void fp2_mulc_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. */
fp_addn_low(t0, a[0], a[1]);
fp_addn_low(t1, b[0], b[1]);
/* c_0 = a_0 * b_0, c_1 = a_1 * b_1, t2 = (a_0 + a_1) * (b_0 + b_1). */
fp_muln_low(c[0], a[0], b[0]);
fp_muln_low(c[1], a[1], b[1]);
fp_muln_low(t2, t0, t1);
/* t0 = (a_0 * b_0) + (a_1 * b_1). */
fp_addd_low(t0, c[0], c[1]);
/* c_0 = (a_0 * b_0) + u^2 * (a_1 * b_1). */
fp_subd_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_subd_low(c[0], c[0], c[1]);
}
#endif
/* c_1 = (t2 - t0). */
fp_subd_low(c[1], t2, t0);
/* c_0 = c_0 + 2^N * p/4. */
bn_lshb_low(c[0] + FP_DIGS - 1, c[0] + FP_DIGS - 1, FP_DIGS + 1, 2);
fp_addn_low(c[0] + FP_DIGS, c[0] + FP_DIGS, fp_prime_get());
bn_rshb_low(c[0] + FP_DIGS - 1, c[0] + FP_DIGS - 1, FP_DIGS + 1, 2);
}
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 fp_subc_low(dig_t *c, const dig_t *a, const dig_t *b) {
dig_t carry = fp_subd_low(c, a, b);
if (carry) {
fp_addn_low(c + FP_DIGS, c + FP_DIGS, fp_prime_get());
}
}
void fp_subm_low(dig_t *c, const dig_t *a, const dig_t *b) {
dig_t carry;
carry = fp_subn_low(c, a, b);
if (carry) {
fp_addn_low(c, c, fp_prime_get());
}
}
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 fp_hlvm_low(dig_t *c, const dig_t *a) {
dig_t carry = 0;
if (a[0] & 1) {
carry = fp_addn_low(c, a, fp_prime_get());
} else {
fp_copy(c, a);
}
fp_rsh1_low(c, c);
if (carry) {
c[FP_DIGS - 1] ^= ((dig_t)1 << (FP_DIGIT - 1));
}
}
void fp_subc_low(dig_t *c, const dig_t *a, const dig_t *b) {
int i;
dig_t carry, r0, diff;
/* Zero the carry. */
carry = 0;
for (i = 0; i < 2 * FP_DIGS; i++, a++, b++) {
diff = (*a) - (*b);
r0 = diff - carry;
carry = ((*a) < (*b)) || (carry && !diff);
c[i] = r0;
}
if (carry) {
fp_addn_low(c + FP_DIGS, c + FP_DIGS, fp_prime_get());
}
}
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 fp_hlvd_low(dig_t *c, const dig_t *a) {
dig_t carry = 0;
if (a[0] & 1) {
carry = fp_addn_low(c, a, fp_prime_get());
} else {
dv_copy(c, a, FP_DIGS);
}
fp_add1_low(c + FP_DIGS, a + FP_DIGS, carry);
carry = fp_rsh1_low(c + FP_DIGS, c + FP_DIGS);
fp_rsh1_low(c, c);
if (carry) {
c[FP_DIGS - 1] ^= ((dig_t)1 << (FP_DIGIT - 1));
}
}
void fp_addm_low(dig_t *c, const dig_t *a, const dig_t *b) {
fp_addn_low(c, a, b);
if (fp_cmp(c, fp_prime_get()) != CMP_LT) {
fp_subn_low(c, c, fp_prime_get());
}
}
void fp2_addn_low(fp2_t c, fp2_t a, fp2_t b) {
fp_addn_low(c[0], a[0], b[0]);
fp_addn_low(c[1], a[1], b[1]);
}
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 fp_rdcs_low(dig_t *c, const dig_t *a, const dig_t *m) {
align dig_t q[2 * FP_DIGS], _q[2 * FP_DIGS];
align dig_t _r[2 * FP_DIGS], r[2 * FP_DIGS], t[2 * FP_DIGS];
const int *sform;
int len, first, i, j, b0, d0, b1, d1;
dig_t carry;
sform = fp_prime_get_sps(&len);
SPLIT(b0, d0, sform[len - 1], FP_DIG_LOG);
first = (d0) + (b0 == 0 ? 0 : 1);
/* q = floor(a/b^k) */
dv_zero(q, 2 * FP_DIGS);
bn_rshd_low(q, a, 2 * FP_DIGS, d0);
if (b0 > 0) {
bn_rshb_low(q, q, 2 * FP_DIGS, b0);
}
/* r = a - qb^k. */
dv_copy(r, a, first);
if (b0 > 0) {
r[first - 1] &= MASK(b0);
}
carry = 0;
while (!fp_is_zero(q)) {
dv_zero(_q, 2 * FP_DIGS);
for (i = len - 2; i > 0; i--) {
j = (sform[i] < 0 ? -sform[i] : sform[i]);
SPLIT(b1, d1, j, FP_DIG_LOG);
dv_zero(t, 2 * FP_DIGS);
bn_lshd_low(t, q, FP_DIGS, d1);
if (b1 > 0) {
bn_lshb_low(t, t, 2 * FP_DIGS, b1);
}
if (sform[i] > 0) {
bn_subn_low(_q, _q, t, 2 * FP_DIGS);
} else {
bn_addn_low(_q, _q, t, 2 * FP_DIGS);
}
}
if (sform[0] > 0) {
bn_subn_low(_q, _q, q, 2 * FP_DIGS);
} else {
bn_addn_low(_q, _q, q, 2 * FP_DIGS);
}
bn_rshd_low(q, _q, 2 * FP_DIGS, d0);
if (b0 > 0) {
bn_rshb_low(q, q, 2 * FP_DIGS, b0);
}
dv_copy(_r, _q, first);
if (b0 > 0) {
_r[first - 1] &= MASK(b0);
}
carry = fp_addn_low(r, r, _r);
if (carry) {
fp_subn_low(r, r, m);
}
}
while (fp_cmpn_low(r, m) != CMP_LT) {
fp_subn_low(r, r, m);
}
fp_copy(c, r);
}
void fp3_addn_low(fp3_t c, fp3_t a, fp3_t b) {
fp_addn_low(c[0], a[0], b[0]);
fp_addn_low(c[1], a[1], b[1]);
fp_addn_low(c[2], a[2], b[2]);
}
void fp2_norh_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[1], a[1], 2 * FP_DIGS);
fp_addd_low(c[1], a[0], a[1]);
/* c_0 = c_0 + 2^N * p/2. */
dv_copy(t[0], a[0], 2 * FP_DIGS);
bn_lshb_low(t[0] + FP_DIGS - 1, t[0] + FP_DIGS - 1, FP_DIGS + 1, 1);
fp_addn_low(t[0] + FP_DIGS, t[0] + FP_DIGS, fp_prime_get());
bn_rshb_low(t[0] + FP_DIGS - 1, t[0] + FP_DIGS - 1, FP_DIGS + 1, 1);
fp_subd_low(c[0], t[0], t[1]);
#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);
}
}
