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);
}
}
static void fp4_mul_unr(dv2_t e, dv2_t f, fp2_t a, fp2_t b, fp2_t c, fp2_t d) {
fp2_t t0, t1;
dv2_t u0, u1;
fp2_null(t0);
fp2_null(t1);
dv2_null(u0);
dv2_null(u1);
TRY {
fp2_new(t0);
fp2_new(t1);
dv2_new(u0);
dv2_new(u1);
#ifdef FP_SPACE
fp2_mulc_low(u0, a, c);
fp2_mulc_low(u1, b, d);
fp2_addn_low(t0, c, d);
fp2_addn_low(t1, a, b);
#else
fp2_muln_low(u0, a, c);
fp2_muln_low(u1, b, d);
fp2_addm_low(t0, c, d);
fp2_addm_low(t1, a, b);
#endif
fp2_muln_low(f, t1, t0);
fp2_subc_low(f, f, u0);
fp2_subc_low(f, f, u1);
fp2_norh_low(e, u1);
fp2_addc_low(e, e, u0);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t0);
dv2_free(t1);
dv2_free(u0);
dv2_free(u1);
}
}
void fp2_mulm_low(fp2_t c, fp2_t a, fp2_t b) {
align dv2_t t;
dv2_null(t);
TRY {
dv2_new(t);
fp2_muln_low(t, a, b);
fp2_rdcn_low(c, t);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
dv2_free(t);
}
}
void fp2_sqrm_low(fp2_t c, fp2_t a) {
align dv2_t t;
dv2_null(t);
TRY {
dv2_new(t);
fp2_sqrn_low(t, a);
fp2_rdcn_low(c, t);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
dv2_free(t);
}
}
static void fp4_sqr_unr(dv2_t c, dv2_t d, fp2_t a, fp2_t b) {
fp2_t t;
dv2_t u0, u1;
fp2_null(t);
dv2_null(u0);
dv2_null(u1);
TRY {
fp2_new(t);
dv2_new(u0);
dv2_new(u1);
/* t0 = a^2. */
fp2_sqrn_low(u0, a);
/* t1 = b^2. */
fp2_sqrn_low(u1, b);
fp2_addm_low(t, a, b);
/* c = a^2 + b^2 * E. */
fp2_norh_low(c, u1);
fp2_addc_low(c, c, u0);
/* d = (a + b)^2 - a^2 - b^2 = 2 * a * b. */
fp2_addc_low(u1, u1, u0);
fp2_sqrn_low(d, t);
fp2_subc_low(d, d, u1);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t);
dv2_free(u0);
dv2_free(u1);
}
}
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);
}
}
inline static void fp6_mul_dxs_unr_lazyr(dv6_t c, fp6_t a, fp6_t b) {
dv2_t u0, u1, u2, u3;
fp2_t t0, t1;
dv2_null(u0);
dv2_null(u1);
dv2_null(u2);
dv2_null(u3);
fp2_null(t0);
fp2_null(t1);
TRY {
dv2_new(u0);
dv2_new(u1);
dv2_new(u2);
dv2_new(u3);
fp2_new(t0);
fp2_new(t1);
#ifdef RLC_FP_ROOM
fp2_mulc_low(u0, a[0], b[0]);
fp2_mulc_low(u1, a[1], b[1]);
fp2_addn_low(t0, a[0], a[1]);
fp2_addn_low(t1, b[0], b[1]);
/* c_1 = (a_0 + a_1)(b_0 + b_1) - a_0b_0 - a_1b_1 */
fp2_muln_low(u2, t0, t1);
fp2_subc_low(u2, u2, u0);
fp2_subc_low(c[1], u2, u1);
/* c_0 = a_0b_0 + E a_2b_1 */
fp2_mulc_low(u2, a[2], b[1]);
fp2_norh_low(c[0], u2);
fp2_addc_low(c[0], u0, c[0]);
/* c_2 = a_0b_2 + a_1b_1 */
fp2_mulc_low(u2, a[2], b[0]);
fp2_addc_low(c[2], u1, u2);
#else
fp2_muln_low(u0, a[0], b[0]);
fp2_muln_low(u1, a[1], b[1]);
fp2_addm_low(t0, a[0], a[1]);
fp2_addm_low(t1, b[0], b[1]);
/* c_1 = (a_0 + a_1)(b_0 + b_1) - a_0b_0 - a_1b_1 */
fp2_muln_low(u2, t0, t1);
fp2_subc_low(u2, u2, u0);
fp2_subc_low(c[1], u2, u1);
/* c_0 = a_0b_0 + E a_2b_1 */
fp2_muln_low(u2, a[2], b[1]);
fp2_nord_low(c[0], u2);
fp2_addc_low(c[0], u0, c[0]);
/* c_2 = a_0b_2 + a_1b_1 */
fp2_muln_low(u2, a[2], b[0]);
fp2_addc_low(c[2], u1, u2);
#endif
#ifdef RLC_FP_ROOM
#else
#endif
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
dv2_free(u0);
dv2_free(u1);
dv2_free(u2);
dv2_free(u3);
fp2_free(t0);
fp2_free(t1);
}
}
void pp_dbl_k12_projc_lazyr(fp12_t l, ep2_t r, ep2_t q, ep_t p) {
fp2_t t0, t1, t2, t3, t4, t5, t6;
dv2_t u0, u1;
int one = 1, zero = 0;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
fp2_null(t3);
fp2_null(t4);
fp2_null(t5);
fp2_null(t6);
dv2_null(u0);
dv2_null(u1);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
fp2_new(t3);
fp2_new(t4);
fp2_new(t5);
fp2_new(t6);
dv2_new(u0);
dv2_new(u1);
if (ep2_curve_is_twist() == EP_MTYPE) {
one ^= 1;
zero ^= 1;
}
if (ep_curve_opt_b() == RLC_TWO) {
/* C = z1^2. */
fp2_sqr(t0, q->z);
/* B = y1^2. */
fp2_sqr(t1, q->y);
/* t5 = B + C. */
fp2_add(t5, t0, t1);
/* t3 = E = 3b'C = 3C * (1 - i). */
fp2_dbl(t3, t0);
fp2_add(t0, t0, t3);
fp_add(t2[0], t0[0], t0[1]);
fp_sub(t2[1], t0[1], t0[0]);
/* t0 = x1^2. */
fp2_sqr(t0, q->x);
/* t4 = A = (x1 * y1)/2. */
fp2_mul(t4, q->x, q->y);
fp_hlv(t4[0], t4[0]);
fp_hlv(t4[1], t4[1]);
/* t3 = F = 3E. */
fp2_dbl(t3, t2);
fp2_add(t3, t3, t2);
/* x3 = A * (B - F). */
fp2_sub(r->x, t1, t3);
fp2_mul(r->x, r->x, t4);
/* G = (B + F)/2. */
fp2_add(t3, t1, t3);
fp_hlv(t3[0], t3[0]);
fp_hlv(t3[1], t3[1]);
/* y3 = G^2 - 3E^2. */
fp2_sqrn_low(u0, t2);
fp2_addd_low(u1, u0, u0);
fp2_addd_low(u1, u1, u0);
fp2_sqrn_low(u0, t3);
fp2_subc_low(u0, u0, u1);
/* H = (Y + Z)^2 - B - C. */
fp2_add(t3, q->y, q->z);
fp2_sqr(t3, t3);
fp2_sub(t3, t3, t5);
fp2_rdcn_low(r->y, u0);
/* z3 = B * H. */
fp2_mul(r->z, t1, t3);
/* l11 = E - B. */
fp2_sub(l[1][1], t2, t1);
/* l10 = (3 * xp) * t0. */
fp_mul(l[one][zero][0], p->x, t0[0]);
fp_mul(l[one][zero][1], p->x, t0[1]);
/* l01 = F * (-yp). */
fp_mul(l[zero][zero][0], t3[0], p->y);
fp_mul(l[zero][zero][1], t3[1], p->y);
} else {
/* A = x1^2. */
fp2_sqr(t0, q->x);
/* B = y1^2. */
fp2_sqr(t1, q->y);
/* C = z1^2. */
fp2_sqr(t2, q->z);
/* D = 3bC, for general b. */
fp2_dbl(t3, t2);
fp2_add(t3, t3, t2);
ep2_curve_get_b(t4);
fp2_mul(t3, t3, t4);
/* E = (x1 + y1)^2 - A - B. */
fp2_add(t4, q->x, q->y);
fp2_sqr(t4, t4);
fp2_sub(t4, t4, t0);
fp2_sub(t4, t4, t1);
/* F = (y1 + z1)^2 - B - C. */
fp2_add(t5, q->y, q->z);
fp2_sqr(t5, t5);
fp2_sub(t5, t5, t1);
fp2_sub(t5, t5, t2);
/* G = 3D. */
fp2_dbl(t6, t3);
fp2_add(t6, t6, t3);
/* x3 = E * (B - G). */
fp2_sub(r->x, t1, t6);
fp2_mul(r->x, r->x, t4);
/* y3 = (B + G)^2 -12D^2. */
fp2_add(t6, t6, t1);
fp2_sqr(t6, t6);
fp2_sqr(t2, t3);
fp2_dbl(r->y, t2);
fp2_dbl(t2, r->y);
fp2_dbl(r->y, t2);
fp2_add(r->y, r->y, t2);
fp2_sub(r->y, t6, r->y);
/* z3 = 4B * F. */
fp2_dbl(r->z, t1);
fp2_dbl(r->z, r->z);
fp2_mul(r->z, r->z, t5);
/* l00 = D - B. */
fp2_sub(l[one][one], t3, t1);
/* l10 = (3 * xp) * A. */
fp_mul(l[one][zero][0], p->x, t0[0]);
fp_mul(l[one][zero][1], p->x, t0[1]);
/* l01 = F * (-yp). */
fp_mul(l[zero][zero][0], t5[0], p->y);
fp_mul(l[zero][zero][1], t5[1], p->y);
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
fp2_free(t3);
fp2_free(t4);
fp2_free(t5);
fp2_free(t6);
dv2_free(u0);
dv2_free(u1);
}
}
void fp12_sqr_pck_lazyr(fp12_t c, fp12_t a) {
fp2_t t0, t1;
dv2_t u0, u1, u2, u3, u4;
fp2_null(t0);
fp2_null(t1);
dv2_null(u0);
dv2_null(u1);
dv2_null(u2);
dv2_null(u3);
dv2_null(u4);
TRY {
fp2_new(t0);
fp2_new(t1);
dv2_new(u0);
dv2_new(u1);
dv2_new(u2);
dv2_new(u3);
dv2_new(u4);
fp2_sqrn_low(u0, a[0][1]);
fp2_sqrn_low(u1, a[1][2]);
fp2_norh_low(u4, u1);
fp2_addm_low(t0, a[0][1], a[1][2]);
fp2_sqrn_low(u2, t0);
fp2_addd_low(u3, u0, u1);
fp2_subc_low(u3, u2, u3);
fp2_rdcn_low(t0, u3);
fp2_sqrn_low(u2, a[1][0]);
fp2_addm_low(t1, a[1][0], a[0][2]);
fp2_sqrn_low(u3, t1);
fp2_norm_low(t1, t0);
fp2_addm_low(t0, t1, a[1][0]);
fp2_dblm_low(t0, t0);
fp2_addm_low(c[1][0], t0, t1);
fp2_sqrn_low(u1, a[0][2]);
fp2_addc_low(u4, u0, u4);
fp2_rdcn_low(t0, u4);
fp2_subm_low(t1, t0, a[0][2]);
fp2_dblm_low(t1, t1);
fp2_norh_low(u4, u1);
fp2_addm_low(c[0][2], t1, t0);
fp2_addc_low(u4, u2, u4);
fp2_rdcn_low(t0, u4);
fp2_subm_low(t1, t0, a[0][1]);
fp2_dblm_low(t1, t1);
fp2_addd_low(u0, u1, u2);
fp2_addm_low(c[0][1], t1, t0);
fp2_subc_low(u3, u3, u0);
fp2_rdcn_low(t0, u3);
fp2_addm_low(t1, t0, a[1][2]);
fp2_dblm_low(t1, t1);
fp2_addm_low(c[1][2], t1, t0);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t0);
fp2_free(t1);
dv2_free(u0);
dv2_free(u1);
dv2_free(u2);
dv2_free(u3);
dv2_free(u4);
}
}
void fp12_sqr_lazyr(fp12_t c, fp12_t a) {
fp2_t t0, t1, t2, t3;
dv2_t u0, u1, u2, u3, u4, u5, u6, u7, u8, u9;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
fp2_null(t3);
dv2_null(u0);
dv2_null(u1);
dv2_null(u2);
dv2_null(u3);
dv2_null(u4);
dv2_null(u5);
dv2_null(u6);
dv2_null(u7);
dv2_null(u8);
dv2_null(u9);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
fp2_new(t3);
dv2_new(u0);
dv2_new(u1);
dv2_new(u2);
dv2_new(u3);
dv2_new(u4);
dv2_new(u5);
dv2_new(u6);
dv2_new(u7);
dv2_new(u8);
dv2_new(u9);
/* a0 = (a00, a11). */
/* a1 = (a10, a02). */
/* a2 = (a01, a12). */
/* (t0,t1) = a0^2 */
fp4_sqr_unr(u0, u1, a[0][0], a[1][1]);
/* (t2,t3) = 2 * a1 * a2 */
fp4_mul_unr(u2, u3, a[1][0], a[0][2], a[0][1], a[1][2]);
fp2_addc_low(u2, u2, u2);
fp2_addc_low(u3, u3, u3);
/* (t4,t5) = a2^2. */
fp4_sqr_unr(u4, u5, a[0][1], a[1][2]);
/* c2 = a0 + a2. */
fp2_addm_low(t2, a[0][0], a[0][1]);
fp2_addm_low(t3, a[1][1], a[1][2]);
/* (t6,t7) = (a0 + a2 + a1)^2. */
fp2_addm_low(t0, t2, a[1][0]);
fp2_addm_low(t1, t3, a[0][2]);
fp4_sqr_unr(u6, u7, t0, t1);
/* c2 = (a0 + a2 - a1)^2. */
fp2_subm_low(t2, t2, a[1][0]);
fp2_subm_low(t3, t3, a[0][2]);
fp4_sqr_unr(u8, u9, t2, t3);
/* c2 = (c2 + (t6,t7))/2. */
#ifdef FP_SPACE
fp2_addd_low(u8, u8, u6);
fp2_addd_low(u9, u9, u7);
#else
fp2_addc_low(u8, u8, u6);
fp2_addc_low(u9, u9, u7);
#endif
fp_hlvd_low(u8[0], u8[0]);
fp_hlvd_low(u8[1], u8[1]);
fp_hlvd_low(u9[0], u9[0]);
fp_hlvd_low(u9[1], u9[1]);
/* (t6,t7) = (t6,t7) - c2 - (t2,t3). */
fp2_subc_low(u6, u6, u8);
fp2_subc_low(u7, u7, u9);
fp2_subc_low(u6, u6, u2);
fp2_subc_low(u7, u7, u3);
/* c2 = c2 - (t0,t1) - (t4,t5). */
fp2_subc_low(u8, u8, u0);
fp2_subc_low(u9, u9, u1);
fp2_subc_low(u8, u8, u4);
fp2_subc_low(u9, u9, u5);
fp2_rdcn_low(c[0][1], u8);
fp2_rdcn_low(c[1][2], u9);
/* c1 = (t6,t7) + (t4,t5) * E. */
fp2_nord_low(u9, u5);
fp2_addc_low(u6, u6, u9);
fp2_addc_low(u7, u7, u4);
fp2_rdcn_low(c[1][0], u6);
fp2_rdcn_low(c[0][2], u7);
/* c0 = (t0,t1) + (t2,t3) * E. */
fp2_nord_low(u9, u3);
fp2_addc_low(u0, u0, u9);
fp2_addc_low(u1, u1, u2);
fp2_rdcn_low(c[0][0], u0);
fp2_rdcn_low(c[1][1], u1);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
fp2_free(t3);
dv2_free(u0);
dv2_free(u1);
dv2_free(u2);
dv2_free(u3);
dv2_free(u4);
dv2_free(u5);
dv2_free(u6);
dv2_free(u7);
dv2_free(u8);
dv2_free(u9);
}
}