void fp2_nord_low(dv2_t c, dv2_t a) { dv2_t t; bn_t b; dv2_null(t); bn_null(b); TRY { dv2_new(t); bn_new(b); #ifdef FP_QNRES /* If p = 3 mod 8, (1 + i) is a QNR/CNR. */ /* (a_0 + a_1 * i) * (1 + i) = (a_0 - a_1) + (a_0 + a_1) * u. */ dv_copy(t[0], a[1], 2 * FP_DIGS); fp_addc_low(c[1], a[0], a[1]); fp_subc_low(c[0], a[0], t[0]); #else switch (fp_prime_get_mod8()) { case 3: /* If p = 3 mod 8, (1 + u) is a QNR, u^2 = -1. */ /* (a_0 + a_1 * u) * (1 + u) = (a_0 - a_1) + (a_0 + a_1) * u. */ dv_copy(t[0], a[1], 2 * FP_DIGS); fp_addc_low(c[1], a[0], a[1]); fp_subc_low(c[0], a[0], t[0]); break; case 1: case 5: /* If p = 1,5 mod 8, (u) is a QNR. */ dv_copy(t[0], a[0], 2 * FP_DIGS); dv_zero(t[1], FP_DIGS); dv_copy(t[1] + FP_DIGS, fp_prime_get(), FP_DIGS); fp_subc_low(c[0], t[1], a[1]); for (int i = -1; i > fp_prime_get_qnr(); i--) { fp_subc_low(c[0], c[0], a[1]); } dv_copy(c[1], t[0], 2 * FP_DIGS); break; case 7: /* If p = 7 mod 8, (2 + u) is a QNR/CNR. */ fp2_addc_low(t, a, a); fp_subc_low(c[0], t[0], a[1]); fp_addc_low(c[1], t[1], a[0]); break; default: THROW(ERR_NO_VALID); break; } #endif } CATCH_ANY { THROW(ERR_CAUGHT); } FINALLY { dv2_free(t); bn_free(b); } }
void fp3_sqrn_low(dv3_t c, fp3_t a) { align dig_t t0[2 * FP_DIGS], t1[2 * FP_DIGS], t2[2 * FP_DIGS]; align dig_t t3[2 * FP_DIGS], t4[2 * FP_DIGS], t5[2 * FP_DIGS]; /* t0 = a_0^2. */ fp_sqrn_low(t0, a[0]); /* t1 = 2 * a_1 * a_2. */ #ifdef FP_SPACE fp_dbln_low(t2, a[1]); #else fp_dblm_low(t2, a[1]); #endif fp_muln_low(t1, t2, a[2]); /* t2 = a_2^2. */ fp_sqrn_low(t2, a[2]); /* t3 = (a_0 + a_2 + a_1)^2, t4 = (a_0 + a_2 - a_1)^2. */ #ifdef FP_SPACE fp_addn_low(t3, a[0], a[2]); fp_addn_low(t4, t3, a[1]); #else fp_addm_low(t3, a[0], a[2]); fp_addm_low(t4, t3, a[1]); #endif fp_subm_low(t5, t3, a[1]); fp_sqrn_low(t3, t4); fp_sqrn_low(t4, t5); /* t4 = (t4 + t3)/2. */ fp_addd_low(t4, t4, t3); fp_hlvd_low(t4, t4); /* t3 = t3 - t4 - t1. */ fp_addc_low(t5, t1, t4); fp_subc_low(t3, t3, t5); /* c_2 = t4 - t0 - t2. */ fp_addc_low(t5, t0, t2); fp_subc_low(c[2], t4, t5); /* c_0 = t0 + t1 * B. */ fp_subc_low(c[0], t0, t1); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(c[0], c[0], t1); } /* c_1 = t3 + t2 * B. */ fp_subc_low(c[1], t3, t2); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(c[1], c[1], t2); } }
void fp2_mul_basic(fp2_t c, fp2_t a, fp2_t b) { dv_t t0, t1, t2, t3, t4; dv_null(t0); dv_null(t1); dv_null(t2); dv_null(t3); dv_null(t4); TRY { dv_new(t0); dv_new(t1); dv_new(t2); dv_new(t3); dv_new(t4); /* Karatsuba algorithm. */ /* t2 = a_0 + a_1, t1 = b0 + b1. */ fp_add(t2, a[0], a[1]); fp_add(t1, b[0], b[1]); /* t3 = (a_0 + a_1) * (b0 + b1). */ fp_muln_low(t3, t2, t1); /* t0 = a_0 * b0, t4 = a_1 * b1. */ fp_muln_low(t0, a[0], b[0]); fp_muln_low(t4, a[1], b[1]); /* t2 = (a_0 * b0) + (a_1 * b1). */ fp_addc_low(t2, t0, t4); /* t1 = (a_0 * b0) + u^2 * (a_1 * b1). */ fp_subc_low(t1, t0, t4); /* t1 = u^2 * (a_1 * b1). */ for (int i = -1; i > fp_prime_get_qnr(); i--) { fp_subc_low(t1, t1, t4); } /* c_0 = t1 mod p. */ fp_rdc(c[0], t1); /* t4 = t3 - t2. */ fp_subc_low(t4, t3, t2); /* c_1 = t4 mod p. */ fp_rdc(c[1], t4); } CATCH_ANY { THROW(ERR_CAUGHT); } FINALLY { dv_free(t0); dv_free(t1); dv_free(t2); dv_free(t3); dv_free(t4); } }
void fp2_muln_low(dv2_t c, fp2_t a, fp2_t b) { align dig_t t0[2 * FP_DIGS], t1[2 * FP_DIGS], t2[2 * FP_DIGS]; /* Karatsuba algorithm. */ /* t0 = a_0 + a_1, t1 = b_0 + b_1. */ #ifdef FP_SPACE fp_addn_low(t0, a[0], a[1]); fp_addn_low(t1, b[0], b[1]); #else fp_addm_low(t0, a[0], a[1]); fp_addm_low(t1, b[0], b[1]); #endif /* c_0 = a_0 * b_0, c_1 = a_1 * b_1. */ fp_muln_low(c[0], a[0], b[0]); fp_muln_low(c[1], a[1], b[1]); /* t2 = (a_0 + a_1) * (b_0 + b_1). */ fp_muln_low(t2, t0, t1); /* t0 = (a_0 * b_0) + (a_1 * b_1). */ #ifdef FP_SPACE fp_addd_low(t0, c[0], c[1]); #else fp_addc_low(t0, c[0], c[1]); #endif /* c_0 = (a_0 * b_0) + u^2 * (a_1 * b_1). */ fp_subc_low(c[0], c[0], c[1]); #ifndef FP_QNRES /* t1 = u^2 * (a_1 * b_1). */ for (int i = -1; i > fp_prime_get_qnr(); i--) { fp_subc_low(c[0], c[0], c[1]); } for (int i = 0; i <= fp_prime_get_qnr(); i++) { fp_addc_low(c[0], c[0], c[1]); } #endif /* c_1 = t2 - t0. */ #ifdef FP_SPACE fp_subd_low(c[1], t2, t0); #else fp_subc_low(c[1], t2, t0); #endif }
void fp3_muln_low(dv3_t c, fp3_t a, fp3_t b) { align dig_t t0[2 * FP_DIGS], t1[2 * FP_DIGS], t2[2 * FP_DIGS], t3[2 * FP_DIGS]; align dig_t t4[2 * FP_DIGS], t5[2 * FP_DIGS], t6[2 * FP_DIGS]; /* Karatsuba algorithm. */ /* t0 = a_0 * b_0, t1 = a_1 * b_1, t2 = a_2 * b_2. */ fp_muln_low(t0, a[0], b[0]); fp_muln_low(t1, a[1], b[1]); fp_muln_low(t2, a[2], b[2]); /* t3 = (a_1 + a_2) * (b_1 + b_2). */ #ifdef FP_SPACE fp_addn_low(t3, a[1], a[2]); fp_addn_low(t4, b[1], b[2]); #else fp_addm_low(t3, a[1], a[2]); fp_addm_low(t4, b[1], b[2]); #endif fp_muln_low(t5, t3, t4); fp_addd_low(t6, t1, t2); fp_subc_low(t4, t5, t6); fp_subc_low(c[0], t0, t4); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(c[0], c[0], t4); } #ifdef FP_SPACE fp_addn_low(t4, a[0], a[1]); fp_addn_low(t5, b[0], b[1]); #else fp_addm_low(t4, a[0], a[1]); fp_addm_low(t5, b[0], b[1]); #endif fp_muln_low(t6, t4, t5); fp_addd_low(t4, t0, t1); fp_subc_low(t4, t6, t4); fp_subc_low(c[1], t4, t2); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(c[1], c[1], t2); } #ifdef FP_SPACE fp_addn_low(t5, a[0], a[2]); fp_addn_low(t6, b[0], b[2]); #else fp_addm_low(t5, a[0], a[2]); fp_addm_low(t6, b[0], b[2]); #endif fp_muln_low(t4, t5, t6); fp_addd_low(t6, t0, t2); fp_subc_low(t5, t4, t6); fp_addc_low(c[2], t5, t1); }
void fp2_addc_low(dv2_t c, dv2_t a, dv2_t b) { fp_addc_low(c[0], a[0], b[0]); fp_addc_low(c[1], a[1], b[1]); }
void fp2_nord_low(dv2_t c, dv2_t a) { dv2_t t; bn_t b; dv2_null(t); bn_null(b); TRY { dv2_new(t); bn_new(b); #if FP_PRIME == 158 fp_addc_low(t[0], a[0], a[0]); fp_addc_low(t[0], t[0], t[0]); fp_subc_low(t[0], t[0], a[1]); fp_addc_low(t[1], a[1], a[1]); fp_addc_low(t[1], t[1], t[1]); fp_addc_low(c[1], a[0], t[1]); dv_copy(c[0], t[0], 2 * FP_DIGS); #elif defined(FP_QNRES) /* If p = 3 mod 8, (1 + i) is a QNR/CNR. */ /* (a_0 + a_1 * i) * (1 + i) = (a_0 - a_1) + (a_0 + a_1) * u. */ dv_copy(t[0], a[1], 2 * FP_DIGS); fp_addc_low(c[1], a[0], a[1]); fp_subc_low(c[0], a[0], t[0]); #else switch (fp_prime_get_mod8()) { case 3: /* If p = 3 mod 8, (1 + u) is a QNR, u^2 = -1. */ /* (a_0 + a_1 * u) * (1 + u) = (a_0 - a_1) + (a_0 + a_1) * u. */ dv_copy(t[0], a[1], 2 * FP_DIGS); fp_addc_low(c[1], a[0], a[1]); fp_subc_low(c[0], a[0], t[0]); break; case 5: /* If p = 5 mod 8, (u) is a QNR. */ dv_copy(t[0], a[0], 2 * FP_DIGS); dv_zero(t[1], FP_DIGS); dv_copy(t[1] + FP_DIGS, fp_prime_get(), FP_DIGS); fp_subc_low(c[0], t[1], a[1]); for (int i = -1; i > fp_prime_get_qnr(); i--) { fp_subc_low(c[0], c[0], a[1]); } dv_copy(c[1], t[0], 2 * FP_DIGS); break; case 7: /* If p = 7 mod 8, (2^lg_4(b-1) + u) is a QNR/CNR. */ /* (a_0 + a_1 * u)(2^lg_4(b-1) + u) = * (2^lg_4(b-1)a_0 - a_1) + (a_0 + 2^lg_4(b-1)a_1 * u. */ fp2_addc_low(t, a, a); fp_prime_back(b, ep_curve_get_b()); for (int i = 1; i < bn_bits(b) / 2; i++) { fp2_addc_low(t, t, t); } fp_subc_low(c[0], t[0], a[1]); fp_addc_low(c[1], t[1], a[0]); break; default: THROW(ERR_NO_VALID); break; } #endif } CATCH_ANY { THROW(ERR_CAUGHT); } FINALLY { dv2_free(t); bn_free(b); } }
void fp3_sqr_basic(fp3_t c, fp3_t a) { dv_t t0, t1, t2, t3, t4, t5; dv_null(t0); dv_null(t1); dv_null(t2); dv_null(t3); dv_null(t4); dv_null(t5); TRY { dv_new(t0); dv_new(t1); dv_new(t2); dv_new(t3); dv_new(t4); dv_new(t5); /* t0 = a_0^2. */ fp_sqrn_low(t0, a[0]); /* t1 = 2 * a_1 * a_2. */ fp_dbl(t2, a[1]); fp_muln_low(t1, t2, a[2]); /* t2 = a_2^2. */ fp_sqrn_low(t2, a[2]); /* t3 = (a_0 + a_2 + a_1)^2, t4 = (a_0 + a_2 - a_1)^2. */ fp_add(t3, a[0], a[2]); fp_add(t4, t3, a[1]); fp_sub(t5, t3, a[1]); fp_sqrn_low(t3, t4); fp_sqrn_low(t4, t5); /* t4 = (t4 + t3)/2. */ fp_addd_low(t4, t4, t3); fp_hlvd_low(t4, t4); /* t3 = t3 - t4 - t1. */ fp_addc_low(t5, t1, t4); fp_subc_low(t3, t3, t5); /* c_2 = t4 - t0 - t2. */ fp_addc_low(t5, t0, t2); fp_subc_low(t4, t4, t5); fp_rdc(c[2], t4); /* c_0 = t0 + t1 * B. */ fp_subc_low(t0, t0, t1); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(t0, t0, t1); } fp_rdc(c[0], t0); /* c_1 = t3 + t2 * B. */ fp_subc_low(t3, t3, t2); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(t3, t3, t2); } fp_rdc(c[1], t3); } CATCH_ANY { THROW(ERR_CAUGHT); } FINALLY { dv_free(t0); dv_free(t1); dv_free(t2); dv_free(t3); dv_free(t4); dv_free(t5); } }
void fp2_sqrn_low(dv2_t c, fp2_t a) { align dig_t t0[2 * FP_DIGS], t1[2 * FP_DIGS], t2[2 * FP_DIGS]; /* t0 = (a0 + a1). */ #ifdef FP_SPACE /* if we have room for carries, we can avoid reductions here. */ fp_addn_low(t0, a[0], a[1]); #else fp_addm_low(t0, a[0], a[1]); #endif /* t1 = (a0 - a1). */ fp_subm_low(t1, a[0], a[1]); #ifdef FP_QNRES #ifdef FP_SPACE fp_dbln_low(t2, a[0]); #else fp_dblm_low(t2, a[0]); #endif /* c1 = 2 * a0 * a1. */ fp_muln_low(c[1], t2, a[1]); /* c_0 = a_0^2 + a_1^2 * u^2. */ fp_muln_low(c[0], t0, t1); #else /* !FP_QNRES */ /* t1 = u^2 * (a1 * b1). */ for (int i = -1; i > fp_prime_get_qnr(); i--) { fp_subm_low(t1, t1, a[1]); } if (fp_prime_get_qnr() == -1) { /* t2 = 2 * a0. */ fp_dbl(t2, a[0]); /* c1 = 2 * a0 * a1. */ fp_muln_low(c[1], t2, a[1]); /* c0 = a0^2 + b_0^2 * u^2. */ fp_muln_low(c[0], t0, t1); } else { /* c1 = a0 * a1. */ fp_muln_low(c[1], a[0], a[1]); /* c0 = a0^2 + b_0^2 * u^2. */ fp_muln_low(c[0], t0, t1); #ifdef FP_SPACE for (int i = -1; i > fp_prime_get_qnr(); i--) { fp_addd_low(c[0], c[0], c[1]); } /* c1 = 2 * a0 * a1. */ fp_addd_low(c[1], c[1], c[1]); #else for (int i = -1; i > fp_prime_get_qnr(); i--) { fp_addc_low(c[0], c[0], c[1]); } /* c1 = 2 * a0 * a1. */ fp_addc_low(c[1], c[1], c[1]); #endif } #endif /* c = c0 + c1 * u. */ }
void pp_dbl_k2_projc_lazyr(fp2_t l, ep_t r, ep_t p, ep_t q) { fp_t t0, t1, t2, t3, t4, t5; dv_t u0, u1; fp_null(t0); fp_null(t1); fp_null(t2); fp_null(t3); fp_null(t4); fp_null(t5); dv_null(u0); dv_null(u1); TRY { fp_new(t0); fp_new(t1); fp_new(t2); fp_new(t3); fp_new(t4); fp_new(t5); dv_new(u0); dv_new(u1); /* For these curves, we always can choose a = -3. */ /* dbl-2001-b formulas: 3M + 5S + 8add + 1*4 + 2*8 + 1*3 */ /* http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b */ /* t0 = delta = z1^2. */ fp_sqr(t0, p->z); /* t1 = gamma = y1^2. */ fp_sqr(t1, p->y); /* t2 = beta = x1 * y1^2. */ fp_mul(t2, p->x, t1); /* t3 = alpha = 3 * (x1 - z1^2) * (x1 + z1^2). */ fp_sub(t3, p->x, t0); fp_add(t4, p->x, t0); fp_mul(t4, t3, t4); fp_dbl(t3, t4); fp_add(t3, t3, t4); /* t2 = 4 * beta. */ fp_dbl(t2, t2); fp_dbl(t2, t2); /* z3 = (y1 + z1)^2 - gamma - delta. */ fp_add(r->z, p->y, p->z); fp_sqr(r->z, r->z); fp_sub(r->z, r->z, t1); fp_sub(r->z, r->z, t0); /* l0 = 2 * gamma - alpha * (delta * xq + x1). */ fp_dbl(t1, t1); fp_mul(t5, t0, q->x); fp_add(t5, t5, p->x); fp_mul(t5, t5, t3); fp_sub(l[0], t1, t5); /* x3 = alpha^2 - 8 * beta. */ fp_dbl(t5, t2); fp_sqr(r->x, t3); fp_sub(r->x, r->x, t5); /* y3 = alpha * (4 * beta - x3) - 8 * gamma^2. */ fp_sqrn_low(u0, t1); fp_addc_low(u0, u0, u0); fp_subm_low(r->y, t2, r->x); fp_muln_low(u1, r->y, t3); fp_subc_low(u1, u1, u0); fp_rdcn_low(r->y, u1); /* l1 = - z3 * delta * yq. */ fp_mul(l[1], r->z, t0); fp_mul(l[1], l[1], q->y); r->norm = 0; } CATCH_ANY { THROW(ERR_CAUGHT); } FINALLY { fp_free(t0); fp_free(t1); fp_free(t2); fp_free(t3); fp_free(t4); fp_free(t5); dv_free(u0); dv_free(u1); } }
void fp3_mul_basic(fp3_t c, fp3_t a, fp3_t b) { dv_t t, t0, t1, t2, t3, t4, t5, t6; dv_null(t); dv_null(t0); dv_null(t1); dv_null(t2); dv_null(t3); dv_null(t4); dv_null(t5); dv_null(t6); TRY { dv_new(t); dv_new(t0); dv_new(t1); dv_new(t2); dv_new(t3); dv_new(t4); dv_new(t5); dv_new(t6); /* Karatsuba algorithm. */ /* t0 = a_0 * b_0, t1 = a_1 * b_1, t2 = a_2 * b_2. */ fp_muln_low(t0, a[0], b[0]); fp_muln_low(t1, a[1], b[1]); fp_muln_low(t2, a[2], b[2]); /* t3 = (a_1 + a_2) * (b_1 + b_2). */ fp_add(t3, a[1], a[2]); fp_add(t4, b[1], b[2]); fp_muln_low(t, t3, t4); fp_addd_low(t6, t1, t2); fp_subc_low(t4, t, t6); fp_subc_low(t3, t0, t4); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(t3, t3, t4); } fp_add(t4, a[0], a[1]); fp_add(t5, b[0], b[1]); fp_muln_low(t, t4, t5); fp_addd_low(t4, t0, t1); fp_subc_low(t4, t, t4); fp_subc_low(t4, t4, t2); for (int i = -1; i > fp_prime_get_cnr(); i--) { fp_subc_low(t4, t4, t2); } fp_add(t5, a[0], a[2]); fp_add(t6, b[0], b[2]); fp_muln_low(t, t5, t6); fp_addd_low(t6, t0, t2); fp_subc_low(t5, t, t6); fp_addc_low(t5, t5, t1); /* c_0 = t3 mod p. */ fp_rdc(c[0], t3); /* c_1 = t4 mod p. */ fp_rdc(c[1], t4); /* c_2 = t5 mod p. */ fp_rdc(c[2], t5); } CATCH_ANY { THROW(ERR_CAUGHT); } FINALLY { dv_free(t); dv_free(t0); dv_free(t1); dv_free(t2); dv_free(t3); dv_free(t4); dv_free(t5); dv_free(t6); } }