/* Recursive routine to prove via ECPP */
static int ecpp_down(int i, mpz_t Ni, int facstage, IV* dlist, mpz_t* sfacs, int* nsfacs, char** prooftextptr)
{
mpz_t a, b, u, v, m, q, minfactor, sqrtn, mD, t, t2;
mpz_t mlist[6];
UV nm1a;
IV np1lp, np1lq;
struct ec_affine_point P;
int k, dnum, nidigits, facresult, curveresult, downresult, stage, D;
int verbose = get_verbose_level();
nidigits = mpz_sizeinbase(Ni, 10);
k = _GMP_is_prob_prime(Ni);
if (k == 0) return 0;
if (k == 2) {
/* No need to put anything in the proof */
if (verbose) printf("%*sN[%d] (%d dig) PRIME\n", i, "", i, nidigits);
return 2;
}
downresult = 0;
if (verbose) {
printf("%*sN[%d] (%d dig)", i, "", i, nidigits);
if (facstage > 1) printf(" FS %d", facstage);
fflush(stdout);
}
mpz_init(a); mpz_init(b);
mpz_init(u); mpz_init(v);
mpz_init(m); mpz_init(q);
mpz_init(mD); mpz_init(minfactor); mpz_init(sqrtn);
mpz_init(t); mpz_init(t2);
mpz_init(P.x);mpz_init(P.y);
for (k = 0; k < 6; k++)
mpz_init(mlist[k]);
/* Any factors q found must be strictly > minfactor.
* See Atkin and Morain, 1992, section 6.4 */
mpz_root(minfactor, Ni, 4);
mpz_add_ui(minfactor, minfactor, 1);
mpz_mul(minfactor, minfactor, minfactor);
mpz_sqrt(sqrtn, Ni);
for (stage = (i == 0) ? facstage : 1; stage <= facstage; stage++) {
int next_stage = (stage > 1) ? stage : 1;
for (dnum = 0; dlist[dnum] != 0; dnum++) {
int poly_type; /* just for debugging/verbose */
int poly_degree;
D = -dlist[dnum];
if (D > 1) continue; /* Marked for skip */
if (D == 1) { /* n-1 test */
mpz_sub_ui(m, Ni, 1); /* m = N-1 */
mpz_sub_ui(t2, sqrtn, 1);
mpz_tdiv_q_2exp(t2, t2, 1); /* t2 = minfactor */
facresult = check_for_factor2(q, m, t2, t, stage, sfacs, nsfacs, 0);
if (facresult <= 0)
continue;
if (verbose)
{ printf(" n-1\n"); fflush(stdout); }
downresult = ecpp_down(i+1, q, next_stage, dlist, sfacs, nsfacs, prooftextptr);
if (downresult == 0) /* composite */
goto end_down;
if (downresult == 1) { /* nothing found at this stage */
if (verbose) {
printf("%*sN[%d] (%d dig)", i, "", i, nidigits);
if (facstage > 1) printf(" FS %d", facstage);
fflush(stdout);
}
continue;
}
if (verbose)
{ printf("%*sN[%d] (%d dig) n-1", i, "", i, nidigits); fflush(stdout); }
curveresult = _GMP_primality_bls_3(Ni, q, &nm1a);
if (verbose) { printf(" %d\n", curveresult); fflush(stdout); }
if ( ! curveresult ) {
/* This ought not happen */
dlist[dnum] = -2; /* skip this D value from now on */
if (verbose) gmp_printf("\n Could not prove n-1 with N = %Zd\n", D, Ni);
downresult = 0;
continue;
}
goto end_down;
}
if (D == -1) { /* n+1 test */
mpz_add_ui(m, Ni, 1); /* m = N+1 */
mpz_add_ui(t2, sqrtn, 1);
mpz_tdiv_q_2exp(t2, t2, 1); /* t2 = minfactor */
facresult = check_for_factor2(q, m, t2, t, stage, sfacs, nsfacs, 0);
if (facresult <= 0)
continue;
if (verbose)
{ printf(" n+1\n"); fflush(stdout); }
downresult = ecpp_down(i+1, q, next_stage, dlist, sfacs, nsfacs, prooftextptr);
if (downresult == 0) /* composite */
goto end_down;
if (downresult == 1) { /* nothing found at this stage */
if (verbose) {
printf("%*sN[%d] (%d dig)", i, "", i, nidigits);
if (facstage > 1) printf(" FS %d", facstage);
fflush(stdout);
}
continue;
}
if (verbose)
{ printf("%*sN[%d] (%d dig) n-1", i, "", i, nidigits); fflush(stdout); }
curveresult = _GMP_primality_bls_15(Ni, q, &np1lp, &np1lq);
if (verbose) { printf(" %d\n", curveresult); fflush(stdout); }
if ( ! curveresult ) {
/* This ought not happen */
dlist[dnum] = -2; /* skip this D value from now on */
if (verbose) gmp_printf("\n Could not prove n-1 with N = %Zd\n", D, Ni);
downresult = 0;
continue;
}
goto end_down;
}
if ( (-D % 4) != 3 && (-D % 16) != 4 && (-D % 16) != 8 )
croak("Invalid discriminant '%d' in list\n", D);
/* D must also be squarefree in odd divisors, but assume it. */
/* Make sure we can get a class polynomial for this D. */
poly_degree = poly_class_poly(D, NULL, &poly_type);
if (poly_degree == 0) continue;
/* We'll save time in the long run by not looking at big polys once
* we've found a good path from the start. TODO: Needs more tuning. */
if (stage == 0 && i >= 0 && poly_degree > 2) break;
if (facstage == 1) {
if (i > 2 && nidigits < 1100 && poly_degree > 24) break;
if (i > 3 && nidigits < 950 && poly_degree > 15) break;
if (i > 4 && nidigits < 800 && poly_degree > 11) break;
if (i > 8 && nidigits < 700 && poly_degree > 9) break;
if (i > 16 && nidigits < 600 && poly_degree > 8) break;
}
mpz_set_si(mD, D);
/* (D/N) must be 1, and we have to have a u,v solution */
if (mpz_jacobi(mD, Ni) != 1)
continue;
if ( ! modified_cornacchia(u, v, mD, Ni) )
continue;
if (verbose > 1)
{ printf(" %d", D); fflush(stdout); }
choose_m(mlist, D, u, v, Ni, t, t2);
for (k = 0; k < 6; k++) {
facresult = check_for_factor2(q, mlist[k], minfactor, t, stage, sfacs, nsfacs, poly_degree);
/* -1 = couldn't find, 0 = no big factors, 1 = found */
if (facresult <= 0)
continue;
mpz_set(m, mlist[k]);
if (verbose)
{ printf(" %d (%s %d)\n", D, (poly_type == 1) ? "Hilbert" : "Weber", poly_degree); fflush(stdout); }
/* Great, now go down. */
downresult = ecpp_down(i+1, q, next_stage, dlist, sfacs, nsfacs, prooftextptr);
if (downresult == 0) /* composite */
goto end_down;
if (downresult == 1) { /* nothing found at this stage */
if (verbose) {
printf("%*sN[%d] (%d dig)", i, "", i, nidigits);
if (facstage > 1) printf(" FS %d", facstage);
fflush(stdout);
}
continue;
}
/* Awesome, we found the q chain and are in STAGE 2 */
if (verbose)
{ printf("%*sN[%d] (%d dig) %d (%s %d)", i, "", i, nidigits, D, (poly_type == 1) ? "Hilbert" : "Weber", poly_degree); fflush(stdout); }
curveresult = find_curve(a, b, P.x, P.y, D, m, q, Ni);
if (verbose) { printf(" %d\n", curveresult); fflush(stdout); }
if (curveresult == 1) {
/* Oh no! We can't find a point on the curve. Something is right
* messed up, and we've wasted a lot of time. Sigh. */
dlist[dnum] = -2; /* skip this D value from now on */
if (verbose) gmp_printf("\n Invalidated D = %d with N = %Zd\n", D, Ni);
downresult = 0;
continue;
}
/* We found it was composite or proved it */
goto end_down;
} /* k loop for D */
} /* D */
} /* fac stage */
/* Nothing at this level */
downresult = 1;
if (verbose) { printf(" ---\n"); fflush(stdout); }
end_down:
if (downresult == 2) {
if (0 && verbose > 1) {
if (D == 1) {
gmp_printf("\n");
gmp_printf("Type BLS3\n");
gmp_printf("N %Zd\n", Ni);
gmp_printf("Q %Zd\n", q);
gmp_printf("A %lu\n", (unsigned long) nm1a);
gmp_printf("\n");
fflush(stdout);
} else if (D == -1) {
gmp_printf("\n");
gmp_printf("Type BLS15\n");
gmp_printf("N %Zd\n", Ni);
gmp_printf("Q %Zd\n", q);
gmp_printf("LP %ld\n", (signed long) np1lp);
gmp_printf("LQ %ld\n", (signed long) np1lq);
gmp_printf("\n");
fflush(stdout);
} else {
gmp_printf("\n");
gmp_printf("Type ECPP\n");
gmp_printf("N %Zd\n", Ni);
gmp_printf("A %Zd\n", a);
gmp_printf("B %Zd\n", b);
gmp_printf("M %Zd\n", m);
gmp_printf("Q %Zd\n", q);
gmp_printf("X %Zd\n", P.x);
gmp_printf("Y %Zd\n", P.y);
gmp_printf("\n");
fflush(stdout);
}
}
/* Prepend our proof to anything that exists. */
if (prooftextptr != 0) {
char *proofstr, *proofptr;
int curprooflen = (*prooftextptr == 0) ? 0 : strlen(*prooftextptr);
if (D == 1) {
int myprooflen = 20 + 2*(4 + mpz_sizeinbase(Ni, 10)) + 1*21;
New(0, proofstr, myprooflen + curprooflen + 1, char);
proofptr = proofstr;
proofptr += gmp_sprintf(proofptr, "Type BLS3\nN %Zd\nQ %Zd\nA %"UVuf"\n", Ni, q, nm1a);
} else if (D == -1) {
/* Recursive routine to prove via ECPP */
static int ecpp_down(int i, mpz_t Ni, int facstage, int *pmaxH, int* dilist, mpz_t* sfacs, int* nsfacs, char** prooftextptr)
{
mpz_t a, b, u, v, m, q, minfactor, sqrtn, mD, t, t2;
mpz_t mlist[6];
mpz_t qlist[6];
UV nm1a;
IV np1lp, np1lq;
struct ec_affine_point P;
int k, dindex, pindex, nidigits, facresult, curveresult, downresult, stage, D;
int verbose = get_verbose_level();
nidigits = mpz_sizeinbase(Ni, 10);
downresult = _GMP_is_prob_prime(Ni);
if (downresult == 0) return 0;
if (downresult == 2) {
if (mpz_sizeinbase(Ni,2) <= 64) {
/* No need to put anything in the proof */
if (verbose) printf("%*sN[%d] (%d dig) PRIME\n", i, "", i, nidigits);
return 2;
}
downresult = 1;
}
if (i == 0 && facstage == 2 && miller_rabin_random(Ni, 2, 0) == 0) {
gmp_printf("\n\n**** BPSW counter-example found? ****\n**** N = %Zd ****\n\n", Ni);
return 0;
}
VERBOSE_PRINT_N(i, nidigits, *pmaxH, facstage);
mpz_init(a); mpz_init(b);
mpz_init(u); mpz_init(v);
mpz_init(m); mpz_init(q);
mpz_init(mD); mpz_init(minfactor); mpz_init(sqrtn);
mpz_init(t); mpz_init(t2);
mpz_init(P.x);mpz_init(P.y);
for (k = 0; k < 6; k++) {
mpz_init(mlist[k]);
mpz_init(qlist[k]);
}
/* Any factors q found must be strictly > minfactor.
* See Atkin and Morain, 1992, section 6.4 */
mpz_root(minfactor, Ni, 4);
mpz_add_ui(minfactor, minfactor, 1);
mpz_mul(minfactor, minfactor, minfactor);
mpz_sqrt(sqrtn, Ni);
stage = 0;
if (nidigits > 700) stage = 1; /* Too rare to find them */
if (i == 0 && facstage > 1) stage = facstage;
for ( ; stage <= facstage; stage++) {
int next_stage = (stage > 1) ? stage : 1;
for (dindex = -1; dindex < 0 || dilist[dindex] != 0; dindex++) {
int poly_type; /* just for debugging/verbose */
int poly_degree;
int allq = (nidigits < 400); /* Do all q values together, or not */
if (dindex == -1) { /* n-1 and n+1 tests */
int nm1_success = 0;
int np1_success = 0;
const char* ptype = "";
mpz_sub_ui(m, Ni, 1);
mpz_sub_ui(t2, sqrtn, 1);
mpz_tdiv_q_2exp(t2, t2, 1); /* t2 = minfactor */
nm1_success = check_for_factor(u, m, t2, t, stage, sfacs, nsfacs, 0);
mpz_add_ui(m, Ni, 1);
mpz_add_ui(t2, sqrtn, 1);
mpz_tdiv_q_2exp(t2, t2, 1); /* t2 = minfactor */
np1_success = check_for_factor(v, m, t2, t, stage, sfacs, nsfacs, 0);
/* If both successful, pick smallest */
if (nm1_success > 0 && np1_success > 0) {
if (mpz_cmp(u, v) <= 0) np1_success = 0;
else nm1_success = 0;
}
if (nm1_success > 0) { ptype = "n-1"; mpz_set(q, u); D = 1; }
else if (np1_success > 0) { ptype = "n+1"; mpz_set(q, v); D = -1; }
else continue;
if (verbose) { printf(" %s\n", ptype); fflush(stdout); }
downresult = ecpp_down(i+1, q, next_stage, pmaxH, dilist, sfacs, nsfacs, prooftextptr);
if (downresult == 0) goto end_down; /* composite */
if (downresult == 1) { /* nothing found at this stage */
VERBOSE_PRINT_N(i, nidigits, *pmaxH, facstage);
continue;
}
if (verbose)
{ printf("%*sN[%d] (%d dig) %s", i, "", i, nidigits, ptype); fflush(stdout); }
curveresult = (nm1_success > 0)
? _GMP_primality_bls_3(Ni, q, &nm1a)
: _GMP_primality_bls_15(Ni, q, &np1lp, &np1lq);
if (verbose) { printf(" %d\n", curveresult); fflush(stdout); }
if ( ! curveresult ) { /* This ought not happen */
if (verbose)
gmp_printf("\n Could not prove %s with N = %Zd\n", ptype, Ni);
downresult = 1;
continue;
}
goto end_down;
}
pindex = dilist[dindex];
if (pindex < 0) continue; /* We marked this for skip */
/* Get the values for D, degree, and poly type */
poly_degree = poly_class_poly_num(pindex, &D, NULL, &poly_type);
if (poly_degree == 0)
croak("Unknown value in dilist[%d]: %d\n", dindex, pindex);
if ( (-D % 4) != 3 && (-D % 16) != 4 && (-D % 16) != 8 )
croak("Invalid discriminant '%d' in list\n", D);
/* D must also be squarefree in odd divisors, but assume it. */
/* Make sure we can get a class polynomial for this D. */
if (poly_degree > 16 && stage == 0) {
if (verbose) printf(" [1]");
break;
}
/* Make the continue-search vs. backtrack decision */
if (*pmaxH > 0 && poly_degree > *pmaxH) break;
mpz_set_si(mD, D);
/* (D/N) must be 1, and we have to have a u,v solution */
if (mpz_jacobi(mD, Ni) != 1)
continue;
if ( ! modified_cornacchia(u, v, mD, Ni) )
continue;
if (verbose > 1)
{ printf(" %d", D); fflush(stdout); }
/* We're going to factor all the values for this discriminant then pick
* the smallest. This adds a little time, but it means we go down
* faster. This makes smaller proofs, and might even save time. */
choose_m(mlist, D, u, v, Ni, t, t2);
if (allq) {
int x, y;
/* We have 0 to 6 m values. Try to factor them, put in qlist. */
for (k = 0; k < 6; k++) {
mpz_set_ui(qlist[k], 0);
if (mpz_sgn(mlist[k])) {
facresult = check_for_factor(qlist[k], mlist[k], minfactor, t, stage, sfacs, nsfacs, poly_degree);
/* -1 = couldn't find, 0 = no big factors, 1 = found */
if (facresult <= 0)
mpz_set_ui(qlist[k], 0);
}
}
/* Sort any q values by size, so we work on the smallest first */
for (x = 0; x < 5; x++)
if (mpz_sgn(qlist[x]))
for (y = x+1; y < 6; y++)
if (mpz_sgn(qlist[y]) && mpz_cmp(qlist[x],qlist[y]) > 0) {
mpz_swap( qlist[x], qlist[y] );
mpz_swap( mlist[x], mlist[y] );
}
}
/* Try to make a proof with the first (smallest) q value.
* Repeat for others if we have to. */
for (k = 0; k < 6; k++) {
int maxH = *pmaxH;
int minH = (nidigits <= 240) ? 7 : (nidigits+39)/40;
if (allq) {
if (mpz_sgn(qlist[k]) == 0) continue;
mpz_set(m, mlist[k]);
mpz_set(q, qlist[k]);
} else {
if (mpz_sgn(mlist[k]) == 0) continue;
mpz_set(m, mlist[k]);
facresult = check_for_factor(q, m, minfactor, t, stage, sfacs, nsfacs, poly_degree);
if (facresult <= 0) continue;
}
if (verbose)
{ printf(" %d (%s %d)\n", D, poly_class_type_name(poly_type), poly_degree); fflush(stdout); }
if (maxH == 0) {
maxH = minH-1 + poly_degree;
if (facstage > 1) /* We worked hard to get here, */
maxH = 2*maxH + 10; /* try hard to make use of it. */
} else if (maxH > minH && maxH > (poly_degree+2)) {
maxH--;
}
/* Great, now go down. */
downresult = ecpp_down(i+1, q, next_stage, &maxH, dilist, sfacs, nsfacs, prooftextptr);
/* Nothing found, look at more polys in the future */
if (downresult == 1 && *pmaxH > 0) *pmaxH = maxH;
if (downresult == 0) goto end_down; /* composite */
if (downresult == 1) { /* nothing found at this stage */
VERBOSE_PRINT_N(i, nidigits, *pmaxH, facstage);
continue;
}
/* Awesome, we found the q chain and are in STAGE 2 */
if (verbose)
{ printf("%*sN[%d] (%d dig) %d (%s %d)", i, "", i, nidigits, D, poly_class_type_name(poly_type), poly_degree); fflush(stdout); }
/* Try with only one or two roots, then 8 if that didn't work. */
/* TODO: This should be done using a root iterator in find_curve() */
curveresult = find_curve(a, b, P.x, P.y, D, pindex, m, q, Ni, 1);
if (curveresult == 1) {
if (verbose) { printf(" [redo roots]"); fflush(stdout); }
curveresult = find_curve(a, b, P.x, P.y, D, pindex, m, q, Ni, 8);
}
if (verbose) { printf(" %d\n", curveresult); fflush(stdout); }
if (curveresult == 1) {
/* Something is wrong. Very likely the class poly coefficients are
incorrect. We've wasted lots of time, and need to try again. */
dilist[dindex] = -2; /* skip this D value from now on */
if (verbose) gmp_printf("\n Invalidated D = %d with N = %Zd\n", D, Ni);
downresult = 1;
continue;
}
/* We found it was composite or proved it */
goto end_down;
} /* k loop for D */
} /* D */
} /* fac stage */
/* Nothing at this level */
if (downresult != 1) croak("ECPP internal error: downresult is %d at end\n", downresult);
if (verbose) {
if (*pmaxH > 0) printf(" (max %d)", *pmaxH);
printf(" ---\n");
fflush(stdout);
}
if (*pmaxH > 0) *pmaxH = *pmaxH + 2;
end_down:
if (downresult == 2) {
if (0 && verbose > 1) {
gmp_printf("\n");
if (D == 1) {
gmp_printf("Type BLS3\nN %Zd\nQ %Zd\nA %"UVuf"\n", Ni, q, nm1a);
} else if (D == -1) {
gmp_printf("Type BLS15\nN %Zd\nQ %Zd\nLP %"IVdf"\nLQ %"IVdf"\n", Ni, q, np1lp, np1lq);
} else {
gmp_printf("Type ECPP\nN %Zd\nA %Zd\nB %Zd\nM %Zd\nQ %Zd\nX %Zd\nY %Zd\n", Ni, a, b, m, q, P.x, P.y);
}
gmp_printf("\n");
fflush(stdout);
}
/* Prepend our proof to anything that exists. */
if (prooftextptr != 0) {
char *proofstr, *proofptr;
int curprooflen = (*prooftextptr == 0) ? 0 : strlen(*prooftextptr);
if (D == 1) {
int myprooflen = 20 + 2*(4 + mpz_sizeinbase(Ni, 10)) + 1*21;
New(0, proofstr, myprooflen + curprooflen + 1, char);
proofptr = proofstr;
proofptr += gmp_sprintf(proofptr, "Type BLS3\nN %Zd\nQ %Zd\n", Ni,q);
proofptr += sprintf(proofptr, "A %"UVuf"\n", nm1a);
} else if (D == -1) {
