLEAN_FALCON1024_CLEAN_iFFT(t0, logn);
PQCLEAN_FALCON1024_CLEAN_iFFT(t1, logn);
/*
* Compute the signature.
*/
s1tmp = (int16_t *)tx;
sqn = 0;
ng = 0;
for (u = 0; u < n; u ++) {
int32_t z;
z = (int32_t)hm[u] - (int32_t)fpr_rint(t0[u]);
sqn += (uint32_t)(z * z);
ng |= sqn;
s1tmp[u] = (int16_t)z;
}
sqn |= -(ng >> 31);
/*
* With "normal" degrees (e.g. 512 or 1024), it is very
* improbable that the computed vector is not short enough;
* however, it may happen in practice for the very reduced
* versions (e.g. degree 16 or below). In that case, the caller
* will loop, and we must not write anything into s2[] because
* s2[] may overlap with the hashed message hm[] and we need
* hm[] for the next iteration.
*/
s2tmp = (int16_t *)tmp;
for (u = 0; u < n; u ++) {
s2tmp[u] = (int16_t) - fpr_rint(t1[u]);
}
if (PQCLEAN_FALCON1024_CLEAN_is_short_half(sqn, s2tmp, logn)) {
memcpy(s2, s2tmp, n * sizeof * s2);
memcpy(tmp, s1tmp, n * sizeof * s1tmp);
return 1;
}
return 0;
}
/*
* Compute a signature: the signature contains two vectors, s1 and s2.
* The s1 vector is not returned. The squared norm of (s1,s2) is
* computed, and if it is short enough, then s2 is returned into the
* s2[] buffer, and 1 is returned; otherwise, s2[] is untouched and 0 is
* returned; the caller should then try again.
*
* tmp[] must have room for at least nine polynomials.
*/
static int
do_sign_dyn(samplerZ samp, void *samp_ctx, int16_t *s2,
const int8_t *f, const int8_t *g,
const int8_t *F, const int8_t *G,
const uint16_t *hm, unsigned logn, fpr *tmp) {
size_t n, u;
fpr *t0, *t1, *tx, *ty;
fpr *b00, *b01, *b10, *b11, *g00, *g01, *g11;
fpr ni;
uint32_t sqn, ng;
int16_t *s1tmp, *s2tmp;
n = MKN(logn);
/*
* Lattice basis is B = [[g, -f], [G, -F]]. We convert it to FFT.
*/
b00 = tmp;
b01 = b00 + n;
b10 = b01 + n;
b11 = b10 + n;
smallints_to_fpr(b01, f, logn);
smallints_to_fpr(b00, g, logn);
smallints_to_fpr(b11, F, logn);
smallints_to_fpr(b10, G, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b01, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b00, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b11, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b10, logn);
PQCLEAN_FALCON1024_CLEAN_poly_neg(b01, logn);
PQCLEAN_FALCON1024_CLEAN_poly_neg(b11, logn);
/*
* Compute the Gram matrix G = B·B*. Formulas are:
* g00 = b00*adj(b00) + b01*adj(b01)
* g01 = b00*adj(b10) + b01*adj(b11)
* g10 = b10*adj(b00) + b11*adj(b01)
* g11 = b10*adj(b10) + b11*adj(b11)
*
* For historical reasons, this implementation uses
* g00, g01 and g11 (upper triangle). g10 is not kept
* since it is equal to adj(g01).
*
* We _replace_ the matrix B with the Gram matrix, but we
* must keep b01 and b11 for computing the target vector.
*/
t0 = b11 + n;
t1 = t0 + n;
memcpy(t0, b01, n * sizeof * b01);
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(t0, logn); // t0 <- b01*adj(b01)
memcpy(t1, b00, n * sizeof * b00);
PQCLEAN_FALCON1024_CLEAN_poly_muladj_fft(t1, b10, logn); // t1 <- b00*adj(b10)
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(b00, logn); // b00 <- b00*adj(b00)
PQCLEAN_FALCON1024_CLEAN_poly_add(b00, t0, logn); // b00 <- g00
memcpy(t0, b01, n * sizeof * b01);
PQCLEAN_FALCON1024_CLEAN_poly_muladj_fft(b01, b11, logn); // b01 <- b01*adj(b11)
PQCLEAN_FALCON1024_CLEAN_poly_add(b01, t1, logn); // b01 <- g01
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(b10, logn); // b10 <- b10*adj(b10)
memcpy(t1, b11, n * sizeof * b11);
PQCLEAN_FALCON1024_CLEAN_poly_mulselfadj_fft(t1, logn); // t1 <- b11*adj(b11)
PQCLEAN_FALCON1024_CLEAN_poly_add(b10, t1, logn); // b10 <- g11
/*
* We rename variables to make things clearer. The three elements
* of the Gram matrix uses the first 3*n slots of tmp[], followed
* by b11 and b01 (in that order).
*/
g00 = b00;
g01 = b01;
g11 = b10;
b01 = t0;
t0 = b01 + n;
t1 = t0 + n;
/*
* Memory layout at that point:
* g00 g01 g11 b11 b01 t0 t1
*/
/*
* Set the target vector to [hm, 0] (hm is the hashed message).
*/
for (u = 0; u < n; u ++) {
t0[u] = fpr_of(hm[u]);
/* This is implicit.
t1[u] = fpr_zero;
*/
}
/*
* Apply the lattice basis to obtain the real target
* vector (after normalization with regards to modulus).
*/
PQCLEAN_FALCON1024_CLEAN_FFT(t0, logn);
ni = fpr_inverse_of_q;
memcpy(t1, t0, n * sizeof * t0);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(t1, b01, logn);
PQCLEAN_FALCON1024_CLEAN_poly_mulconst(t1, fpr_neg(ni), logn);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(t0, b11, logn);
PQCLEAN_FALCON1024_CLEAN_poly_mulconst(t0, ni, logn);
/*
* b01 and b11 can be discarded, so we move back (t0,t1).
* Memory layout is now:
* g00 g01 g11 t0 t1
*/
memcpy(b11, t0, n * 2 * sizeof * t0);
t0 = g11 + n;
t1 = t0 + n;
/*
* Apply sampling; result is written over (t0,t1).
*/
ffSampling_fft_dyntree(samp, samp_ctx,
t0, t1, g00, g01, g11, logn, t1 + n);
/*
* We arrange the layout back to:
* b00 b01 b10 b11 t0 t1
*
* We did not conserve the matrix basis, so we must recompute
* it now.
*/
b00 = tmp;
b01 = b00 + n;
b10 = b01 + n;
b11 = b10 + n;
memmove(b11 + n, t0, n * 2 * sizeof * t0);
t0 = b11 + n;
t1 = t0 + n;
smallints_to_fpr(b01, f, logn);
smallints_to_fpr(b00, g, logn);
smallints_to_fpr(b11, F, logn);
smallints_to_fpr(b10, G, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b01, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b00, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b11, logn);
PQCLEAN_FALCON1024_CLEAN_FFT(b10, logn);
PQCLEAN_FALCON1024_CLEAN_poly_neg(b01, logn);
PQCLEAN_FALCON1024_CLEAN_poly_neg(b11, logn);
tx = t1 + n;
ty = tx + n;
/*
* Get the lattice point corresponding to that tiny vector.
*/
memcpy(tx, t0, n * sizeof * t0);
memcpy(ty, t1, n * sizeof * t1);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(tx, b00, logn);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(ty, b10, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(tx, ty, logn);
memcpy(ty, t0, n * sizeof * t0);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(ty, b01, logn);
memcpy(t0, tx, n * sizeof * tx);
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(t1, b11, logn);
PQCLEAN_FALCON1024_CLEAN_poly_add(t1, ty, logn);
PQCLEAN_FALCON1024_CLEAN_iFFT(t0, logn);
PQCLEAN_FALCON1024_CLEAN_iFFT(t1, logn);
s1tmp = (int16_t *)tx;
sqn = 0;
ng = 0;
for (u = 0; u < n; u ++) {
int32_t z;
z = (int32_t)hm[u] - (int32_t)fpr_rint(t0[u]);
sqn += (uint32_t)(z * z);
ng |= sqn;
s1tmp[u] = (int16_t)z;
}
sqn |= -(ng >> 31);
/*
* With "normal" degrees (e.g. 512 or 1024), it is very
* improbable that the computed vector is not short enough;
* however, it may happen in practice for the very reduced
* versions (e.g. degree 16 or below). In that case, the caller
* will loop, and we must not write anything into s2[] because
* s2[] may overlap with the hashed message hm[] and we need
* hm[] for the next iteration.
*/
s2tmp = (int16_t *)tmp;
for (u = 0; u < n; u ++) {
s2tmp[u] = (int16_t) - fpr_rint(t1[u]);
}
if (PQCLEAN_FALCON1024_CLEAN_is_short_half(sqn, s2tmp, logn)) {
memcpy(s2, s2tmp, n * sizeof * s2);
memcpy(tmp, s1tmp, n * sizeof * s1tmp);
return 1;
}
return 0;
}
/*
* Sample an integer value along a half-gaussian distribution centered
* on zero and standard deviation 1.8205, with a precision of 72 bits.
*/
int
PQCLEAN_FALCON1024_CLEAN_gaussian0_sampler(prng *p) {
static const uint32_t dist[] = {
10745844u, 3068844u, 3741698u,
5559083u, 1580863u, 8248194u,
2260429u, 13669192u, 2736639u,
708981u, 4421575u, 10046180u,
169348u, 7122675u, 4136815u,
30538u, 13063405u, 7650655u,
4132u, 14505003u, 7826148u,
417u, 16768101u, 11363290u,
31u, 8444042u, 8086568u,
1u, 12844466u, 265321u,
0u, 1232676u, 13644283u,
0u, 38047u, 9111839u,
0u, 870u, 6138264u,
0u, 14u, 12545723u,
0u, 0u, 3104126u,
0u, 0u, 28824u,
0u, 0u, 198u,
0u, 0u, 1u
};
uint32_t v0, v1, v2, hi;
uint64_t lo;
size_t u;
int z;
/*
* Get a random 72-bit value, into three 24-bit limbs v0..v2.
*/
lo = prng_get_u64(p);
hi = prng_get_u8(p);
v0 = (uint32_t)lo & 0xFFFFFF;
v1 = (uint32_t)(lo >> 24) & 0xFFFFFF;
v2 = (uint32_t)(lo >> 48) | (hi << 16);
/*
* Sampled value is z, such that v0..v2 is lower than the first
* z elements of the table.
*/
z = 0;
for (u = 0; u < (sizeof dist) / sizeof(dist[0]); u += 3) {
uint32_t w0, w1, w2, cc;
w0 = dist[u + 2];
w1 = dist[u + 1];
w2 = dist[u + 0];
cc = (v0 - w0) >> 31;
cc = (v1 - w1 - cc) >> 31;
cc = (v2 - w2 - cc) >> 31;
z += (int)cc;
}
return z;
}
/*
* Sample a bit with probability exp(-x) for some x >= 0.
*/
static int
BerExp(prng *p, fpr x, fpr ccs) {
int s, i;
fpr r;
uint32_t sw, w;
uint64_t z;
/*
* Reduce x modulo log(2): x = s*log(2) + r, with s an integer,
* and 0 <= r < log(2). Since x >= 0, we can use fpr_trunc().
*/
s = (int)fpr_trunc(fpr_mul(x, fpr_inv_log2));
r = fpr_sub(x, fpr_mul(fpr_of(s), fpr_log2));
/*
* It may happen (quite rarely) that s >= 64; if sigma = 1.2
* (the minimum value for sigma), r = 0 and b = 1, then we get
* s >= 64 if the half-Gaussian produced a z >= 13, which happens
* with probability about 0.000000000230383991, which is
* approximatively equal to 2^(-32). In any case, if s >= 64,
* then BerExp will be non-zero with probability less than
* 2^(-64), so we can simply saturate s at 63.
*/
sw = (uint32_t)s;
sw ^= (sw ^ 63) & -((63 - sw) >> 31);
s = (int)sw;
/*
* Compute exp(-r); we know that 0 <= r < log(2) at this point, so
* we can use fpr_expm_p63(), which yields a result scaled to 2^63.
* We scale it up to 2^64, then right-shift it by s bits because
* we really want exp(-x) = 2^(-s)*exp(-r).
*
* The "-1" operation makes sure that the value fits on 64 bits
* (i.e. if r = 0, we may get 2^64, and we prefer 2^64-1 in that
* case). The bias is negligible since fpr_expm_p63() only computes
* with 51 bits of precision or so.
*/
z = ((fpr_expm_p63(r, ccs) << 1) - 1) >> s;
/*
* Sample a bit with probability exp(-x). Since x = s*log(2) + r,
* exp(-x) = 2^-s * exp(-r), we compare lazily exp(-x) with the
* PRNG output to limit its consumption, the sign of the difference
* yields the expected result.
*/
i = 64;
do {
i -= 8;
w = prng_get_u8(p) - ((uint32_t)(z >> i) & 0xFF);
} while (!w && i > 0);
return (int)(w >> 31);
}
/*
* The sampler produces a random integer that follows a discrete Gaussian
* distribution, centered on mu, and with standard deviation sigma. The
* provided parameter isigma is equal to 1/sigma.
*
* The value of sigma MUST lie between 1 and 2 (i.e. isigma lies between
* 0.5 and 1); in Falcon, sigma should always be between 1.2 and 1.9.
*/
int
PQCLEAN_FALCON1024_CLEAN_sampler(void *ctx, fpr mu, fpr isigma) {
sampler_context *spc;
int s;
fpr r, dss, ccs;
spc = ctx;
/*
* Center is mu. We compute mu = s + r where s is an integer
* and 0 <= r < 1.
*/
s = (int)fpr_floor(mu);
r = fpr_sub(mu, fpr_of(s));
/*
* dss = 1/(2*sigma^2) = 0.5*(isigma^2).
*/
dss = fpr_half(fpr_sqr(isigma));
/*
* ccs = sigma_min / sigma = sigma_min * isigma.
*/
ccs = fpr_mul(isigma, spc->sigma_min);
/*
* We now need to sample on center r.
*/
for (;;) {
int z0, z, b;
fpr x;
/*
* Sample z for a Gaussian distribution. Then get a
* random bit b to turn the sampling into a bimodal
* distribution: if b = 1, we use z+1, otherwise we
* use -z. We thus have two situations:
*
* - b = 1: z >= 1 and sampled against a Gaussian
* centered on 1.
* - b = 0: z <= 0 and sampled against a Gaussian
* centered on 0.
*/
z0 = PQCLEAN_FALCON1024_CLEAN_gaussian0_sampler(&spc->p);
b = (int)prng_get_u8(&spc->p) & 1;
z = b + ((b << 1) - 1) * z0;
/*
* Rejection sampling. We want a Gaussian centered on r;
* but we sampled against a Gaussian centered on b (0 or
* 1). But we know that z is always in the range where
* our sampling distribution is greater than the Gaussian
* distribution, so rejection works.
*
* We got z with distribution:
* G(z) = exp(-((z-b)^2)/(2*sigma0^2))
* We target distribution:
* S(z) = exp(-((z-r)^2)/(2*sigma^2))
* Rejection sampling works by keeping the value z with
* probability S(z)/G(z), and starting again otherwise.
* This requires S(z) <= G(z), which is the case here.
* Thus, we simply need to keep our z with probability:
* P = exp(-x)
* where:
* x = ((z-r)^2)/(2*sigma^2) - ((z-b)^2)/(2*sigma0^2)
*
* Here, we scale up the Bernouilli distribution, which
* makes rejection more probable, but makes rejection
* rate sufficiently decorrelated from the Gaussian
* center and standard deviation that the whole sampler
* can be said to be constant-time.
*/
x = fpr_mul(fpr_sqr(fpr_sub(fpr_of(z), r)), dss);
x = fpr_sub(x, fpr_mul(fpr_of(z0 * z0), fpr_inv_2sqrsigma0));
if (BerExp(&spc->p, x, ccs)) {
/*
* Rejection sampling was centered on r, but the
* actual center is mu = s + r.
*/
return s + z;
}
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_sign_tree(int16_t *sig, inner_shake256_context *rng,
const fpr *expanded_key,
const uint16_t *hm, unsigned logn, uint8_t *tmp) {
fpr *ftmp;
ftmp = (fpr *)tmp;
for (;;) {
/*
* Signature produces short vectors s1 and s2. The
* signature is acceptable only if the aggregate vector
* s1,s2 is short; we must use the same bound as the
* verifier.
*
* If the signature is acceptable, then we return only s2
* (the verifier recomputes s1 from s2, the hashed message,
* and the public key).
*/
sampler_context spc;
samplerZ samp;
void *samp_ctx;
/*
* Normal sampling. We use a fast PRNG seeded from our
* SHAKE context ('rng').
*/
spc.sigma_min = (logn == 10)
? fpr_sigma_min_10
: fpr_sigma_min_9;
PQCLEAN_FALCON1024_CLEAN_prng_init(&spc.p, rng);
samp = PQCLEAN_FALCON1024_CLEAN_sampler;
samp_ctx = &spc;
/*
* Do the actual signature.
*/
if (do_sign_tree(samp, samp_ctx, sig,
expanded_key, hm, logn, ftmp)) {
break;
}
}
}
/* see inner.h */
void
PQCLEAN_FALCON1024_CLEAN_sign_dyn(int16_t *sig, inner_shake256_context *rng,
const int8_t *f, const int8_t *g,
const int8_t *F, const int8_t *G,
const uint16_t *hm, unsigned logn, uint8_t *tmp) {
fpr *ftmp;
ftmp = (fpr *)tmp;
for (;;) {
/*
* Signature produces short vectors s1 and s2. The
* signature is acceptable only if the aggregate vector
* s1,s2 is short; we must use the same bound as the
* verifier.
*
* If the signature is acceptable, then we return only s2
* (the verifier recomputes s1 from s2, the hashed message,
* and the public key).
*/
sampler_context spc;
samplerZ samp;
void *samp_ctx;
/*
* Normal sampling. We use a fast PRNG seeded from our
* SHAKE context ('rng').
*/
spc.sigma_min = (logn == 10)
? fpr_sigma_min_10
: fpr_sigma_min_9;
PQCLEAN_FALCON1024_CLEAN_prng_init(&spc.p, rng);
samp = PQCLEAN_FALCON1024_CLEAN_sampler;
samp_ctx = &spc;
/*
* Do the actual signature.
*/
if (do_sign_dyn(samp, samp_ctx, sig,
f, g, F, G, hm, logn, ftmp)) {
break;
}
}
}
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