/*-------------------------------------------------------------------------*/
static void
do_sieving(xydata_t *curr_xydata)
{
uint32 i, j;
uint16 *sieve = curr_xydata->sieve;
xypower_t *curr_xypower = curr_xydata->powers + 0;
uint32 p = curr_xypower->power;
uint16 contrib = curr_xypower->contrib;
uint32 num_roots = curr_xypower->num_roots;
xyprog_t *roots = curr_xypower->roots;
for (i = 0; i < num_roots; i++) {
uint16 *row = sieve;
xyprog_t *curr_prog = roots + i;
uint32 start = curr_prog->start;
uint8 *invtable_y = curr_prog->invtable_y;
for (j = 0; j < p; j++) {
uint32 curr_start = mp_modsub_1(start,
invtable_y[j], p);
row[curr_start] += contrib;
row += p;
}
curr_prog->start = mp_modsub_1(start,
curr_prog->stride_z, p);
}
}
/*-------------------------------------------------------------------------*/
static void
xydata_init(xydata_t *xydata, uint32 num_lattice_primes,
lattice_t *lattice_xyz, int64 z_base)
{
uint32 i, j, k, m, n;
for (i = 0; i < num_lattice_primes; i++) {
xydata_t *curr_xydata = xydata + i;
uint32 num_powers = curr_xydata->num_powers;
for (j = 0; j < num_powers; j++) {
xypower_t *curr_xypower = curr_xydata->powers + j;
uint32 num_roots = curr_xypower->num_roots;
uint32 p = curr_xypower->power;
uint32 latsize_mod = curr_xypower->latsize_mod;
uint32 y_mod_p = lattice_xyz->y % p;
int64 z_start = z_base + lattice_xyz->z;
int32 z_start_mod = z_start % p;
uint32 z_mod_p = (z_start_mod < 0) ?
(z_start_mod + (int32)p) : z_start_mod;
for (k = 0; k < num_roots; k++) {
xyprog_t *curr_xyprog = curr_xypower->roots + k;
uint8 *invtable_y = curr_xyprog->invtable_y;
uint32 start = curr_xyprog->base_start;
uint32 resclass = curr_xyprog->resclass;
uint32 resclass2 = mp_modmul_1(resclass,
resclass, p);
uint32 ytmp = y_mod_p;
uint32 stride_y = mp_modmul_1(resclass,
latsize_mod, p);
curr_xyprog->stride_z = mp_modmul_1(resclass2,
latsize_mod, p);
start = mp_modsub_1(start,
mp_modmul_1(resclass,
y_mod_p, p), p);
curr_xyprog->start = mp_modsub_1(start,
mp_modmul_1(resclass2,
z_mod_p, p), p);
for (m = n = 0; m < p; m++) {
invtable_y[ytmp] = n;
ytmp = mp_modadd_1(ytmp,
latsize_mod, p);
n = mp_modadd_1(n, stride_y, p);
}
}
}
}
}
/*-------------------------------------------------------------------------*/
static void
do_sieving(sieve_root_t *r, uint16 *sieve,
uint32 contrib, uint32 dim)
{
uint32 i;
uint32 start = r->start;
uint32 step = r->step;
uint32 resclass = r->resclass;
if (resclass >= step)
resclass %= step;
for (i = 0; i < dim; i++) {
uint32 ri = start;
do {
sieve[ri] += contrib;
ri += step;
} while (ri < dim);
sieve += dim;
start = mp_modsub_1(start, resclass, step);
}
}
/*-------------------------------------------------------------------------*/
static void
do_sieving_powers(xydata_t *curr_xydata)
{
uint32 i, j, k, m;
uint16 *sieve = curr_xydata->sieve;
uint32 p = curr_xydata->p;
uint32 num_powers = curr_xydata->num_powers;
xypower_t *powers = curr_xydata->powers;
for (i = 0; i < num_powers; i++) {
xypower_t *curr_xypower = powers + i;
uint32 power = curr_xypower->power;
uint16 contrib = curr_xypower->contrib;
uint32 num_roots = curr_xypower->num_roots;
xyprog_t *roots = curr_xypower->roots;
for (j = 0; j < num_roots; j++) {
uint16 *row = sieve;
xyprog_t *curr_prog = roots + j;
uint32 start = curr_prog->start;
uint8 *invtable_y = curr_prog->invtable_y;
for (k = 0; k < p; k += power) {
for (m = 0; m < power; m++) {
uint32 curr_start = mp_modsub_1(start,
invtable_y[m],
power);
do {
row[curr_start] += contrib;
curr_start += power;
} while (curr_start < p);
row += p;
}
}
curr_prog->start = mp_modsub_1(start,
curr_prog->stride_z, p);
}
}
}
/*------------------------------------------------------------------------*/
static uint32
lift_root_32(uint32 n, uint32 r, uint32 old_power,
uint32 p, uint32 d)
{
uint32 q;
uint32 p2 = old_power * p;
uint64 rsave = r;
q = mp_modsub_1(n % p2, mp_expo_1(r, d, p2), p2) / old_power;
r = mp_modmul_1(d, mp_expo_1(r % p, d - 1, p), p);
r = mp_modmul_1(q, mp_modinv_1(r, p), p);
return rsave + old_power * r;
}
u_int32_t is_irreducible(mpzpoly_t poly, u_int32_t p)
{
/* this uses Proposition 3.4.4 of H. Cohen, "A Course
in Computational Algebraic Number Theory". The tests
below are much simpler than trying to factor 'poly' */
u_int32_t i;
poly_t f, tmp;
poly_reduce_mod_p(f, poly, p);
poly_make_monic(f, f, p);
/* in practice, the degree of f will be 8 or less,
and we want to compute GCDs for all prime numbers
that divide the degree. For this limited range
the loop below avoids duplicated code */
for (i = 2; i < f->degree; i++) {
if (f->degree % i)
continue;
/* for degree d, compute x^(p^(d/i)) - x */
poly_xpow_pd(tmp, p, f->degree / i, f);
if (tmp->degree == 0) {
tmp->degree = 1;
tmp->coef[1] = p - 1;
}
else {
tmp->coef[1] = mp_modsub_1(tmp->coef[1],
(u_int32_t)1, p);
poly_fix_degree(tmp);
}
/* this must be relatively prime to f */
poly_gcd(tmp, f, p);
if (tmp->degree > 0 || tmp->coef[0] != 1) {
return 0;
}
}
/* final test: x^(p^d) mod f must equal x */
poly_xpow_pd(tmp, p, f->degree, f);
if (tmp->degree == 1 && tmp->coef[0] == 0 && tmp->coef[1] == 1)
return 1;
return 0;
}
/*------------------------------------------------------------------------*/
static uint32
lift_root_32(uint32 n, uint32 r, uint32 old_power,
uint32 p, uint32 d)
{
/* given r, a d_th root of n mod old_power, compute
the corresponding root mod (old_power*p) via Hensel lifting */
uint32 q;
uint32 p2 = old_power * p;
uint64 rsave = r;
q = mp_modsub_1(n % p2, mp_expo_1(r, d, p2), p2) / old_power;
r = mp_modmul_1(d, mp_expo_1(r % p, d - 1, p), p);
r = mp_modmul_1(q, mp_modinv_1(r, p), p);
return rsave + old_power * r;
}
static void poly_mod(poly_t res, poly_t op, poly_t _mod, u_int32_t p)
{
/* divide the polynomial 'op' by the polynomial '_mod'
and write the remainder to 'res'. All polynomial
coefficients are reduced modulo 'p' */
int32_t i;
u_int32_t msw;
poly_t tmp, mod;
if(_mod->degree == 0)
{
memset(res, 0, sizeof(res[0]));
return;
}
poly_cp(tmp, op);
poly_make_monic(mod, _mod, p);
while(tmp->degree >= mod->degree)
{
/* tmp <-- tmp - msw * mod * x^{deg(tmp)- deg(mod)} */
msw = tmp->coef[tmp->degree];
tmp->coef[tmp->degree] = 0;
for(i = mod->degree-1; i >= 0; i--)
{
u_int32_t c = mp_modmul_1(msw, mod->coef[i], p);
u_int32_t j = tmp->degree - (mod->degree - i);
tmp->coef[j] = mp_modsub_1(tmp->coef[j], c, p);
}
poly_fix_degree(tmp);
}
poly_cp(res, tmp);
return;
}
/*-------------------------------------------------------------------*/
static uint32 verify_product(gmp_poly_t *gmp_prod, abpair_t *abpairs,
uint32 num_relations, uint32 q, mp_t *c,
mp_poly_t *alg_poly) {
/* a sanity check on the computed value of S(x): for
a small prime q for which alg_poly is irreducible,
verify that gmp_prod mod q equals the product
mod q of the relations in abpairs[]. The latter can
be computed very quickly */
uint32 i, j;
uint32 c_mod_q = mp_mod_1(c, q);
uint32 d = alg_poly->degree;
uint32 ref_prod[MAX_POLY_DEGREE];
uint32 prod[MAX_POLY_DEGREE];
uint32 mod[MAX_POLY_DEGREE];
uint32 accum[MAX_POLY_DEGREE + 1];
/* compute the product mod q directly. First initialize
and reduce the coefficients of alg_poly and gmp_prod mod q */
for (i = 0; i < d; i++) {
prod[i] = 0;
ref_prod[i] = mpz_fdiv_ui(gmp_prod->coeff[i],
(unsigned long)q);
mod[i] = mp_mod_1(&alg_poly->coeff[i].num, q);
if (alg_poly->coeff[i].sign == NEGATIVE && mod[i] > 0) {
mod[i] = q - mod[i];
}
}
prod[0] = 1;
/* multiply the product by each relation in
turn, modulo q */
for (i = 0; i < num_relations; i++) {
int64 a = abpairs[i].a;
uint32 b = q - (abpairs[i].b % q);
uint32 ac;
a = a % (int64)q;
if (a < 0)
a += q;
ac = mp_modmul_1((uint32)a, c_mod_q, q);
for (j = accum[0] = 0; j < d; j++) {
accum[j+1] = mp_modmul_1(prod[j], b, q);
accum[j] = mp_modadd_1(accum[j],
mp_modmul_1(ac, prod[j], q), q);
}
for (j = 0; j < d; j++) {
prod[j] = mp_modsub_1(accum[j],
mp_modmul_1(accum[d], mod[j], q), q);
}
}
/* do the polynomial compare */
for (i = 0; i < d; i++) {
if (ref_prod[i] != prod[i])
break;
}
if (i == d)
return 1;
return 0;
}
/*------------------------------------------------------------------*/
u_int32_t poly_get_zeros(u_int32_t *zeros, mpzpoly_t _f, u_int32_t p, u_int32_t count_only)
{
/* Find all roots of multiplicity 1 for polynomial _f,
when the coefficients of _f are reduced mod p.
The leading coefficient of _f mod p is returned
Make count_only nonzero if only the number of roots
and not their identity matters; this is much faster */
poly_t g, f;
u_int32_t i, j, num_zeros;
/* reduce the coefficients mod p */
poly_reduce_mod_p(f, _f, p);
/* bail out if the polynomial is zero */
if (f->degree == 0)
return 0;
/* pull out roots of zero. We do this early to
avoid having to handle degree-1 polynomials
in later code */
num_zeros = 0;
if (f->coef[0] == 0) {
for (i = 1; i <= f->degree; i++) {
if (f->coef[i])
break;
}
for (j = i; i <= f->degree; i++) {
f->coef[i - j] = f->coef[i];
}
f->degree = i - j - 1;
zeros[num_zeros++] = 0;
}
/* handle trivial cases */
if (f->degree == 0) {
return num_zeros;
}
else if (f->degree == 1) {
u_int32_t w = f->coef[1];
if (count_only)
return num_zeros + 1;
if (w != 1) {
w = mp_modinv_1(w, p);
zeros[num_zeros++] = mp_modmul_1(p - f->coef[0],
w, p);
}
else {
zeros[num_zeros++] = (f->coef[0] == 0 ?
0 : p - f->coef[0]);
}
return num_zeros;
}
/* the rest of the algorithm assumes p is odd, which
will not work for p=2. Fortunately, in that case
there are only two possible roots, 0 and 1. The above
already tried 0, so try 1 here */
if (p == 2) {
u_int32_t parity = 0;
for (i = 0; i <= f->degree; i++)
parity ^= f->coef[i];
if (parity == 0)
zeros[num_zeros++] = 1;
return num_zeros;
}
/* Compute g = gcd(f, x^(p-1) - 1). The result is
a polynomial that is the product of all the linear
factors of f. A given factor only occurs once in
this polynomial */
poly_xpow(g, 0, p-1, f, p);
g->coef[0] = mp_modsub_1(g->coef[0], 1, p);
poly_fix_degree(g);
poly_gcd(g, f, p);
/* no linear factors, no service */
if (g->degree < 1 || count_only)
return num_zeros + g->degree;
/* isolate the linear factors */
get_zeros_rec(zeros, 0, &num_zeros, g, p);
return num_zeros;
}
/*------------------------------------------------------------------*/
static void get_zeros_rec(u_int32_t *zeros, u_int32_t shift,
u_int32_t *num_zeros, poly_t f, u_int32_t p) {
/* get the zeros of a poly, f, that is known to split
completely over Z/pZ. Many thanks to Bob Silverman
for a neat implementation of Cantor-Zassenhaus splitting */
poly_t g, xpow;
u_int32_t degree1, degree2;
/* base cases of the recursion: we can find the roots
of linear and quadratic polynomials immediately */
if (f->degree == 1) {
u_int32_t w = f->coef[1];
if (w != 1) {
w = mp_modinv_1(w, p);
zeros[(*num_zeros)++] = mp_modmul_1(p - f->coef[0],w,p);
}
else {
zeros[(*num_zeros)++] = (f->coef[0] == 0 ? 0 :
p - f->coef[0]);
}
return;
}
else if (f->degree == 2) {
/* if f is a quadratic polynomial, then it will
always have two distinct nonzero roots or else
we wouldn't have gotten to this point. The two
roots are the solution of a general quadratic
equation, mod p */
u_int32_t d = mp_modmul_1(f->coef[0], f->coef[2], p);
u_int32_t root1 = p - f->coef[1];
u_int32_t root2 = root1;
u_int32_t ainv = mp_modinv_1(
mp_modadd_1(f->coef[2], f->coef[2], p),
p);
d = mp_modsub_1(mp_modmul_1(f->coef[1], f->coef[1], p),
mp_modmul_1(4, d, p),
p);
d = mp_modsqrt_1(d, p);
root1 = mp_modadd_1(root1, d, p);
root2 = mp_modsub_1(root2, d, p);
zeros[(*num_zeros)++] = mp_modmul_1(root1, ainv, p);
zeros[(*num_zeros)++] = mp_modmul_1(root2, ainv, p);
return;
}
/* For an increasing sequence of integers 's', compute
the polynomial gcd((x-s)^(p-1)/2 - 1, f). If the result is
not g = 1 or g = f, this is a nontrivial splitting
of f. References require choosing s randomly, but however
s is chosen there is a 50% chance that it will split f.
Since only 0 <= s < p is valid, we choose each s in turn;
choosing random s allows the possibility that the same
s gets chosen twice (mod p), which would waste time */
while (shift < p) {
poly_xpow(xpow, shift, (p-1)/2, f, p);
poly_cp(g, xpow);
g->coef[0] = mp_modsub_1(g->coef[0], 1, p);
poly_fix_degree(g);
poly_gcd(g, f, p);
if (g->degree > 0)
break;
shift++;
}
/* f was split; repeat the splitting process on
the two halves of f. The linear factors of f are
either somewhere in x^((p-1)/2) - 1, in
x^((p-1)/2) + 1, or 'shift' itself is a linear
factor. Test each of these possibilities in turn.
In the first two cases, begin trying values of s
strictly greater than have been tried thus far */
degree1 = g->degree;
get_zeros_rec(zeros, shift + 1, num_zeros, g, p);
poly_cp(g, xpow);
g->coef[0] = mp_modadd_1(g->coef[0], 1, p);
poly_fix_degree(g);
poly_gcd(g, f, p);
degree2 = g->degree;
if (degree2 > 0)
get_zeros_rec(zeros, shift + 1, num_zeros, g, p);
if (degree1 + degree2 < f->degree)
zeros[(*num_zeros)++] = (shift == 0 ? 0 : p - shift);
}
static inline u_int32_t mp_modadd_1(u_int32_t a, u_int32_t b, u_int32_t n)
{
return mp_modsub_1(a, n - b, n);
}
