/** Double an ECC point @param P The point to double @param R [out] The destination of the double @param modulus The modulus of the field the ECC curve is in @param mp The "b" value from montgomery_setup() @return CRYPT_OK on success */ int ltc_ecc_projective_dbl_point(ecc_point *P, ecc_point *R, void *modulus, void *mp) { void *t1, *t2; int err; LTC_ARGCHK(P != NULL); LTC_ARGCHK(R != NULL); LTC_ARGCHK(modulus != NULL); LTC_ARGCHK(mp != NULL); if ((err = mp_init_multi(&t1, &t2, NULL)) != CRYPT_OK) { return err; } if (P != R) { if ((err = mp_copy(P->x, R->x)) != CRYPT_OK) { goto done; } if ((err = mp_copy(P->y, R->y)) != CRYPT_OK) { goto done; } if ((err = mp_copy(P->z, R->z)) != CRYPT_OK) { goto done; } } /* t1 = Z * Z */ if ((err = mp_sqr(R->z, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* Z = Y * Z */ if ((err = mp_mul(R->z, R->y, R->z)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(R->z, modulus, mp)) != CRYPT_OK) { goto done; } /* Z = 2Z */ if ((err = mp_add(R->z, R->z, R->z)) != CRYPT_OK) { goto done; } if (mp_cmp(R->z, modulus) != LTC_MP_LT) { if ((err = mp_sub(R->z, modulus, R->z)) != CRYPT_OK) { goto done; } } /* T2 = X - T1 */ if ((err = mp_sub(R->x, t1, t2)) != CRYPT_OK) { goto done; } if (mp_cmp_d(t2, 0) == LTC_MP_LT) { if ((err = mp_add(t2, modulus, t2)) != CRYPT_OK) { goto done; } } /* T1 = X + T1 */ if ((err = mp_add(t1, R->x, t1)) != CRYPT_OK) { goto done; } if (mp_cmp(t1, modulus) != LTC_MP_LT) { if ((err = mp_sub(t1, modulus, t1)) != CRYPT_OK) { goto done; } } /* T2 = T1 * T2 */ if ((err = mp_mul(t1, t2, t2)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t2, modulus, mp)) != CRYPT_OK) { goto done; } /* T1 = 2T2 */ if ((err = mp_add(t2, t2, t1)) != CRYPT_OK) { goto done; } if (mp_cmp(t1, modulus) != LTC_MP_LT) { if ((err = mp_sub(t1, modulus, t1)) != CRYPT_OK) { goto done; } } /* T1 = T1 + T2 */ if ((err = mp_add(t1, t2, t1)) != CRYPT_OK) { goto done; } if (mp_cmp(t1, modulus) != LTC_MP_LT) { if ((err = mp_sub(t1, modulus, t1)) != CRYPT_OK) { goto done; } } /* Y = 2Y */ if ((err = mp_add(R->y, R->y, R->y)) != CRYPT_OK) { goto done; } if (mp_cmp(R->y, modulus) != LTC_MP_LT) { if ((err = mp_sub(R->y, modulus, R->y)) != CRYPT_OK) { goto done; } } /* Y = Y * Y */ if ((err = mp_sqr(R->y, R->y)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(R->y, modulus, mp)) != CRYPT_OK) { goto done; } /* T2 = Y * Y */ if ((err = mp_sqr(R->y, t2)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t2, modulus, mp)) != CRYPT_OK) { goto done; } /* T2 = T2/2 */ if (mp_isodd(t2)) { if ((err = mp_add(t2, modulus, t2)) != CRYPT_OK) { goto done; } } if ((err = mp_div_2(t2, t2)) != CRYPT_OK) { goto done; } /* Y = Y * X */ if ((err = mp_mul(R->y, R->x, R->y)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(R->y, modulus, mp)) != CRYPT_OK) { goto done; } /* X = T1 * T1 */ if ((err = mp_sqr(t1, R->x)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(R->x, modulus, mp)) != CRYPT_OK) { goto done; } /* X = X - Y */ if ((err = mp_sub(R->x, R->y, R->x)) != CRYPT_OK) { goto done; } if (mp_cmp_d(R->x, 0) == LTC_MP_LT) { if ((err = mp_add(R->x, modulus, R->x)) != CRYPT_OK) { goto done; } } /* X = X - Y */ if ((err = mp_sub(R->x, R->y, R->x)) != CRYPT_OK) { goto done; } if (mp_cmp_d(R->x, 0) == LTC_MP_LT) { if ((err = mp_add(R->x, modulus, R->x)) != CRYPT_OK) { goto done; } } /* Y = Y - X */ if ((err = mp_sub(R->y, R->x, R->y)) != CRYPT_OK) { goto done; } if (mp_cmp_d(R->y, 0) == LTC_MP_LT) { if ((err = mp_add(R->y, modulus, R->y)) != CRYPT_OK) { goto done; } } /* Y = Y * T1 */ if ((err = mp_mul(R->y, t1, R->y)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(R->y, modulus, mp)) != CRYPT_OK) { goto done; } /* Y = Y - T2 */ if ((err = mp_sub(R->y, t2, R->y)) != CRYPT_OK) { goto done; } if (mp_cmp_d(R->y, 0) == LTC_MP_LT) { if ((err = mp_add(R->y, modulus, R->y)) != CRYPT_OK) { goto done; } } err = CRYPT_OK; done: mp_clear_multi(t1, t2, NULL); return err; }
/* this is a shell function that calls either the normal or Montgomery * exptmod functions. Originally the call to the montgomery code was * embedded in the normal function but that wasted alot of stack space * for nothing (since 99% of the time the Montgomery code would be called) */ int mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y) { int dr; /* modulus P must be positive */ if (P->sign == MP_NEG) { return MP_VAL; } /* if exponent X is negative we have to recurse */ if (X->sign == MP_NEG) { #ifdef BN_MP_INVMOD_C mp_int tmpG, tmpX; int err; /* first compute 1/G mod P */ if ((err = mp_init(&tmpG)) != MP_OKAY) { return err; } if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { mp_clear(&tmpG); return err; } /* now get |X| */ if ((err = mp_init(&tmpX)) != MP_OKAY) { mp_clear(&tmpG); return err; } if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { mp_clear_multi(&tmpG, &tmpX, NULL); return err; } /* and now compute (1/G)**|X| instead of G**X [X < 0] */ err = mp_exptmod(&tmpG, &tmpX, P, Y); mp_clear_multi(&tmpG, &tmpX, NULL); return err; #else /* no invmod */ return MP_VAL; #endif } /* modified diminished radix reduction */ #if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defined(BN_S_MP_EXPTMOD_C) if (mp_reduce_is_2k_l(P) == MP_YES) { return s_mp_exptmod(G, X, P, Y, 1); } #endif #ifdef BN_MP_DR_IS_MODULUS_C /* is it a DR modulus? */ dr = mp_dr_is_modulus(P); #else /* default to no */ dr = 0; #endif #ifdef BN_MP_REDUCE_IS_2K_C /* if not, is it a unrestricted DR modulus? */ if (dr == 0) { dr = mp_reduce_is_2k(P) << 1; } #endif /* if the modulus is odd or dr != 0 use the montgomery method */ #ifdef BN_MP_EXPTMOD_FAST_C if ((mp_isodd(P) == MP_YES) || (dr != 0)) { return mp_exptmod_fast(G, X, P, Y, dr); } else { #endif #ifdef BN_S_MP_EXPTMOD_C /* otherwise use the generic Barrett reduction technique */ return s_mp_exptmod(G, X, P, Y, 0); #else /* no exptmod for evens */ return MP_VAL; #endif #ifdef BN_MP_EXPTMOD_FAST_C } #endif }
/* hac 14.61, pp608 */ int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c) { mp_int x, y, u, v, A, B, C, D; int res; /* b cannot be negative */ if (b->sign == MP_NEG || mp_iszero(b) == 1) { return MP_VAL; } /* init temps */ if ((res = mp_init_multi(&x, &y, &u, &v, &A, &B, &C, &D, NULL)) != MP_OKAY) { return res; } /* x = a, y = b */ if ((res = mp_mod(a, b, &x)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_copy (b, &y)) != MP_OKAY) { goto LBL_ERR; } /* 2. [modified] if x,y are both even then return an error! */ if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) { res = MP_VAL; goto LBL_ERR; } /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ if ((res = mp_copy (&x, &u)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_copy (&y, &v)) != MP_OKAY) { goto LBL_ERR; } mp_set (&A, 1); mp_set (&D, 1); top: /* 4. while u is even do */ while (mp_iseven (&u) == 1) { /* 4.1 u = u/2 */ if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { goto LBL_ERR; } /* 4.2 if A or B is odd then */ if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) { /* A = (A+y)/2, B = (B-x)/2 */ if ((res = mp_add (&A, &y, &A)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { goto LBL_ERR; } } /* A = A/2, B = B/2 */ if ((res = mp_div_2 (&A, &A)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { goto LBL_ERR; } } /* 5. while v is even do */ while (mp_iseven (&v) == 1) { /* 5.1 v = v/2 */ if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { goto LBL_ERR; } /* 5.2 if C or D is odd then */ if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) { /* C = (C+y)/2, D = (D-x)/2 */ if ((res = mp_add (&C, &y, &C)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { goto LBL_ERR; } } /* C = C/2, D = D/2 */ if ((res = mp_div_2 (&C, &C)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { goto LBL_ERR; } } /* 6. if u >= v then */ if (mp_cmp (&u, &v) != MP_LT) { /* u = u - v, A = A - C, B = B - D */ if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { goto LBL_ERR; } } else { /* v - v - u, C = C - A, D = D - B */ if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { goto LBL_ERR; } } /* if not zero goto step 4 */ if (mp_iszero (&u) == 0) goto top; /* now a = C, b = D, gcd == g*v */ /* if v != 1 then there is no inverse */ if (mp_cmp_d (&v, 1) != MP_EQ) { res = MP_VAL; goto LBL_ERR; } /* if its too low */ while (mp_cmp_d(&C, 0) == MP_LT) { if ((res = mp_add(&C, b, &C)) != MP_OKAY) { goto LBL_ERR; } } /* too big */ while (mp_cmp_mag(&C, b) != MP_LT) { if ((res = mp_sub(&C, b, &C)) != MP_OKAY) { goto LBL_ERR; } } /* C is now the inverse */ mp_exch (&C, c); res = MP_OKAY; LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL); return res; }
/** Add two ECC points @param P The point to add @param Q The point to add @param R [out] The destination of the double @param modulus The modulus of the field the ECC curve is in @param mp The "b" value from montgomery_setup() @return CRYPT_OK on success */ int ltc_ecc_projective_add_point(ecc_point *P, ecc_point *Q, ecc_point *R, void *modulus, void *mp) { void *t1, *t2, *x, *y, *z; int err; LTC_ARGCHK(P != NULL); LTC_ARGCHK(Q != NULL); LTC_ARGCHK(R != NULL); LTC_ARGCHK(modulus != NULL); LTC_ARGCHK(mp != NULL); if ((err = mp_init_multi(&t1, &t2, &x, &y, &z, NULL)) != CRYPT_OK) { return err; } /* should we dbl instead? */ if ((err = mp_sub(modulus, Q->y, t1)) != CRYPT_OK) { goto done; } if ( (mp_cmp(P->x, Q->x) == LTC_MP_EQ) && (Q->z != NULL && mp_cmp(P->z, Q->z) == LTC_MP_EQ) && (mp_cmp(P->y, Q->y) == LTC_MP_EQ || mp_cmp(P->y, t1) == LTC_MP_EQ)) { mp_clear_multi(t1, t2, x, y, z, NULL); return ltc_ecc_projective_dbl_point(P, R, modulus, mp); } if ((err = mp_copy(P->x, x)) != CRYPT_OK) { goto done; } if ((err = mp_copy(P->y, y)) != CRYPT_OK) { goto done; } if ((err = mp_copy(P->z, z)) != CRYPT_OK) { goto done; } /* if Z is one then these are no-operations */ if (Q->z != NULL) { /* T1 = Z' * Z' */ if ((err = mp_sqr(Q->z, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* X = X * T1 */ if ((err = mp_mul(t1, x, x)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(x, modulus, mp)) != CRYPT_OK) { goto done; } /* T1 = Z' * T1 */ if ((err = mp_mul(Q->z, t1, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* Y = Y * T1 */ if ((err = mp_mul(t1, y, y)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(y, modulus, mp)) != CRYPT_OK) { goto done; } } /* T1 = Z*Z */ if ((err = mp_sqr(z, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* T2 = X' * T1 */ if ((err = mp_mul(Q->x, t1, t2)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t2, modulus, mp)) != CRYPT_OK) { goto done; } /* T1 = Z * T1 */ if ((err = mp_mul(z, t1, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* T1 = Y' * T1 */ if ((err = mp_mul(Q->y, t1, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* Y = Y - T1 */ if ((err = mp_sub(y, t1, y)) != CRYPT_OK) { goto done; } if (mp_cmp_d(y, 0) == LTC_MP_LT) { if ((err = mp_add(y, modulus, y)) != CRYPT_OK) { goto done; } } /* T1 = 2T1 */ if ((err = mp_add(t1, t1, t1)) != CRYPT_OK) { goto done; } if (mp_cmp(t1, modulus) != LTC_MP_LT) { if ((err = mp_sub(t1, modulus, t1)) != CRYPT_OK) { goto done; } } /* T1 = Y + T1 */ if ((err = mp_add(t1, y, t1)) != CRYPT_OK) { goto done; } if (mp_cmp(t1, modulus) != LTC_MP_LT) { if ((err = mp_sub(t1, modulus, t1)) != CRYPT_OK) { goto done; } } /* X = X - T2 */ if ((err = mp_sub(x, t2, x)) != CRYPT_OK) { goto done; } if (mp_cmp_d(x, 0) == LTC_MP_LT) { if ((err = mp_add(x, modulus, x)) != CRYPT_OK) { goto done; } } /* T2 = 2T2 */ if ((err = mp_add(t2, t2, t2)) != CRYPT_OK) { goto done; } if (mp_cmp(t2, modulus) != LTC_MP_LT) { if ((err = mp_sub(t2, modulus, t2)) != CRYPT_OK) { goto done; } } /* T2 = X + T2 */ if ((err = mp_add(t2, x, t2)) != CRYPT_OK) { goto done; } if (mp_cmp(t2, modulus) != LTC_MP_LT) { if ((err = mp_sub(t2, modulus, t2)) != CRYPT_OK) { goto done; } } /* if Z' != 1 */ if (Q->z != NULL) { /* Z = Z * Z' */ if ((err = mp_mul(z, Q->z, z)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(z, modulus, mp)) != CRYPT_OK) { goto done; } } /* Z = Z * X */ if ((err = mp_mul(z, x, z)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(z, modulus, mp)) != CRYPT_OK) { goto done; } /* T1 = T1 * X */ if ((err = mp_mul(t1, x, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* X = X * X */ if ((err = mp_sqr(x, x)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(x, modulus, mp)) != CRYPT_OK) { goto done; } /* T2 = T2 * x */ if ((err = mp_mul(t2, x, t2)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t2, modulus, mp)) != CRYPT_OK) { goto done; } /* T1 = T1 * X */ if ((err = mp_mul(t1, x, t1)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t1, modulus, mp)) != CRYPT_OK) { goto done; } /* X = Y*Y */ if ((err = mp_sqr(y, x)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(x, modulus, mp)) != CRYPT_OK) { goto done; } /* X = X - T2 */ if ((err = mp_sub(x, t2, x)) != CRYPT_OK) { goto done; } if (mp_cmp_d(x, 0) == LTC_MP_LT) { if ((err = mp_add(x, modulus, x)) != CRYPT_OK) { goto done; } } /* T2 = T2 - X */ if ((err = mp_sub(t2, x, t2)) != CRYPT_OK) { goto done; } if (mp_cmp_d(t2, 0) == LTC_MP_LT) { if ((err = mp_add(t2, modulus, t2)) != CRYPT_OK) { goto done; } } /* T2 = T2 - X */ if ((err = mp_sub(t2, x, t2)) != CRYPT_OK) { goto done; } if (mp_cmp_d(t2, 0) == LTC_MP_LT) { if ((err = mp_add(t2, modulus, t2)) != CRYPT_OK) { goto done; } } /* T2 = T2 * Y */ if ((err = mp_mul(t2, y, t2)) != CRYPT_OK) { goto done; } if ((err = mp_montgomery_reduce(t2, modulus, mp)) != CRYPT_OK) { goto done; } /* Y = T2 - T1 */ if ((err = mp_sub(t2, t1, y)) != CRYPT_OK) { goto done; } if (mp_cmp_d(y, 0) == LTC_MP_LT) { if ((err = mp_add(y, modulus, y)) != CRYPT_OK) { goto done; } } /* Y = Y/2 */ if (mp_isodd(y)) { if ((err = mp_add(y, modulus, y)) != CRYPT_OK) { goto done; } } if ((err = mp_div_2(y, y)) != CRYPT_OK) { goto done; } if ((err = mp_copy(x, R->x)) != CRYPT_OK) { goto done; } if ((err = mp_copy(y, R->y)) != CRYPT_OK) { goto done; } if ((err = mp_copy(z, R->z)) != CRYPT_OK) { goto done; } err = CRYPT_OK; done: mp_clear_multi(t1, t2, x, y, z, NULL); return err; }
/* Computes R = 2P. Elliptic curve points P and R can be identical. Uses * Jacobian coordinates. * * Assumes input is already field-encoded using field_enc, and returns * output that is still field-encoded. * * This routine implements Point Doubling in the Jacobian Projective * space as described in the paper "Efficient elliptic curve exponentiation * using mixed coordinates", by H. Cohen, A Miyaji, T. Ono. */ mp_err ec_GFp_pt_dbl_jac(const mp_int *px, const mp_int *py, const mp_int *pz, mp_int *rx, mp_int *ry, mp_int *rz, const ECGroup *group) { mp_err res = MP_OKAY; mp_int t0, t1, M, S; MP_DIGITS(&t0) = 0; MP_DIGITS(&t1) = 0; MP_DIGITS(&M) = 0; MP_DIGITS(&S) = 0; MP_CHECKOK(mp_init(&t0)); MP_CHECKOK(mp_init(&t1)); MP_CHECKOK(mp_init(&M)); MP_CHECKOK(mp_init(&S)); /* P == inf or P == -P */ if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES || mp_cmp_z(py) == 0) { MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz)); goto CLEANUP; } if (mp_cmp_d(pz, 1) == 0) { /* M = 3 * px^2 + a */ MP_CHECKOK(group->meth->field_sqr(px, &t0, group->meth)); MP_CHECKOK(group->meth->field_add(&t0, &t0, &M, group->meth)); MP_CHECKOK(group->meth->field_add(&t0, &M, &t0, group->meth)); MP_CHECKOK(group->meth-> field_add(&t0, &group->curvea, &M, group->meth)); } else if (MP_SIGN(&group->curvea) == MP_NEG && MP_USED(&group->curvea) == 1 && MP_DIGIT(&group->curvea, 0) == 3) { /* M = 3 * (px + pz^2) * (px - pz^2) */ MP_CHECKOK(group->meth->field_sqr(pz, &M, group->meth)); MP_CHECKOK(group->meth->field_add(px, &M, &t0, group->meth)); MP_CHECKOK(group->meth->field_sub(px, &M, &t1, group->meth)); MP_CHECKOK(group->meth->field_mul(&t0, &t1, &M, group->meth)); MP_CHECKOK(group->meth->field_add(&M, &M, &t0, group->meth)); MP_CHECKOK(group->meth->field_add(&t0, &M, &M, group->meth)); } else { /* M = 3 * (px^2) + a * (pz^4) */ MP_CHECKOK(group->meth->field_sqr(px, &t0, group->meth)); MP_CHECKOK(group->meth->field_add(&t0, &t0, &M, group->meth)); MP_CHECKOK(group->meth->field_add(&t0, &M, &t0, group->meth)); MP_CHECKOK(group->meth->field_sqr(pz, &M, group->meth)); MP_CHECKOK(group->meth->field_sqr(&M, &M, group->meth)); MP_CHECKOK(group->meth-> field_mul(&M, &group->curvea, &M, group->meth)); MP_CHECKOK(group->meth->field_add(&M, &t0, &M, group->meth)); } /* rz = 2 * py * pz */ /* t0 = 4 * py^2 */ if (mp_cmp_d(pz, 1) == 0) { MP_CHECKOK(group->meth->field_add(py, py, rz, group->meth)); MP_CHECKOK(group->meth->field_sqr(rz, &t0, group->meth)); } else { MP_CHECKOK(group->meth->field_add(py, py, &t0, group->meth)); MP_CHECKOK(group->meth->field_mul(&t0, pz, rz, group->meth)); MP_CHECKOK(group->meth->field_sqr(&t0, &t0, group->meth)); } /* S = 4 * px * py^2 = px * (2 * py)^2 */ MP_CHECKOK(group->meth->field_mul(px, &t0, &S, group->meth)); /* rx = M^2 - 2 * S */ MP_CHECKOK(group->meth->field_add(&S, &S, &t1, group->meth)); MP_CHECKOK(group->meth->field_sqr(&M, rx, group->meth)); MP_CHECKOK(group->meth->field_sub(rx, &t1, rx, group->meth)); /* ry = M * (S - rx) - 8 * py^4 */ MP_CHECKOK(group->meth->field_sqr(&t0, &t1, group->meth)); if (mp_isodd(&t1)) { MP_CHECKOK(mp_add(&t1, &group->meth->irr, &t1)); } MP_CHECKOK(mp_div_2(&t1, &t1)); MP_CHECKOK(group->meth->field_sub(&S, rx, &S, group->meth)); MP_CHECKOK(group->meth->field_mul(&M, &S, &M, group->meth)); MP_CHECKOK(group->meth->field_sub(&M, &t1, ry, group->meth)); CLEANUP: mp_clear(&t0); mp_clear(&t1); mp_clear(&M); mp_clear(&S); return res; }
/** Create DSA parameters (INTERNAL ONLY, not part of public API) @param prng An active PRNG state @param wprng The index of the PRNG desired @param group_size Size of the multiplicative group (octets) @param modulus_size Size of the modulus (octets) @param p [out] bignum where generated 'p' is stored (must be initialized by caller) @param q [out] bignum where generated 'q' is stored (must be initialized by caller) @param g [out] bignum where generated 'g' is stored (must be initialized by caller) @return CRYPT_OK if successful, upon error this function will free all allocated memory */ static int _dsa_make_params(prng_state *prng, int wprng, int group_size, int modulus_size, void *p, void *q, void *g) { unsigned long L, N, n, outbytes, seedbytes, counter, j, i; int err, res, mr_tests_q, mr_tests_p, found_p, found_q, hash; unsigned char *wbuf, *sbuf, digest[MAXBLOCKSIZE]; void *t2L1, *t2N1, *t2q, *t2seedlen, *U, *W, *X, *c, *h, *e, *seedinc; /* check size */ if (group_size >= LTC_MDSA_MAX_GROUP || group_size < 1 || group_size >= modulus_size) { return CRYPT_INVALID_ARG; } /* FIPS-186-4 A.1.1.2 Generation of the Probable Primes p and q Using an Approved Hash Function * * L = The desired length of the prime p (in bits e.g. L = 1024) * N = The desired length of the prime q (in bits e.g. N = 160) * seedlen = The desired bit length of the domain parameter seed; seedlen shallbe equal to or greater than N * outlen = The bit length of Hash function * * 1. Check that the (L, N) * 2. If (seedlen <N), then return INVALID. * 3. n = ceil(L / outlen) - 1 * 4. b = L- 1 - (n * outlen) * 5. domain_parameter_seed = an arbitrary sequence of seedlen bits * 6. U = Hash (domain_parameter_seed) mod 2^(N-1) * 7. q = 2^(N-1) + U + 1 - (U mod 2) * 8. Test whether or not q is prime as specified in Appendix C.3 * 9. If qis not a prime, then go to step 5. * 10. offset = 1 * 11. For counter = 0 to (4L- 1) do { * For j=0 to n do { * Vj = Hash ((domain_parameter_seed+ offset + j) mod 2^seedlen * } * W = V0 + (V1 *2^outlen) + ... + (Vn-1 * 2^((n-1) * outlen)) + ((Vn mod 2^b) * 2^(n * outlen)) * X = W + 2^(L-1) Comment: 0 <= W < 2^(L-1); hence 2^(L-1) <= X < 2^L * c = X mod 2*q * p = X - (c - 1) Comment: p ~ 1 (mod 2*q) * If (p >= 2^(L-1)) { * Test whether or not p is prime as specified in Appendix C.3. * If p is determined to be prime, then return VALID and the values of p, qand (optionally) the values of domain_parameter_seed and counter * } * offset = offset + n + 1 Comment: Increment offset * } */ seedbytes = group_size; L = (unsigned long)modulus_size * 8; N = (unsigned long)group_size * 8; /* XXX-TODO no Lucas test */ #ifdef LTC_MPI_HAS_LUCAS_TEST /* M-R tests (when followed by one Lucas test) according FIPS-186-4 - Appendix C.3 - table C.1 */ mr_tests_p = (L <= 2048) ? 3 : 2; if (N <= 160) { mr_tests_q = 19; } else if (N <= 224) { mr_tests_q = 24; } else { mr_tests_q = 27; } #else /* M-R tests (without Lucas test) according FIPS-186-4 - Appendix C.3 - table C.1 */ if (L <= 1024) { mr_tests_p = 40; } else if (L <= 2048) { mr_tests_p = 56; } else { mr_tests_p = 64; } if (N <= 160) { mr_tests_q = 40; } else if (N <= 224) { mr_tests_q = 56; } else { mr_tests_q = 64; } #endif if (N <= 256) { hash = register_hash(&sha256_desc); } else if (N <= 384) { hash = register_hash(&sha384_desc); } else if (N <= 512) { hash = register_hash(&sha512_desc); } else { return CRYPT_INVALID_ARG; /* group_size too big */ } if ((err = hash_is_valid(hash)) != CRYPT_OK) { return err; } outbytes = hash_descriptor[hash].hashsize; n = ((L + outbytes*8 - 1) / (outbytes*8)) - 1; if ((wbuf = XMALLOC((n+1)*outbytes)) == NULL) { err = CRYPT_MEM; goto cleanup3; } if ((sbuf = XMALLOC(seedbytes)) == NULL) { err = CRYPT_MEM; goto cleanup2; } err = mp_init_multi(&t2L1, &t2N1, &t2q, &t2seedlen, &U, &W, &X, &c, &h, &e, &seedinc, NULL); if (err != CRYPT_OK) { goto cleanup1; } if ((err = mp_2expt(t2L1, L-1)) != CRYPT_OK) { goto cleanup; } /* t2L1 = 2^(L-1) */ if ((err = mp_2expt(t2N1, N-1)) != CRYPT_OK) { goto cleanup; } /* t2N1 = 2^(N-1) */ if ((err = mp_2expt(t2seedlen, seedbytes*8)) != CRYPT_OK) { goto cleanup; } /* t2seedlen = 2^seedlen */ for(found_p=0; !found_p;) { /* q */ for(found_q=0; !found_q;) { if (prng_descriptor[wprng].read(sbuf, seedbytes, prng) != seedbytes) { err = CRYPT_ERROR_READPRNG; goto cleanup; } i = outbytes; if ((err = hash_memory(hash, sbuf, seedbytes, digest, &i)) != CRYPT_OK) { goto cleanup; } if ((err = mp_read_unsigned_bin(U, digest, outbytes)) != CRYPT_OK) { goto cleanup; } if ((err = mp_mod(U, t2N1, U)) != CRYPT_OK) { goto cleanup; } if ((err = mp_add(t2N1, U, q)) != CRYPT_OK) { goto cleanup; } if (!mp_isodd(q)) mp_add_d(q, 1, q); if ((err = mp_prime_is_prime(q, mr_tests_q, &res)) != CRYPT_OK) { goto cleanup; } if (res == LTC_MP_YES) found_q = 1; } /* p */ if ((err = mp_read_unsigned_bin(seedinc, sbuf, seedbytes)) != CRYPT_OK) { goto cleanup; } if ((err = mp_add(q, q, t2q)) != CRYPT_OK) { goto cleanup; } for(counter=0; counter < 4*L && !found_p; counter++) { for(j=0; j<=n; j++) { if ((err = mp_add_d(seedinc, 1, seedinc)) != CRYPT_OK) { goto cleanup; } if ((err = mp_mod(seedinc, t2seedlen, seedinc)) != CRYPT_OK) { goto cleanup; } /* seedinc = (seedinc+1) % 2^seed_bitlen */ if ((i = mp_unsigned_bin_size(seedinc)) > seedbytes) { err = CRYPT_INVALID_ARG; goto cleanup; } zeromem(sbuf, seedbytes); if ((err = mp_to_unsigned_bin(seedinc, sbuf + seedbytes-i)) != CRYPT_OK) { goto cleanup; } i = outbytes; err = hash_memory(hash, sbuf, seedbytes, wbuf+(n-j)*outbytes, &i); if (err != CRYPT_OK) { goto cleanup; } } if ((err = mp_read_unsigned_bin(W, wbuf, (n+1)*outbytes)) != CRYPT_OK) { goto cleanup; } if ((err = mp_mod(W, t2L1, W)) != CRYPT_OK) { goto cleanup; } if ((err = mp_add(W, t2L1, X)) != CRYPT_OK) { goto cleanup; } if ((err = mp_mod(X, t2q, c)) != CRYPT_OK) { goto cleanup; } if ((err = mp_sub_d(c, 1, p)) != CRYPT_OK) { goto cleanup; } if ((err = mp_sub(X, p, p)) != CRYPT_OK) { goto cleanup; } if (mp_cmp(p, t2L1) != LTC_MP_LT) { /* p >= 2^(L-1) */ if ((err = mp_prime_is_prime(p, mr_tests_p, &res)) != CRYPT_OK) { goto cleanup; } if (res == LTC_MP_YES) { found_p = 1; } } } } /* FIPS-186-4 A.2.1 Unverifiable Generation of the Generator g * 1. e = (p - 1)/q * 2. h = any integer satisfying: 1 < h < (p - 1) * h could be obtained from a random number generator or from a counter that changes after each use * 3. g = h^e mod p * 4. if (g == 1), then go to step 2. * */ if ((err = mp_sub_d(p, 1, e)) != CRYPT_OK) { goto cleanup; } if ((err = mp_div(e, q, e, c)) != CRYPT_OK) { goto cleanup; } /* e = (p - 1)/q */ i = mp_count_bits(p); do { do { if ((err = rand_bn_bits(h, i, prng, wprng)) != CRYPT_OK) { goto cleanup; } } while (mp_cmp(h, p) != LTC_MP_LT || mp_cmp_d(h, 2) != LTC_MP_GT); if ((err = mp_sub_d(h, 1, h)) != CRYPT_OK) { goto cleanup; } /* h is randon and 1 < h < (p-1) */ if ((err = mp_exptmod(h, e, p, g)) != CRYPT_OK) { goto cleanup; } } while (mp_cmp_d(g, 1) == LTC_MP_EQ); err = CRYPT_OK; cleanup: mp_clear_multi(t2L1, t2N1, t2q, t2seedlen, U, W, X, c, h, e, seedinc, NULL); cleanup1: XFREE(sbuf); cleanup2: XFREE(wbuf); cleanup3: return err; }
/* computes the modular inverse via binary extended euclidean algorithm, * that is c = 1/a mod b * * Based on slow invmod except this is optimized for the case where b is * odd as per HAC Note 14.64 on pp. 610 */ int fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c) { mp_int x, y, u, v, B, D; int res, neg; /* 2. [modified] b must be odd */ if (mp_iseven (b) == 1) { return MP_VAL; } /* init all our temps */ if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) { return res; } /* x == modulus, y == value to invert */ if ((res = mp_copy (b, &x)) != MP_OKAY) { goto LBL_ERR; } /* we need y = |a| */ if ((res = mp_mod (a, b, &y)) != MP_OKAY) { goto LBL_ERR; } /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ if ((res = mp_copy (&x, &u)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_copy (&y, &v)) != MP_OKAY) { goto LBL_ERR; } mp_set (&D, 1); top: /* 4. while u is even do */ while (mp_iseven (&u) == 1) { /* 4.1 u = u/2 */ if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { goto LBL_ERR; } /* 4.2 if B is odd then */ if (mp_isodd (&B) == 1) { if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { goto LBL_ERR; } } /* B = B/2 */ if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { goto LBL_ERR; } } /* 5. while v is even do */ while (mp_iseven (&v) == 1) { /* 5.1 v = v/2 */ if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { goto LBL_ERR; } /* 5.2 if D is odd then */ if (mp_isodd (&D) == 1) { /* D = (D-x)/2 */ if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { goto LBL_ERR; } } /* D = D/2 */ if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { goto LBL_ERR; } } /* 6. if u >= v then */ if (mp_cmp (&u, &v) != MP_LT) { /* u = u - v, B = B - D */ if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { goto LBL_ERR; } } else { /* v - v - u, D = D - B */ if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { goto LBL_ERR; } if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { goto LBL_ERR; } } /* if not zero goto step 4 */ if (mp_iszero (&u) == 0) { goto top; } /* now a = C, b = D, gcd == g*v */ /* if v != 1 then there is no inverse */ if (mp_cmp_d (&v, 1) != MP_EQ) { res = MP_VAL; goto LBL_ERR; } /* b is now the inverse */ neg = a->sign; while (D.sign == MP_NEG) { if ((res = mp_add (&D, b, &D)) != MP_OKAY) { goto LBL_ERR; } } mp_exch (&D, c); c->sign = neg; res = MP_OKAY; LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL); return res; }
/* Greatest Common Divisor using the binary method [Algorithm B, page 338, vol2 of TAOCP] */ int mp_gcd (mp_int * a, mp_int * b, mp_int * c) { mp_int u, v, t; int k, res, neg; /* either zero than gcd is the largest */ if (mp_iszero (a) == 1 && mp_iszero (b) == 0) { return mp_copy (b, c); } if (mp_iszero (a) == 0 && mp_iszero (b) == 1) { return mp_copy (a, c); } if (mp_iszero (a) == 1 && mp_iszero (b) == 1) { mp_set (c, 1); return MP_OKAY; } /* if both are negative they share (-1) as a common divisor */ neg = (a->sign == b->sign) ? a->sign : MP_ZPOS; if ((res = mp_init_copy (&u, a)) != MP_OKAY) { return res; } if ((res = mp_init_copy (&v, b)) != MP_OKAY) { goto __U; } /* must be positive for the remainder of the algorithm */ u.sign = v.sign = MP_ZPOS; if ((res = mp_init (&t)) != MP_OKAY) { goto __V; } /* B1. Find power of two */ k = 0; while (mp_iseven(&u) == 1 && mp_iseven(&v) == 1) { ++k; if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { goto __T; } if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { goto __T; } } /* B2. Initialize */ if (mp_isodd(&u) == 1) { /* t = -v */ if ((res = mp_copy (&v, &t)) != MP_OKAY) { goto __T; } t.sign = MP_NEG; } else { /* t = u */ if ((res = mp_copy (&u, &t)) != MP_OKAY) { goto __T; } } do { /* B3 (and B4). Halve t, if even */ while (t.used != 0 && mp_iseven(&t) == 1) { if ((res = mp_div_2 (&t, &t)) != MP_OKAY) { goto __T; } } /* B5. if t>0 then u=t otherwise v=-t */ if (t.used != 0 && t.sign != MP_NEG) { if ((res = mp_copy (&t, &u)) != MP_OKAY) { goto __T; } } else { if ((res = mp_copy (&t, &v)) != MP_OKAY) { goto __T; } v.sign = (v.sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; } /* B6. t = u - v, if t != 0 loop otherwise terminate */ if ((res = mp_sub (&u, &v, &t)) != MP_OKAY) { goto __T; } } while (mp_iszero(&t) == 0); /* multiply by 2^k which we divided out at the beginning */ if ((res = mp_mul_2d (&u, k, &u)) != MP_OKAY) { goto __T; } mp_exch (&u, c); c->sign = neg; res = MP_OKAY; __T: mp_clear (&t); __V: mp_clear (&u); __U: mp_clear (&v); return res; }
mp_err mp_exptmod(const mp_int *inBase, const mp_int *exponent, const mp_int *modulus, mp_int *result) { const mp_int *base; mp_size bits_in_exponent, i, window_bits, odd_ints; mp_err res; int nLen; mp_int montBase, goodBase; mp_mont_modulus mmm; #ifdef MP_USING_CACHE_SAFE_MOD_EXP static unsigned int max_window_bits; #endif /* function for computing n0prime only works if n0 is odd */ if (!mp_isodd(modulus)) return s_mp_exptmod(inBase, exponent, modulus, result); MP_DIGITS(&montBase) = 0; MP_DIGITS(&goodBase) = 0; if (mp_cmp(inBase, modulus) < 0) { base = inBase; } else { MP_CHECKOK( mp_init(&goodBase) ); base = &goodBase; MP_CHECKOK( mp_mod(inBase, modulus, &goodBase) ); } nLen = MP_USED(modulus); MP_CHECKOK( mp_init_size(&montBase, 2 * nLen + 2) ); mmm.N = *modulus; /* a copy of the mp_int struct */ /* compute n0', given n0, n0' = -(n0 ** -1) mod MP_RADIX ** where n0 = least significant mp_digit of N, the modulus. */ mmm.n0prime = 0 - s_mp_invmod_radix( MP_DIGIT(modulus, 0) ); MP_CHECKOK( s_mp_to_mont(base, &mmm, &montBase) ); bits_in_exponent = mpl_significant_bits(exponent); #ifdef MP_USING_CACHE_SAFE_MOD_EXP if (mp_using_cache_safe_exp) { if (bits_in_exponent > 780) window_bits = 6; else if (bits_in_exponent > 256) window_bits = 5; else if (bits_in_exponent > 20) window_bits = 4; /* RSA public key exponents are typically under 20 bits (common values * are: 3, 17, 65537) and a 4-bit window is inefficient */ else window_bits = 1; } else #endif if (bits_in_exponent > 480) window_bits = 6; else if (bits_in_exponent > 160) window_bits = 5; else if (bits_in_exponent > 20) window_bits = 4; /* RSA public key exponents are typically under 20 bits (common values * are: 3, 17, 65537) and a 4-bit window is inefficient */ else window_bits = 1; #ifdef MP_USING_CACHE_SAFE_MOD_EXP /* * clamp the window size based on * the cache line size. */ if (!max_window_bits) { unsigned long cache_size = s_mpi_getProcessorLineSize(); /* processor has no cache, use 'fast' code always */ if (cache_size == 0) { mp_using_cache_safe_exp = 0; } if ((cache_size == 0) || (cache_size >= 64)) { max_window_bits = 6; } else if (cache_size >= 32) { max_window_bits = 5; } else if (cache_size >= 16) { max_window_bits = 4; } else max_window_bits = 1; /* should this be an assert? */ } /* clamp the window size down before we caclulate bits_in_exponent */ if (mp_using_cache_safe_exp) { if (window_bits > max_window_bits) { window_bits = max_window_bits; } } #endif odd_ints = 1 << (window_bits - 1); i = bits_in_exponent % window_bits; if (i != 0) { bits_in_exponent += window_bits - i; } #ifdef MP_USING_MONT_MULF if (mp_using_mont_mulf) { MP_CHECKOK( s_mp_pad(&montBase, nLen) ); res = mp_exptmod_f(&montBase, exponent, modulus, result, &mmm, nLen, bits_in_exponent, window_bits, odd_ints); } else #endif #ifdef MP_USING_CACHE_SAFE_MOD_EXP if (mp_using_cache_safe_exp) { res = mp_exptmod_safe_i(&montBase, exponent, modulus, result, &mmm, nLen, bits_in_exponent, window_bits, 1 << window_bits); } else #endif res = mp_exptmod_i(&montBase, exponent, modulus, result, &mmm, nLen, bits_in_exponent, window_bits, odd_ints); CLEANUP: mp_clear(&montBase); mp_clear(&goodBase); /* Don't mp_clear mmm.N because it is merely a copy of modulus. ** Just zap it. */ memset(&mmm, 0, sizeof mmm); return res; }
/* Strong Lucas-Selfridge test. returns MP_YES if it is a strong L-S prime, MP_NO if it is composite Code ported from Thomas Ray Nicely's implementation of the BPSW test at http://www.trnicely.net/misc/bpsw.html Freeware copyright (C) 2016 Thomas R. Nicely <http://www.trnicely.net>. Released into the public domain by the author, who disclaims any legal liability arising from its use The multi-line comments are made by Thomas R. Nicely and are copied verbatim. Additional comments marked "CZ" (without the quotes) are by the code-portist. (If that name sounds familiar, he is the guy who found the fdiv bug in the Pentium (P5x, I think) Intel processor) */ int mp_prime_strong_lucas_selfridge(const mp_int *a, int *result) { /* CZ TODO: choose better variable names! */ mp_int Dz, gcd, Np1, Uz, Vz, U2mz, V2mz, Qmz, Q2mz, Qkdz, T1z, T2z, T3z, T4z, Q2kdz; /* CZ TODO: Some of them need the full 32 bit, hence the (temporary) exclusion of MP_8BIT */ int32_t D, Ds, J, sign, P, Q, r, s, u, Nbits; int e; int isset; *result = MP_NO; /* Find the first element D in the sequence {5, -7, 9, -11, 13, ...} such that Jacobi(D,N) = -1 (Selfridge's algorithm). Theory indicates that, if N is not a perfect square, D will "nearly always" be "small." Just in case, an overflow trap for D is included. */ if ((e = mp_init_multi(&Dz, &gcd, &Np1, &Uz, &Vz, &U2mz, &V2mz, &Qmz, &Q2mz, &Qkdz, &T1z, &T2z, &T3z, &T4z, &Q2kdz, NULL)) != MP_OKAY) { return e; } D = 5; sign = 1; for (;;) { Ds = sign * D; sign = -sign; if ((e = mp_set_long(&Dz, (unsigned long)D)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_gcd(a, &Dz, &gcd)) != MP_OKAY) { goto LBL_LS_ERR; } /* if 1 < GCD < N then N is composite with factor "D", and Jacobi(D,N) is technically undefined (but often returned as zero). */ if ((mp_cmp_d(&gcd, 1uL) == MP_GT) && (mp_cmp(&gcd, a) == MP_LT)) { goto LBL_LS_ERR; } if (Ds < 0) { Dz.sign = MP_NEG; } if ((e = mp_kronecker(&Dz, a, &J)) != MP_OKAY) { goto LBL_LS_ERR; } if (J == -1) { break; } D += 2; if (D > (INT_MAX - 2)) { e = MP_VAL; goto LBL_LS_ERR; } } P = 1; /* Selfridge's choice */ Q = (1 - Ds) / 4; /* Required so D = P*P - 4*Q */ /* NOTE: The conditions (a) N does not divide Q, and (b) D is square-free or not a perfect square, are included by some authors; e.g., "Prime numbers and computer methods for factorization," Hans Riesel (2nd ed., 1994, Birkhauser, Boston), p. 130. For this particular application of Lucas sequences, these conditions were found to be immaterial. */ /* Now calculate N - Jacobi(D,N) = N + 1 (even), and calculate the odd positive integer d and positive integer s for which N + 1 = 2^s*d (similar to the step for N - 1 in Miller's test). The strong Lucas-Selfridge test then returns N as a strong Lucas probable prime (slprp) if any of the following conditions is met: U_d=0, V_d=0, V_2d=0, V_4d=0, V_8d=0, V_16d=0, ..., etc., ending with V_{2^(s-1)*d}=V_{(N+1)/2}=0 (all equalities mod N). Thus d is the highest index of U that must be computed (since V_2m is independent of U), compared to U_{N+1} for the standard Lucas-Selfridge test; and no index of V beyond (N+1)/2 is required, just as in the standard Lucas-Selfridge test. However, the quantity Q^d must be computed for use (if necessary) in the latter stages of the test. The result is that the strong Lucas-Selfridge test has a running time only slightly greater (order of 10 %) than that of the standard Lucas-Selfridge test, while producing only (roughly) 30 % as many pseudoprimes (and every strong Lucas pseudoprime is also a standard Lucas pseudoprime). Thus the evidence indicates that the strong Lucas-Selfridge test is more effective than the standard Lucas-Selfridge test, and a Baillie-PSW test based on the strong Lucas-Selfridge test should be more reliable. */ if ((e = mp_add_d(a, 1uL, &Np1)) != MP_OKAY) { goto LBL_LS_ERR; } s = mp_cnt_lsb(&Np1); /* CZ * This should round towards zero because * Thomas R. Nicely used GMP's mpz_tdiv_q_2exp() * and mp_div_2d() is equivalent. Additionally: * dividing an even number by two does not produce * any leftovers. */ if ((e = mp_div_2d(&Np1, s, &Dz, NULL)) != MP_OKAY) { goto LBL_LS_ERR; } /* We must now compute U_d and V_d. Since d is odd, the accumulated values U and V are initialized to U_1 and V_1 (if the target index were even, U and V would be initialized instead to U_0=0 and V_0=2). The values of U_2m and V_2m are also initialized to U_1 and V_1; the FOR loop calculates in succession U_2 and V_2, U_4 and V_4, U_8 and V_8, etc. If the corresponding bits (1, 2, 3, ...) of t are on (the zero bit having been accounted for in the initialization of U and V), these values are then combined with the previous totals for U and V, using the composition formulas for addition of indices. */ mp_set(&Uz, 1uL); /* U=U_1 */ mp_set(&Vz, (mp_digit)P); /* V=V_1 */ mp_set(&U2mz, 1uL); /* U_1 */ mp_set(&V2mz, (mp_digit)P); /* V_1 */ if (Q < 0) { Q = -Q; if ((e = mp_set_long(&Qmz, (unsigned long)Q)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) { goto LBL_LS_ERR; } /* Initializes calculation of Q^d */ if ((e = mp_set_long(&Qkdz, (unsigned long)Q)) != MP_OKAY) { goto LBL_LS_ERR; } Qmz.sign = MP_NEG; Q2mz.sign = MP_NEG; Qkdz.sign = MP_NEG; Q = -Q; } else { if ((e = mp_set_long(&Qmz, (unsigned long)Q)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) { goto LBL_LS_ERR; } /* Initializes calculation of Q^d */ if ((e = mp_set_long(&Qkdz, (unsigned long)Q)) != MP_OKAY) { goto LBL_LS_ERR; } } Nbits = mp_count_bits(&Dz); for (u = 1; u < Nbits; u++) { /* zero bit off, already accounted for */ /* Formulas for doubling of indices (carried out mod N). Note that * the indices denoted as "2m" are actually powers of 2, specifically * 2^(ul-1) beginning each loop and 2^ul ending each loop. * * U_2m = U_m*V_m * V_2m = V_m*V_m - 2*Q^m */ if ((e = mp_mul(&U2mz, &V2mz, &U2mz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mod(&U2mz, a, &U2mz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_sqr(&V2mz, &V2mz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_sub(&V2mz, &Q2mz, &V2mz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mod(&V2mz, a, &V2mz)) != MP_OKAY) { goto LBL_LS_ERR; } /* Must calculate powers of Q for use in V_2m, also for Q^d later */ if ((e = mp_sqr(&Qmz, &Qmz)) != MP_OKAY) { goto LBL_LS_ERR; } /* prevents overflow */ /* CZ still necessary without a fixed prealloc'd mem.? */ if ((e = mp_mod(&Qmz, a, &Qmz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((isset = mp_get_bit(&Dz, u)) == MP_VAL) { e = isset; goto LBL_LS_ERR; } if (isset == MP_YES) { /* Formulas for addition of indices (carried out mod N); * * U_(m+n) = (U_m*V_n + U_n*V_m)/2 * V_(m+n) = (V_m*V_n + D*U_m*U_n)/2 * * Be careful with division by 2 (mod N)! */ if ((e = mp_mul(&U2mz, &Vz, &T1z)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mul(&Uz, &V2mz, &T2z)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mul(&V2mz, &Vz, &T3z)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mul(&U2mz, &Uz, &T4z)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = s_mp_mul_si(&T4z, (long)Ds, &T4z)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_add(&T1z, &T2z, &Uz)) != MP_OKAY) { goto LBL_LS_ERR; } if (mp_isodd(&Uz) != MP_NO) { if ((e = mp_add(&Uz, a, &Uz)) != MP_OKAY) { goto LBL_LS_ERR; } } /* CZ * This should round towards negative infinity because * Thomas R. Nicely used GMP's mpz_fdiv_q_2exp(). * But mp_div_2() does not do so, it is truncating instead. */ if ((e = mp_div_2(&Uz, &Uz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((Uz.sign == MP_NEG) && (mp_isodd(&Uz) != MP_NO)) { if ((e = mp_sub_d(&Uz, 1uL, &Uz)) != MP_OKAY) { goto LBL_LS_ERR; } } if ((e = mp_add(&T3z, &T4z, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } if (mp_isodd(&Vz) != MP_NO) { if ((e = mp_add(&Vz, a, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } } if ((e = mp_div_2(&Vz, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((Vz.sign == MP_NEG) && (mp_isodd(&Vz) != MP_NO)) { if ((e = mp_sub_d(&Vz, 1uL, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } } if ((e = mp_mod(&Uz, a, &Uz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mod(&Vz, a, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } /* Calculating Q^d for later use */ if ((e = mp_mul(&Qkdz, &Qmz, &Qkdz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) { goto LBL_LS_ERR; } } } /* If U_d or V_d is congruent to 0 mod N, then N is a prime or a strong Lucas pseudoprime. */ if ((mp_iszero(&Uz) != MP_NO) || (mp_iszero(&Vz) != MP_NO)) { *result = MP_YES; goto LBL_LS_ERR; } /* NOTE: Ribenboim ("The new book of prime number records," 3rd ed., 1995/6) omits the condition V0 on p.142, but includes it on p. 130. The condition is NECESSARY; otherwise the test will return false negatives---e.g., the primes 29 and 2000029 will be returned as composite. */ /* Otherwise, we must compute V_2d, V_4d, V_8d, ..., V_{2^(s-1)*d} by repeated use of the formula V_2m = V_m*V_m - 2*Q^m. If any of these are congruent to 0 mod N, then N is a prime or a strong Lucas pseudoprime. */ /* Initialize 2*Q^(d*2^r) for V_2m */ if ((e = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) { goto LBL_LS_ERR; } for (r = 1; r < s; r++) { if ((e = mp_sqr(&Vz, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_sub(&Vz, &Q2kdz, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mod(&Vz, a, &Vz)) != MP_OKAY) { goto LBL_LS_ERR; } if (mp_iszero(&Vz) != MP_NO) { *result = MP_YES; goto LBL_LS_ERR; } /* Calculate Q^{d*2^r} for next r (final iteration irrelevant). */ if (r < (s - 1)) { if ((e = mp_sqr(&Qkdz, &Qkdz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) { goto LBL_LS_ERR; } if ((e = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) { goto LBL_LS_ERR; } } } LBL_LS_ERR: mp_clear_multi(&Q2kdz, &T4z, &T3z, &T2z, &T1z, &Qkdz, &Q2mz, &Qmz, &V2mz, &U2mz, &Vz, &Uz, &Np1, &gcd, &Dz, NULL); return e; }