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#include "library/number/deterministic_miller_rabin.hpp"
#ifndef SUISEN_DETERMINISTIC_MILLER_RABIN #define SUISEN_DETERMINISTIC_MILLER_RABIN #include <array> #include <cassert> #include <cstdint> #include <iterator> #include <tuple> #include <type_traits> #include "library/number/montogomery.hpp" namespace suisen::miller_rabin { namespace internal { constexpr uint64_t THRESHOLD_1 = 341531ULL; constexpr uint64_t BASE_1[]{ 9345883071009581737ULL }; constexpr uint64_t THRESHOLD_2 = 1050535501ULL; constexpr uint64_t BASE_2[]{ 336781006125ULL, 9639812373923155ULL }; constexpr uint64_t THRESHOLD_3 = 350269456337ULL; constexpr uint64_t BASE_3[]{ 4230279247111683200ULL, 14694767155120705706ULL, 16641139526367750375ULL }; constexpr uint64_t THRESHOLD_4 = 55245642489451ULL; constexpr uint64_t BASE_4[]{ 2ULL, 141889084524735ULL, 1199124725622454117ULL, 11096072698276303650ULL }; constexpr uint64_t THRESHOLD_5 = 7999252175582851ULL; constexpr uint64_t BASE_5[]{ 2ULL, 4130806001517ULL, 149795463772692060ULL, 186635894390467037ULL, 3967304179347715805ULL }; constexpr uint64_t THRESHOLD_6 = 585226005592931977ULL; constexpr uint64_t BASE_6[]{ 2ULL, 123635709730000ULL, 9233062284813009ULL, 43835965440333360ULL, 761179012939631437ULL, 1263739024124850375ULL }; constexpr uint64_t BASE_7[]{ 2U, 325U, 9375U, 28178U, 450775U, 9780504U, 1795265022U }; template <auto BASE, std::size_t SIZE> constexpr bool miller_rabin(uint64_t n) { if (n == 2 or n == 3 or n == 5 or n == 7) return true; if (n <= 1 or n % 2 == 0 or n % 3 == 0 or n % 5 == 0 or n % 7 == 0) return false; if (n < 121) return true; const uint32_t s = __builtin_ctzll(n - 1); // >= 1 const uint64_t d = (n - 1) >> s; const Montgomery64 mg{ n }; const uint64_t one = mg.make(1), minus_one = mg.make(n - 1); for (std::size_t i = 0; i < SIZE; ++i) { uint64_t a = BASE[i] % n; if (a == 0) continue; uint64_t Y = mg.pow(mg.make(a), d); if (Y == one) continue; for (uint32_t r = 0;; ++r, Y = mg.mul(Y, Y)) { // Y = a^(d 2^r) if (Y == minus_one) break; if (r == s - 1) return false; } } return true; } } template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr> constexpr bool is_prime(T n) { if constexpr (std::is_signed_v<T>) { assert(n >= 0); } const std::make_unsigned_t<T> n_unsigned = n; assert(n_unsigned <= std::numeric_limits<uint64_t>::max()); // n < 2^64 using namespace internal; if (n_unsigned < THRESHOLD_1) return miller_rabin<BASE_1, 1>(n_unsigned); if (n_unsigned < THRESHOLD_2) return miller_rabin<BASE_2, 2>(n_unsigned); if (n_unsigned < THRESHOLD_3) return miller_rabin<BASE_3, 3>(n_unsigned); if (n_unsigned < THRESHOLD_4) return miller_rabin<BASE_4, 4>(n_unsigned); if (n_unsigned < THRESHOLD_5) return miller_rabin<BASE_5, 5>(n_unsigned); if (n_unsigned < THRESHOLD_6) return miller_rabin<BASE_6, 6>(n_unsigned); return miller_rabin<BASE_7, 7>(n_unsigned); } } // namespace suisen::miller_rabin #endif // SUISEN_DETERMINISTIC_MILLER_RABIN
#line 1 "library/number/deterministic_miller_rabin.hpp" #include <array> #include <cassert> #include <cstdint> #include <iterator> #include <tuple> #include <type_traits> #line 1 "library/number/montogomery.hpp" #line 6 "library/number/montogomery.hpp" #include <limits> namespace suisen { namespace internal::montgomery { template <typename Int, typename DInt> struct Montgomery { private: static constexpr uint32_t bits = std::numeric_limits<Int>::digits; static constexpr Int mask = ~Int(0); // R = 2**32 or 2**64 // 1. N is an odd number // 2. N < R // 3. gcd(N, R) = 1 // 4. R * R2 - N * N2 = 1 // 5. 0 < R2 < N // 6. 0 < N2 < R Int N, N2, R2; // RR = R * R (mod N) Int RR; public: constexpr Montgomery() = default; explicit constexpr Montgomery(Int N) : N(N), N2(calcN2(N)), R2(calcR2(N, N2)), RR(calcRR(N)) { assert(N & 1); } // @returns t * R (mod N) constexpr Int make(Int t) const { return reduce(static_cast<DInt>(t) * RR); } // @returns T * R^(-1) (mod N) constexpr Int reduce(DInt T) const { // 0 <= T < RN // Note: // 1. m = T * N2 (mod R) // 2. 0 <= m < R DInt m = modR(static_cast<DInt>(modR(T)) * N2); // Note: // T + m * N = T + T * N * N2 = T + T * (R * R2 - 1) = 0 (mod R) // => (T + m * N) / R is an integer. // => t * R = T + m * N = T (mod N) // => t = T R^(-1) (mod N) DInt t = divR(T + m * N); // Note: // 1. 0 <= T < RN // 2. 0 <= mN < RN (because 0 <= m < R) // => 0 <= T + mN < 2RN // => 0 <= t < 2N return t >= N ? t - N : t; } constexpr Int add(Int A, Int B) const { return (A += B) >= N ? A - N : A; } constexpr Int sub(Int A, Int B) const { return (A -= B) < 0 ? A + N : A; } constexpr Int mul(Int A, Int B) const { return reduce(static_cast<DInt>(A) * B); } constexpr Int div(Int A, Int B) const { return reduce(static_cast<DInt>(A) * inv(B)); } constexpr Int inv(Int A) const; // TODO: Implement constexpr Int pow(Int A, long long b) const { Int P = make(1); for (; b; b >>= 1) { if (b & 1) P = mul(P, A); A = mul(A, A); } return P; } private: static constexpr Int divR(DInt t) { return t >> bits; } static constexpr Int modR(DInt t) { return t & mask; } static constexpr Int calcN2(Int N) { // - N * N2 = 1 (mod R) // N2 = -N^{-1} (mod R) // calculates N^{-1} (mod R) by Newton's method DInt invN = N; // = N^{-1} (mod 2^2) for (uint32_t cur_bits = 2; cur_bits < bits; cur_bits *= 2) { // loop invariant: invN = N^{-1} (mod 2^cur_bits) // x = a^{-1} mod m => x(2-ax) = a^{-1} mod m^2 because: // ax = 1 (mod m) // => (ax-1)^2 = 0 (mod m^2) // => 2ax - a^2x^2 = 1 (mod m^2) // => a(x(2-ax)) = 1 (mod m^2) invN = modR(invN * modR(2 - N * invN)); } assert(modR(N * invN) == 1); return modR(-invN); } static constexpr Int calcR2(Int N, Int N2) { // R * R2 - N * N2 = 1 // => R2 = (1 + N * N2) / R return divR(1 + static_cast<DInt>(N) * N2); } static constexpr Int calcRR(Int N) { return -DInt(N) % N; } }; } // namespace internal::montgomery using Montgomery32 = internal::montgomery::Montgomery<uint32_t, uint64_t>; using Montgomery64 = internal::montgomery::Montgomery<uint64_t, __uint128_t>; } // namespace suisen #line 12 "library/number/deterministic_miller_rabin.hpp" namespace suisen::miller_rabin { namespace internal { constexpr uint64_t THRESHOLD_1 = 341531ULL; constexpr uint64_t BASE_1[]{ 9345883071009581737ULL }; constexpr uint64_t THRESHOLD_2 = 1050535501ULL; constexpr uint64_t BASE_2[]{ 336781006125ULL, 9639812373923155ULL }; constexpr uint64_t THRESHOLD_3 = 350269456337ULL; constexpr uint64_t BASE_3[]{ 4230279247111683200ULL, 14694767155120705706ULL, 16641139526367750375ULL }; constexpr uint64_t THRESHOLD_4 = 55245642489451ULL; constexpr uint64_t BASE_4[]{ 2ULL, 141889084524735ULL, 1199124725622454117ULL, 11096072698276303650ULL }; constexpr uint64_t THRESHOLD_5 = 7999252175582851ULL; constexpr uint64_t BASE_5[]{ 2ULL, 4130806001517ULL, 149795463772692060ULL, 186635894390467037ULL, 3967304179347715805ULL }; constexpr uint64_t THRESHOLD_6 = 585226005592931977ULL; constexpr uint64_t BASE_6[]{ 2ULL, 123635709730000ULL, 9233062284813009ULL, 43835965440333360ULL, 761179012939631437ULL, 1263739024124850375ULL }; constexpr uint64_t BASE_7[]{ 2U, 325U, 9375U, 28178U, 450775U, 9780504U, 1795265022U }; template <auto BASE, std::size_t SIZE> constexpr bool miller_rabin(uint64_t n) { if (n == 2 or n == 3 or n == 5 or n == 7) return true; if (n <= 1 or n % 2 == 0 or n % 3 == 0 or n % 5 == 0 or n % 7 == 0) return false; if (n < 121) return true; const uint32_t s = __builtin_ctzll(n - 1); // >= 1 const uint64_t d = (n - 1) >> s; const Montgomery64 mg{ n }; const uint64_t one = mg.make(1), minus_one = mg.make(n - 1); for (std::size_t i = 0; i < SIZE; ++i) { uint64_t a = BASE[i] % n; if (a == 0) continue; uint64_t Y = mg.pow(mg.make(a), d); if (Y == one) continue; for (uint32_t r = 0;; ++r, Y = mg.mul(Y, Y)) { // Y = a^(d 2^r) if (Y == minus_one) break; if (r == s - 1) return false; } } return true; } } template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr> constexpr bool is_prime(T n) { if constexpr (std::is_signed_v<T>) { assert(n >= 0); } const std::make_unsigned_t<T> n_unsigned = n; assert(n_unsigned <= std::numeric_limits<uint64_t>::max()); // n < 2^64 using namespace internal; if (n_unsigned < THRESHOLD_1) return miller_rabin<BASE_1, 1>(n_unsigned); if (n_unsigned < THRESHOLD_2) return miller_rabin<BASE_2, 2>(n_unsigned); if (n_unsigned < THRESHOLD_3) return miller_rabin<BASE_3, 3>(n_unsigned); if (n_unsigned < THRESHOLD_4) return miller_rabin<BASE_4, 4>(n_unsigned); if (n_unsigned < THRESHOLD_5) return miller_rabin<BASE_5, 5>(n_unsigned); if (n_unsigned < THRESHOLD_6) return miller_rabin<BASE_6, 6>(n_unsigned); return miller_rabin<BASE_7, 7>(n_unsigned); } } // namespace suisen::miller_rabin