void
mpi_divexact(const mpi *a, const mpi *b, mpi *q)
{
ASSERT(b->size != 0);
if (a->size == 0 ||
a->size < b->size) {
/* FIXME: raise an error here? */
mpi_zero(q);
return;
}
if (a == b) {
mpi_set_u32(q, 1);
return;
}
mp_size qsize = a->size - b->size + 1;
if (a == q || b == q) {
mp_digit *quot = MP_TMP_ALLOC(qsize);
mp_divexact(a->digits, a->size,
b->digits, b->size, quot);
MP_NORMALIZE(quot, qsize);
MPI_SIZE(q, qsize);
mp_copy(quot, qsize, q->digits);
MP_TMP_FREE(quot);
} else {
MPI_MIN_ALLOC(q, qsize);
mp_divexact(a->digits, a->size,
b->digits, b->size, q->digits);
MP_NORMALIZE(q->digits, qsize);
}
q->size = qsize;
q->sign = a->sign ^ b->sign;
}
// References : Cohen H., A course in computational algebraic number theory
// (1996), page 25.
bool multiplicative_order(const Ptr<RCP<const Integer>> &o,
const RCP<const Integer> &a,
const RCP<const Integer> &n)
{
integer_class order, p, t;
integer_class _a = a->as_integer_class(),
_n = mp_abs(n->as_integer_class());
mp_gcd(t, _a, _n);
if (t != 1)
return false;
RCP<const Integer> lambda = carmichael(n);
map_integer_uint prime_mul;
prime_factor_multiplicities(prime_mul, *lambda);
_a %= _n;
order = lambda->as_integer_class();
for (const auto it : prime_mul) {
p = it.first->as_integer_class();
mp_pow_ui(t, p, it.second);
mp_divexact(order, order, t);
mp_powm(t, _a, order, _n);
while (t != 1) {
mp_powm(t, t, p, _n);
order *= p;
}
}
*o = integer(std::move(order));
return true;
}
RCP<const Integer> totient(const RCP<const Integer> &n)
{
if (n->is_zero())
return integer(1);
integer_class phi = n->as_integer_class(), p;
if (phi < 0)
phi = -phi;
map_integer_uint prime_mul;
prime_factor_multiplicities(prime_mul, *n);
for (const auto &it : prime_mul) {
p = it.first->as_integer_class();
mp_divexact(phi, phi, p);
// phi is exactly divisible by p.
phi *= p - 1;
}
return integer(std::move(phi));
}
