static val_ptr run_order(val_ptr v[]) { field_ptr f = v[0]->field; element_ptr e = (element_ptr)pbc_malloc(sizeof(*e)); element_init(e, M); element_set_mpz(e, f->order); return val_new_element(e); }
static val_ptr v_field_cast(val_ptr v, tree_ptr t) { // TODO: Check args, x is an element. val_ptr x = tree_eval((tree_ptr)darray_at(t->child, 0)); element_ptr e = x->elem; if (e->field == M) { if (v->field == M) return x; element_ptr e2 = element_new(v->field); if (element_is0(e)) // if 'set0' is not 'set1' in base field of GT, but we hope 'GT(0)' calls 'set1', we may directly call 'element_set0' here element_set0(e2); else if (element_is1(e)) // reason is same as above element_set1(e2); else element_set_multiz(e2, (multiz)e->data); x->elem = e2; return x; } if (v->field == M) { // Map to/from integer. TODO: Map to/from multiz instead. mpz_t z; mpz_init(z); element_to_mpz(z, e); element_clear(e); element_init(e, v->field); element_set_mpz(e, z); mpz_clear(z); } return x; }
static val_ptr run_nextprime(val_ptr v[]) { element_ptr e = v[0]->elem; mpz_t z; mpz_init(z); element_to_mpz(z, e); mpz_nextprime(z, z); element_set_mpz(e, z); return v[0]; }
static void e_init_pairing(pairing_t pairing, void *data) { e_param_ptr param = data; e_pairing_data_ptr p; element_t a, b; mpz_init(pairing->r); mpz_set(pairing->r, param->r); field_init_fp(pairing->Zr, pairing->r); pairing->map = e_pairing; e_miller_fn = e_miller_proj; p = pairing->data = pbc_malloc(sizeof(e_pairing_data_t)); p->exp2 = param->exp2; p->exp1 = param->exp1; p->sign1 = param->sign1; p->sign0 = param->sign0; field_init_fp(p->Fq, param->q); element_init(a, p->Fq); element_init(b, p->Fq); element_set_mpz(a, param->a); element_set_mpz(b, param->b); field_init_curve_ab(p->Eq, a, b, pairing->r, param->h); //k=1, hence phikonr = (p-1)/r mpz_init(pairing->phikonr); mpz_sub_ui(pairing->phikonr, p->Fq->order, 1); mpz_divexact(pairing->phikonr, pairing->phikonr, pairing->r); pairing->G2 = pairing->G1 = p->Eq; pairing_GT_init(pairing, p->Fq); pairing->finalpow = e_finalpow; pairing->phi = phi_identity; pairing->option_set = e_pairing_option_set; pairing->clear_func = e_pairing_clear; element_init(p->R, p->Eq); curve_set_gen_no_cofac(p->R); element_clear(a); element_clear(b); }
// g, h in some group of order r // finds x such that g^x = h // will hang if no such x exists // x in some field_t that set_mpz makes sense for void element_dlog_brute_force(element_t x, element_t g, element_t h) { element_t g0; mpz_t count; mpz_init(count); element_init_same_as(g0, g); element_set(g0, g); mpz_set_ui(count, 1); while (element_cmp(g0, h)) { element_mul(g0, g0, g); //element_printf("g0^%Zd = %B\n", count, g0); mpz_add_ui(count, count, 1); } element_set_mpz(x, count); mpz_clear(count); element_clear(g0); }
static val_ptr v_field_cast(val_ptr v, tree_ptr t) { // TODO: Check args, x is an element. val_ptr x = tree_eval(darray_at(t->child, 0)); element_ptr e = x->elem; if (e->field == M) { if (v->field == M) return x; element_ptr e2 = element_new(v->field); element_set_multiz(e2, e->data); x->elem = e2; return x; } if (v->field == M) { // Map to/from integer. TODO: Map to/from multiz instead. mpz_t z; mpz_init(z); element_to_mpz(z, e); element_clear(e); element_init(e, v->field); element_set_mpz(e, z); mpz_clear(z); } return x; }
// x in Z_r, g, h in some group of order r // finds x such that g^x = h void element_dlog_pollard_rho(element_t x, element_t g, element_t h) { // see Blake, Seroussi and Smart // only one snark for this implementation int i, s = 20; field_ptr Zr = x->field, G = g->field; element_t asum; element_t bsum; element_t a[s]; element_t b[s]; element_t m[s]; element_t g0, snark; darray_t hole; int interval = 5; mpz_t counter; int found = 0; mpz_init(counter); element_init(g0, G); element_init(snark, G); element_init(asum, Zr); element_init(bsum, Zr); darray_init(hole); //set up multipliers for (i = 0; i < s; i++) { element_init(a[i], Zr); element_init(b[i], Zr); element_init(m[i], G); element_random(a[i]); element_random(b[i]); element_pow_zn(g0, g, a[i]); element_pow_zn(m[i], h, b[i]); element_mul(m[i], m[i], g0); } element_random(asum); element_random(bsum); element_pow_zn(g0, g, asum); element_pow_zn(snark, h, bsum); element_mul(snark, snark, g0); record(asum, bsum, snark, hole, counter); for (;;) { int len = element_length_in_bytes(snark); unsigned char *buf = pbc_malloc(len); unsigned char hash = 0; element_to_bytes(buf, snark); for (i = 0; i < len; i++) { hash += buf[i]; } i = hash % s; pbc_free(buf); element_mul(snark, snark, m[i]); element_add(asum, asum, a[i]); element_add(bsum, bsum, b[i]); for (i = 0; i < hole->count; i++) { snapshot_ptr ss = hole->item[i]; if (!element_cmp(snark, ss->snark)) { element_sub(bsum, bsum, ss->b); element_sub(asum, ss->a, asum); //answer is x such that x * bsum = asum //complications arise if gcd(bsum, r) > 1 //which can happen if r is not prime if (!mpz_probab_prime_p(Zr->order, 10)) { mpz_t za, zb, zd, zm; mpz_init(za); mpz_init(zb); mpz_init(zd); mpz_init(zm); element_to_mpz(za, asum); element_to_mpz(zb, bsum); mpz_gcd(zd, zb, Zr->order); mpz_divexact(zm, Zr->order, zd); mpz_divexact(zb, zb, zd); //if zd does not divide za there is no solution mpz_divexact(za, za, zd); mpz_invert(zb, zb, zm); mpz_mul(zb, za, zb); mpz_mod(zb, zb, zm); do { element_pow_mpz(g0, g, zb); if (!element_cmp(g0, h)) { element_set_mpz(x, zb); break; } mpz_add(zb, zb, zm); mpz_sub_ui(zd, zd, 1); } while (mpz_sgn(zd)); mpz_clear(zm); mpz_clear(za); mpz_clear(zb); mpz_clear(zd); } else { element_div(x, asum, bsum); } found = 1; break; } } if (found) break; mpz_add_ui(counter, counter, 1); if (mpz_tstbit(counter, interval)) { record(asum, bsum, snark, hole, counter); interval++; } } for (i = 0; i < s; i++) { element_clear(a[i]); element_clear(b[i]); element_clear(m[i]); } element_clear(g0); element_clear(snark); for (i = 0; i < hole->count; i++) { snapshot_ptr ss = hole->item[i]; element_clear(ss->a); element_clear(ss->b); element_clear(ss->snark); pbc_free(ss); } darray_clear(hole); element_clear(asum); element_clear(bsum); mpz_clear(counter); }
static void fq_set_mpz(element_ptr e, mpz_t z) { eptr p = e->data; element_set_mpz(p->x, z); element_set0(p->y); }
int main(void) { mpz_t p, q, N, d; mpz_t dmp1, dmq1; mpz_t ipmq, iqmp; mpz_t adq, adp; field_t f; element_t a, b; double t0, t1, tnaive = 0, tcrt=0; int i, n; mpz_init(p); mpz_init(q); mpz_init(N); mpz_init(d); mpz_init(dmp1); mpz_init(dmq1); mpz_init(ipmq); mpz_init(iqmp); mpz_init(adp); mpz_init(adq); pbc_mpz_randomb(p, 512); pbc_mpz_randomb(q, 512); mpz_nextprime(p, p); mpz_nextprime(q, q); mpz_mul(N, p, q); mpz_invert(ipmq, p, q); mpz_invert(iqmp, q, p); field_init_fp(f, N); element_init(a, f); element_init(b, f); n = 10; for (i=0; i<n; i++) { pbc_mpz_random(d, N); element_random(a); t0 = pbc_get_time(); element_pow_mpz(b, a, d); t1 = pbc_get_time(); tnaive += t1 - t0; mpz_sub_ui(p, p, 1); mpz_sub_ui(q, q, 1); mpz_mod(dmp1, d, p); mpz_mod(dmq1, d, q); mpz_add_ui(p, p, 1); mpz_add_ui(q, q, 1); element_to_mpz(adq, a); element_to_mpz(adp, a); t0 = pbc_get_time(); mpz_powm(adp, adp, d, p); mpz_powm(adq, adq, d, q); /* textbook CRT mpz_mul(adp, adp, q); mpz_mul(adp, adp, iqmp); mpz_mul(adq, adq, p); mpz_mul(adq, adq, ipmq); mpz_add(adp, adp, adq); */ // Garner's algorithm mpz_sub(adq, adq, adp); mpz_mul(adq, adq, ipmq); mpz_mod(adq, adq, q); mpz_mul(adq, adq, p); mpz_add(adp, adp, adq); t1 = pbc_get_time(); tcrt += t1 - t0; element_set_mpz(b, adp); } printf("average RSA exp time = %lf\n", tnaive / n); printf("average RSA exp time (CRT) = %lf\n", tcrt / n); return 0; }
void pbc_param_init_f_gen(pbc_param_t p, int bits) { f_init(p); f_param_ptr fp = p->data; //36 is a 6-bit number int xbit = (bits - 6) / 4; //TODO: use binary search to find smallest appropriate x mpz_t x, t; mpz_ptr q = fp->q; mpz_ptr r = fp->r; mpz_ptr b = fp->b; field_t Fq, Fq2, Fq2x; element_t e1; element_t f; field_t c; element_t P; mpz_init(x); mpz_init(t); mpz_setbit(x, xbit); for (;;) { mpz_mul(t, x, x); mpz_mul_ui(t, t, 6); mpz_add_ui(t, t, 1); tryminusx(q, x); mpz_sub(r, q, t); mpz_add_ui(r, r, 1); if (mpz_probab_prime_p(q, 10) && mpz_probab_prime_p(r, 10)) break; tryplusx(q, x); mpz_sub(r, q, t); mpz_add_ui(r, r, 1); if (mpz_probab_prime_p(q, 10) && mpz_probab_prime_p(r, 10)) break; mpz_add_ui(x, x, 1); } field_init_fp(Fq, q); element_init(e1, Fq); for (;;) { element_random(e1); field_init_curve_b(c, e1, r, NULL); element_init(P, c); element_random(P); element_mul_mpz(P, P, r); if (element_is0(P)) break; element_clear(P); field_clear(c); } element_to_mpz(b, e1); element_clear(e1); field_init_quadratic(Fq2, Fq); element_to_mpz(fp->beta, field_get_nqr(Fq)); field_init_poly(Fq2x, Fq2); element_init(f, Fq2x); // Find an irreducible polynomial of the form f = x^6 + alpha. // Call poly_set_coeff1() first so we can use element_item() for the other // coefficients. poly_set_coeff1(f, 6); for (;;) { element_random(element_item(f, 0)); if (poly_is_irred(f)) break; } //extend F_q^2 using f = x^6 + alpha //see if sextic twist contains a subgroup of order r //if not, it's the wrong twist: replace alpha with alpha^5 { field_t ctest; element_t Ptest; mpz_t z0, z1; mpz_init(z0); mpz_init(z1); element_init(e1, Fq2); element_set_mpz(e1, fp->b); element_mul(e1, e1, element_item(f, 0)); element_neg(e1, e1); field_init_curve_b(ctest, e1, r, NULL); element_init(Ptest, ctest); element_random(Ptest); //I'm not sure what the #E'(F_q^2) is, but //it definitely divides n_12 = #E(F_q^12). It contains a //subgroup of order r if and only if //(n_12 / r^2)P != O for some (in fact most) P in E'(F_q^6) mpz_pow_ui(z0, q, 12); mpz_add_ui(z0, z0, 1); pbc_mpz_trace_n(z1, q, t, 12); mpz_sub(z1, z0, z1); mpz_mul(z0, r, r); mpz_divexact(z1, z1, z0); element_mul_mpz(Ptest, Ptest, z1); if (element_is0(Ptest)) { mpz_set_ui(z0, 5); element_pow_mpz(element_item(f, 0), element_item(f, 0), z0); } element_clear(e1); element_clear(Ptest); field_clear(ctest); mpz_clear(z0); mpz_clear(z1); } element_to_mpz(fp->alpha0, element_x(element_item(f, 0))); element_to_mpz(fp->alpha1, element_y(element_item(f, 0))); element_clear(f); field_clear(Fq2x); field_clear(Fq2); field_clear(Fq); mpz_clear(t); mpz_clear(x); }
static void f_init_pairing(pairing_t pairing, void *data) { f_param_ptr param = data; f_pairing_data_ptr p; element_t irred; element_t e0, e1, e2; p = pairing->data = pbc_malloc(sizeof(f_pairing_data_t)); mpz_init(pairing->r); mpz_set(pairing->r, param->r); field_init_fp(pairing->Zr, pairing->r); field_init_fp(p->Fq, param->q); p->Fq->nqr = pbc_malloc(sizeof(element_t)); element_init(p->Fq->nqr, p->Fq); element_set_mpz(p->Fq->nqr, param->beta); field_init_quadratic(p->Fq2, p->Fq); field_init_poly(p->Fq2x, p->Fq2); element_init(irred, p->Fq2x); // Call poly_set_coeff1() first so we can use element_item() for the other // coefficients. poly_set_coeff1(irred, 6); element_init(p->negalpha, p->Fq2); element_init(p->negalphainv, p->Fq2); element_set_mpz(element_x(p->negalpha), param->alpha0); element_set_mpz(element_y(p->negalpha), param->alpha1); element_set(element_item(irred, 0), p->negalpha); field_init_polymod(p->Fq12, irred); element_neg(p->negalpha, p->negalpha); element_invert(p->negalphainv, p->negalpha); element_clear(irred); element_init(e0, p->Fq); element_init(e1, p->Fq); element_init(e2, p->Fq2); // Initialize the curve Y^2 = X^3 + b. element_set_mpz(e1, param->b); field_init_curve_ab(p->Eq, e0, e1, pairing->r, NULL); // Initialize the curve Y^2 = X^3 - alpha0 b - alpha1 sqrt(beta) b. element_set_mpz(e0, param->alpha0); element_neg(e0, e0); element_mul(element_x(e2), e0, e1); element_set_mpz(e0, param->alpha1); element_neg(e0, e0); element_mul(element_y(e2), e0, e1); element_clear(e0); element_init(e0, p->Fq2); field_init_curve_ab(p->Etwist, e0, e2, pairing->r, NULL); element_clear(e0); element_clear(e1); element_clear(e2); mpz_t ndonr; mpz_init(ndonr); // ndonr temporarily holds the trace. mpz_sub(ndonr, param->q, param->r); mpz_add_ui(ndonr, ndonr, 1); // TODO: We can use a smaller quotient_cmp, but I have to figure out // BN curves again. pbc_mpz_curve_order_extn(ndonr, param->q, ndonr, 12); mpz_divexact(ndonr, ndonr, param->r); mpz_divexact(ndonr, ndonr, param->r); field_curve_set_quotient_cmp(p->Etwist, ndonr); mpz_clear(ndonr); pairing->G1 = p->Eq; pairing->G2 = p->Etwist; pairing_GT_init(pairing, p->Fq12); pairing->finalpow = f_finalpow; pairing->map = f_pairing; pairing->clear_func = f_pairing_clear; mpz_init(p->tateexp); /* unoptimized tate exponent mpz_pow_ui(p->tateexp, param->q, 12); mpz_sub_ui(p->tateexp, p->tateexp, 1); mpz_divexact(p->tateexp, p->tateexp, param->r); */ mpz_ptr z = p->tateexp; mpz_mul(z, param->q, param->q); mpz_sub_ui(z, z, 1); mpz_mul(z, z, param->q); mpz_mul(z, z, param->q); mpz_add_ui(z, z, 1); mpz_divexact(z, z, param->r); element_init(p->xpowq2, p->Fq2); element_init(p->xpowq6, p->Fq2); element_init(p->xpowq8, p->Fq2); element_t xpowq; element_init(xpowq, p->Fq12); //there are smarter ways since we know q = 1 mod 6 //and that x^6 = -alpha //but this is fast enough element_set1(element_item(xpowq, 1)); element_pow_mpz(xpowq, xpowq, param->q); element_pow_mpz(xpowq, xpowq, param->q); element_set(p->xpowq2, element_item(xpowq, 1)); element_pow_mpz(xpowq, xpowq, param->q); element_pow_mpz(xpowq, xpowq, param->q); element_pow_mpz(xpowq, xpowq, param->q); element_pow_mpz(xpowq, xpowq, param->q); element_set(p->xpowq6, element_item(xpowq, 1)); element_pow_mpz(xpowq, xpowq, param->q); element_pow_mpz(xpowq, xpowq, param->q); element_set(p->xpowq8, element_item(xpowq, 1)); element_clear(xpowq); }
static void d_init_pairing(pairing_ptr pairing, void *data) { d_param_ptr param = data; pptr p; element_t a, b; element_t irred; int d = param->k / 2; int i; if (param->k % 2) pbc_die("k must be even"); mpz_init(pairing->r); mpz_set(pairing->r, param->r); field_init_fp(pairing->Zr, pairing->r); pairing->map = cc_pairing; pairing->prod_pairings = cc_pairings_affine; pairing->is_almost_coddh = cc_is_almost_coddh; p = pairing->data = pbc_malloc(sizeof(*p)); field_init_fp(p->Fq, param->q); element_init(a, p->Fq); element_init(b, p->Fq); element_set_mpz(a, param->a); element_set_mpz(b, param->b); field_init_curve_ab(p->Eq, a, b, pairing->r, param->h); field_init_poly(p->Fqx, p->Fq); element_init(irred, p->Fqx); poly_set_coeff1(irred, d); for (i = 0; i < d; i++) { element_set_mpz(element_item(irred, i), param->coeff[i]); } field_init_polymod(p->Fqd, irred); element_clear(irred); p->Fqd->nqr = pbc_malloc(sizeof(element_t)); element_init(p->Fqd->nqr, p->Fqd); element_set_mpz(((element_t *) p->Fqd->nqr->data)[0], param->nqr); field_init_quadratic(p->Fqk, p->Fqd); // Compute constants involved in the final powering. if (param->k == 6) { mpz_ptr q = param->q; mpz_ptr z = pairing->phikonr; mpz_init(z); mpz_mul(z, q, q); mpz_sub(z, z, q); mpz_add_ui(z, z, 1); mpz_divexact(z, z, pairing->r); element_ptr e = p->xpowq; element_init(e, p->Fqd); element_set1(((element_t *) e->data)[1]); element_pow_mpz(e, e, q); element_init(p->xpowq2, p->Fqd); element_square(p->xpowq2, e); } else { mpz_init(p->tateexp); mpz_sub_ui(p->tateexp, p->Fqk->order, 1); mpz_divexact(p->tateexp, p->tateexp, pairing->r); } field_init_curve_ab_map(p->Etwist, p->Eq, element_field_to_polymod, p->Fqd, pairing->r, NULL); field_reinit_curve_twist(p->Etwist); mpz_t ndonr; mpz_init(ndonr); // ndonr temporarily holds the trace. mpz_sub(ndonr, param->q, param->n); mpz_add_ui(ndonr, ndonr, 1); // Negate it because we want the trace of the twist. mpz_neg(ndonr, ndonr); pbc_mpz_curve_order_extn(ndonr, param->q, ndonr, d); mpz_divexact(ndonr, ndonr, param->r); field_curve_set_quotient_cmp(p->Etwist, ndonr); mpz_clear(ndonr); element_init(p->nqrinv, p->Fqd); element_invert(p->nqrinv, field_get_nqr(p->Fqd)); element_init(p->nqrinv2, p->Fqd); element_square(p->nqrinv2, p->nqrinv); pairing->G1 = p->Eq; pairing->G2 = p->Etwist; p->k = param->k; pairing_GT_init(pairing, p->Fqk); pairing->finalpow = cc_finalpow; // By default use affine coordinates. cc_miller_no_denom_fn = cc_miller_no_denom_affine; pairing->option_set = d_pairing_option_set; pairing->pp_init = d_pairing_pp_init; pairing->pp_clear = d_pairing_pp_clear; pairing->pp_apply = d_pairing_pp_apply; pairing->clear_func = d_pairing_clear; element_clear(a); element_clear(b); }
// Computes a curve and sets fp to the field it is defined over using the // complex multiplication method, where cm holds the appropriate information // (e.g. discriminant, field order). static void compute_cm_curve(d_param_ptr param, pbc_cm_ptr cm) { element_t hp, root; field_t fp, fpx; field_t cc; field_init_fp(fp, cm->q); field_init_poly(fpx, fp); element_init(hp, fpx); mpz_t *coefflist; int n = (int)pbc_hilbert(&coefflist, cm->D); // Temporarily set the coefficient of x^{n-1} to 1 so hp has degree n - 1, // allowing us to use poly_coeff(). poly_set_coeff1(hp, n - 1); int i; for (i = 0; i < n; i++) { element_set_mpz(element_item(hp, i), coefflist[i]); } pbc_hilbert_free(coefflist, n); // TODO: Remove x = 0, 1728 roots. // TODO: What if there are no roots? //printf("hp "); //element_out_str(stdout, 0, hp); //printf("\n"); element_init(root, fp); poly_findroot(root, hp); //printf("root = "); //element_out_str(stdout, 0, root); //printf("\n"); element_clear(hp); field_clear(fpx); // The root is the j-invariant of the desired curve. field_init_curve_j(cc, root, cm->n, NULL); element_clear(root); // We may need to twist it. { // Pick a random point P and twist the curve if it has the wrong order. element_t P; element_init(P, cc); element_random(P); element_mul_mpz(P, P, cm->n); if (!element_is0(P)) field_reinit_curve_twist(cc); element_clear(P); } mpz_set(param->q, cm->q); mpz_set(param->n, cm->n); mpz_set(param->h, cm->h); mpz_set(param->r, cm->r); element_to_mpz(param->a, curve_field_a_coeff(cc)); element_to_mpz(param->b, curve_field_b_coeff(cc)); param->k = cm->k; { mpz_t z; mpz_init(z); // Compute order of curve in F_q^k. // n = q - t + 1 hence t = q - n + 1 mpz_sub(z, param->q, param->n); mpz_add_ui(z, z, 1); pbc_mpz_trace_n(z, param->q, z, param->k); mpz_pow_ui(param->nk, param->q, param->k); mpz_sub_ui(z, z, 1); mpz_sub(param->nk, param->nk, z); mpz_mul(z, param->r, param->r); mpz_divexact(param->hk, param->nk, z); mpz_clear(z); } field_clear(cc); field_clear(fp); }