mod_sqrt.hpp¶
Compute the smaller square root modulo a prime, or nullopt if no square root exists.
Verified by sqrt_mod.
\[
\displaystyle x^2\equiv a\pmod p
\]
Implementation¶
#ifndef NOYA_MOD_SQRT_HPP
#define NOYA_MOD_SQRT_HPP 1
/// @complexity Time: O(log^2 p).
/// Space: O(1).
#include "noya/factorize.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <optional>
namespace noya {
/// @brief Compute the smaller square root modulo a prime, or nullopt if no
/// square root exists.
inline std::optional<std::uint64_t> mod_sqrt(std::uint64_t value,
std::uint64_t modulus) {
assert(modulus >= 2 && is_prime(modulus));
value %= modulus;
if (modulus == 2 || value == 0) {
return value;
}
using factorize_internal::multiply_mod;
using factorize_internal::power_mod;
if (power_mod(value, (modulus - 1) / 2, modulus) != 1) {
return std::nullopt;
}
if (modulus % 4 == 3) {
std::uint64_t root = power_mod(value, (modulus + 1) / 4, modulus);
return std::min(root, modulus - root);
}
std::uint64_t odd = modulus - 1;
int exponent = 0;
while ((odd & 1) == 0) {
odd >>= 1;
exponent++;
}
std::uint64_t non_residue = 2;
while (power_mod(non_residue, (modulus - 1) / 2, modulus) != modulus - 1) {
non_residue++;
}
std::uint64_t root = power_mod(value, (odd + 1) / 2, modulus);
std::uint64_t remainder = power_mod(value, odd, modulus);
std::uint64_t step = power_mod(non_residue, odd, modulus);
int remaining = exponent;
while (remainder != 1) {
std::uint64_t squared = remainder;
int shift = 0;
while (squared != 1 && shift < remaining) {
squared = multiply_mod(squared, squared, modulus);
shift++;
}
assert(shift < remaining);
std::uint64_t multiplier =
power_mod(step, std::uint64_t(1) << (remaining - shift - 1), modulus);
root = multiply_mod(root, multiplier, modulus);
step = multiply_mod(multiplier, multiplier, modulus);
remainder = multiply_mod(remainder, step, modulus);
remaining = shift;
}
return std::min(root, modulus - root);
}
} // namespace noya
#endif // NOYA_MOD_SQRT_HPP
#include <algorithm>
#include <array>
#include <cassert>
#include <cstdint>
#include <numeric>
#include <optional>
#include <utility>
#include <vector>
/// @complexity Time: O(log^2 p).
/// Space: O(1).
/// @complexity Time: O(log^3 n) primality testing; Pollard-rho factorization is expected about O(n^(1/4)).
/// Space: O(log n) recursion and factors.
namespace noya {
namespace factorize_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
inline u64 multiply_mod(u64 a, u64 b, u64 mod) {
return u64(u128(a) * b % mod);
}
inline u64 power_mod(u64 a, u64 exponent, u64 mod) {
u64 result = 1;
while (exponent > 0) {
if (exponent & 1) {
result = multiply_mod(result, a, mod);
}
a = multiply_mod(a, a, mod);
exponent >>= 1;
}
return result;
}
inline bool miller_rabin(u64 n) {
if (n < 2) {
return false;
}
for (u64 p :
std::array<u64, 12>{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}) {
if (n % p == 0) {
return n == p;
}
}
int shift = __builtin_ctzll(n - 1);
u64 odd = (n - 1) >> shift;
for (u64 base :
std::array<u64, 7>{2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
if (base % n == 0) {
continue;
}
u64 value = power_mod(base % n, odd, n);
if (value == 1 || value == n - 1) {
continue;
}
bool composite = true;
for (int i = 1; i < shift; i++) {
value = multiply_mod(value, value, n);
if (value == n - 1) {
composite = false;
break;
}
}
if (composite) {
return false;
}
}
return true;
}
inline u64 splitmix64(u64 &state) {
u64 z = (state += 0x9e3779b97f4a7c15ULL);
z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9ULL;
z = (z ^ (z >> 27)) * 0x94d049bb133111ebULL;
return z ^ (z >> 31);
}
inline u64 pollard_rho(u64 n) {
if (n % 2 == 0) {
return 2;
}
if (n % 3 == 0) {
return 3;
}
static u64 state = 0x123456789abcdef0ULL;
while (true) {
u64 y = splitmix64(state) % (n - 1) + 1;
u64 c = splitmix64(state) % (n - 1) + 1;
constexpr u64 block = 128;
u64 g = 1;
u64 r = 1;
u64 q = 1;
u64 x = 0;
u64 saved_y = 0;
auto next = [&](u64 value) {
return u64((u128(multiply_mod(value, value, n)) + c) % n);
};
while (g == 1) {
x = y;
for (u64 i = 0; i < r; i++) {
y = next(y);
}
for (u64 offset = 0; offset < r && g == 1; offset += block) {
saved_y = y;
for (u64 i = 0; i < std::min(block, r - offset); i++) {
y = next(y);
u64 difference = x > y ? x - y : y - x;
q = multiply_mod(q, difference, n);
}
g = std::gcd(q, n);
}
r <<= 1;
}
if (g == n) {
do {
saved_y = next(saved_y);
u64 difference = x > saved_y ? x - saved_y : saved_y - x;
g = std::gcd(difference, n);
} while (g == 1);
}
if (g != n) {
return g;
}
}
}
inline void collect_factors(u64 n, std::vector<u64> &result) {
if (n == 1) {
return;
}
if (miller_rabin(n)) {
result.push_back(n);
return;
}
u64 factor = pollard_rho(n);
collect_factors(factor, result);
collect_factors(n / factor, result);
}
} // namespace factorize_internal
/// @brief Deterministic Miller-Rabin primality test for unsigned 64-bit
/// integers.
inline bool is_prime(std::uint64_t n) {
return factorize_internal::miller_rabin(n);
}
/// @brief Return the prime factors of n with multiplicity in increasing order.
inline std::vector<std::uint64_t> prime_factors(std::uint64_t n) {
assert(n >= 1);
std::vector<std::uint64_t> result;
factorize_internal::collect_factors(n, result);
std::sort(result.begin(), result.end());
return result;
}
/// @brief Return the prime factorization of n as (prime, exponent) pairs.
inline std::vector<std::pair<std::uint64_t, int>> factorize(std::uint64_t n) {
std::vector<std::pair<std::uint64_t, int>> result;
for (std::uint64_t p : prime_factors(n)) {
if (result.empty() || result.back().first != p) {
result.emplace_back(p, 1);
} else {
result.back().second++;
}
}
return result;
}
} // namespace noya
namespace noya {
/// @brief Compute the smaller square root modulo a prime, or nullopt if no
/// square root exists.
inline std::optional<std::uint64_t> mod_sqrt(std::uint64_t value,
std::uint64_t modulus) {
assert(modulus >= 2 && is_prime(modulus));
value %= modulus;
if (modulus == 2 || value == 0) {
return value;
}
using factorize_internal::multiply_mod;
using factorize_internal::power_mod;
if (power_mod(value, (modulus - 1) / 2, modulus) != 1) {
return std::nullopt;
}
if (modulus % 4 == 3) {
std::uint64_t root = power_mod(value, (modulus + 1) / 4, modulus);
return std::min(root, modulus - root);
}
std::uint64_t odd = modulus - 1;
int exponent = 0;
while ((odd & 1) == 0) {
odd >>= 1;
exponent++;
}
std::uint64_t non_residue = 2;
while (power_mod(non_residue, (modulus - 1) / 2, modulus) != modulus - 1) {
non_residue++;
}
std::uint64_t root = power_mod(value, (odd + 1) / 2, modulus);
std::uint64_t remainder = power_mod(value, odd, modulus);
std::uint64_t step = power_mod(non_residue, odd, modulus);
int remaining = exponent;
while (remainder != 1) {
std::uint64_t squared = remainder;
int shift = 0;
while (squared != 1 && shift < remaining) {
squared = multiply_mod(squared, squared, modulus);
shift++;
}
assert(shift < remaining);
std::uint64_t multiplier =
power_mod(step, std::uint64_t(1) << (remaining - shift - 1), modulus);
root = multiply_mod(root, multiplier, modulus);
step = multiply_mod(multiplier, multiplier, modulus);
remainder = multiply_mod(remainder, step, modulus);
remaining = shift;
}
return std::min(root, modulus - root);
}
} // namespace noya