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#include "mod/mod_kth_root.hpp"
#include "nt/primetest.hpp" #include "mod/primitive_root.hpp" #include "mod/mod_inv.hpp" #include "ds/hashmap.hpp" // mod は int int mod_kth_root(ll k, ll a, int mod) { assert(primetest(mod) && 0 <= a && a < mod); if (k == 0) return (a == 1 ? 1 : -1); if (a == 0) return 0; if (mod == 2) return a; k %= mod - 1; Barrett bt(mod); ll g = gcd(k, mod - 1); if (mod_pow(a, (mod - 1) / g, mod) != 1) return -1; ll c = mod_inv(k / g, (mod - 1) / g); a = mod_pow(a, c, mod); k = (k * c) % (mod - 1); if (k == 0) return 1; g = primitive_root(mod); auto solve_pp = [&](ll p, int e, ll a) -> ll { int f = 0; ll pf = 1; while ((mod - 1) % (pf * p) == 0) ++f, pf *= p; ll m = (mod - 1) / pf; /* ・位数 Qm の巡回群 ・a の p^e 乗根をとりたい。持つことは分かっている ・a / x^{p^e} = b を維持する。まずは、b が p で割れる回数を増やしていく。 */ ll x = 1, b = a, c = f - e; // b ^ {mp^c} = 1 int pc = 1; FOR(c) pc *= p; int pe = 1; FOR(e) pe *= p; // 必要ならば原始 p 乗根に関する離散対数問題のセットアップ ll G = mod_pow(g, (mod - 1) / p, mod); int M = 0; HashMap<int> MP; ll GM_inv = -1; if (c) { while (M * M < p) ++M; MP.build(M); ll Gpow = 1; FOR(m, M) { MP[Gpow] = m; Gpow = bt.mul(Gpow, G); } GM_inv = mod_pow(Gpow, mod - 2, mod); } while (c) { /* b^{mp^c} = 1 が分かっている。(b/x^{p^e}})^{mp^{c-1}} = 1 にしたい。 x = g^{p^{f-c-e}*k} として探す。原始 p 乗根 B, G に対する B = G^k に帰着。 */ ll B = mod_pow(b, m * pc / p, mod); int k = [&](ll B) -> int { FOR(m, M + 1) { if (MP.count(B)) return m * M + MP[B]; B = bt.mul(B, GM_inv); } return -1; }(B); x = bt.mul(x, mod_pow(g, pf / pc / pe * k, mod)); ll exp = pf / pc * k % (mod - 1); b = bt.mul(b, mod_pow(g, mod - 1 - exp, mod)); --c; pc /= p; } int k = pe - mod_inv(m, pe); k = (k * m + 1) / pe; ll y = mod_pow(b, k, mod); x = bt.mul(x, y); return x; }; auto pf = factor(k); for (auto&& [p, e]: pf) a = solve_pp(p, e, a); return a; } ll mod_kth_root_64(ll k, ll a, ll mod) { assert(primetest(mod) && 0 <= a && a < mod); if (k == 0) return (a == 1 ? 1 : -1); if (a == 0) return 0; if (mod == 2) return a; k %= mod - 1; ll g = gcd(k, mod - 1); if (mod_pow_64(a, (mod - 1) / g, mod) != 1) return -1; ll c = mod_inv(k / g, (mod - 1) / g); a = mod_pow_64(a, c, mod); k = i128(k) * c % (mod - 1); if (k == 0) return 1; g = primitive_root_64(mod); auto solve_pp = [&](ll p, ll e, ll a) -> ll { ll f = 0; ll pf = 1; while (((mod - 1) / pf) % p == 0) ++f, pf *= p; ll m = (mod - 1) / pf; /* ・位数 Qm の巡回群 ・a の p^e 乗根をとりたい。持つことは分かっている ・a / x^{p^e} = b を維持する。まずは、b が p で割れる回数を増やしていく。 */ ll x = 1, b = a, c = f - e; // b ^ {mp^c} = 1 ll pc = 1; FOR(c) pc *= p; ll pe = 1; FOR(e) pe *= p; // 必要ならば原始 p 乗根に関する離散対数問題のセットアップ ll G = mod_pow_64(g, (mod - 1) / p, mod); ll M = 0; ll GM_inv = -1; HashMap<ll> MP; if (c) { while (M * M < p) ++M; MP.build(M); ll Gpow = 1; FOR(m, M) { MP[Gpow] = m; Gpow = i128(Gpow) * G % mod; } GM_inv = mod_pow_64(Gpow, mod - 2, mod); } while (c) { /* b^{mp^c} = 1 が分かっている。(b/x^{p^e}})^{mp^{c-1}} = 1 にしたい。 x = g^{p^{f-c-e}*k} として探す。原始 p 乗根 B, G に対する B = G^k に帰着。 */ ll B = mod_pow_64(b, pc / p * m, mod); ll k = [&](ll B) -> ll { FOR(m, M + 1) { if (MP.count(B)) return m * M + MP[B]; B = i128(B) * GM_inv % mod; } return -1; }(B); x = i128(x) * mod_pow_64(g, pf / pc / pe * k, mod) % mod; ll exp = pf / pc * i128(k) % (mod - 1); b = i128(b) * mod_pow_64(g, mod - 1 - exp, mod) % mod; --c; pc /= p; } ll k = pe - mod_inv(m, pe); k = (i128(k) * m + 1) / pe; ll y = mod_pow_64(b, k, mod); x = i128(x) * y % mod; return x; }; auto pf = factor(k); for (auto&& [p, e]: pf) a = solve_pp(p, e, a); return a; }
#line 2 "mod/mongomery_modint.hpp" // odd mod. // x の代わりに rx を持つ template <int id, typename U1, typename U2> struct Mongomery_modint { using mint = Mongomery_modint; inline static U1 m, r, n2; static constexpr int W = numeric_limits<U1>::digits; static void set_mod(U1 mod) { assert(mod & 1 && mod <= U1(1) << (W - 2)); m = mod, n2 = -U2(m) % m, r = m; FOR(5) r *= 2 - m * r; r = -r; assert(r * m == U1(-1)); } static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; } U1 x; Mongomery_modint() : x(0) {} Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){}; U1 val() const { U1 y = reduce(x); return y >= m ? y - m : y; } mint &operator+=(mint y) { x = ((x += y.x) >= m ? x - m : x); return *this; } mint &operator-=(mint y) { x -= (x >= y.x ? y.x : y.x - m); return *this; } mint &operator*=(mint y) { x = reduce(U2(x) * y.x); return *this; } mint operator+(mint y) const { return mint(*this) += y; } mint operator-(mint y) const { return mint(*this) -= y; } mint operator*(mint y) const { return mint(*this) *= y; } bool operator==(mint y) const { return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x); } bool operator!=(mint y) const { return not operator==(y); } mint pow(ll n) const { assert(n >= 0); mint y = 1, z = *this; for (; n; n >>= 1, z *= z) if (n & 1) y *= z; return y; } }; template <int id> using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>; template <int id> using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>; #line 3 "nt/primetest.hpp" bool primetest(const u64 x) { assert(x < u64(1) << 62); if (x == 2 or x == 3 or x == 5 or x == 7) return true; if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false; if (x < 121) return x > 1; const u64 d = (x - 1) >> lowbit(x - 1); using mint = Mongomery_modint_64<202311020>; mint::set_mod(x); const mint one(u64(1)), minus_one(x - 1); auto ok = [&](u64 a) -> bool { auto y = mint(a).pow(d); u64 t = d; while (y != one && y != minus_one && t != x - 1) y *= y, t <<= 1; if (y != minus_one && t % 2 == 0) return false; return true; }; if (x < (u64(1) << 32)) { for (u64 a: {2, 7, 61}) if (!ok(a)) return false; } else { for (u64 a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) { if (!ok(a)) return false; } } return true; } #line 2 "mod/primitive_root.hpp" #line 2 "nt/factor.hpp" #line 2 "random/base.hpp" u64 RNG_64() { static uint64_t x_ = uint64_t(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL; x_ ^= x_ << 7; return x_ ^= x_ >> 9; } u64 RNG(u64 lim) { return RNG_64() % lim; } ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); } #line 5 "nt/factor.hpp" template <typename mint> ll rho(ll n, ll c) { assert(n > 1); const mint cc(c); auto f = [&](mint x) { return x * x + cc; }; mint x = 1, y = 2, z = 1, q = 1; ll g = 1; const ll m = 1LL << (__lg(n) / 5); for (ll r = 1; g == 1; r <<= 1) { x = y; FOR(r) y = f(y); for (ll k = 0; k < r && g == 1; k += m) { z = y; FOR(min(m, r - k)) y = f(y), q *= x - y; g = gcd(q.val(), n); } } if (g == n) do { z = f(z); g = gcd((x - z).val(), n); } while (g == 1); return g; } ll find_prime_factor(ll n) { assert(n > 1); if (primetest(n)) return n; FOR(100) { ll m = 0; if (n < (1 << 30)) { using mint = Mongomery_modint_32<20231025>; mint::set_mod(n); m = rho<mint>(n, RNG(0, n)); } else { using mint = Mongomery_modint_64<20231025>; mint::set_mod(n); m = rho<mint>(n, RNG(0, n)); } if (primetest(m)) return m; n = m; } assert(0); return -1; } // ソートしてくれる vc<pair<ll, int>> factor(ll n) { assert(n >= 1); vc<pair<ll, int>> pf; FOR(p, 2, 100) { if (p * p > n) break; if (n % p == 0) { ll e = 0; do { n /= p, e += 1; } while (n % p == 0); pf.eb(p, e); } } while (n > 1) { ll p = find_prime_factor(n); ll e = 0; do { n /= p, e += 1; } while (n % p == 0); pf.eb(p, e); } sort(all(pf)); return pf; } vc<pair<ll, int>> factor_by_lpf(ll n, vc<int>& lpf) { vc<pair<ll, int>> res; while (n > 1) { int p = lpf[n]; int e = 0; while (n % p == 0) { n /= p; ++e; } res.eb(p, e); } return res; } #line 2 "mod/mod_pow.hpp" #line 2 "mod/barrett.hpp" // https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp struct Barrett { u32 m; u64 im; explicit Barrett(u32 m = 1) : m(m), im(u64(-1) / m + 1) {} u32 umod() const { return m; } u32 modulo(u64 z) { if (m == 1) return 0; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; return (z - y + (z < y ? m : 0)); } u64 floor(u64 z) { if (m == 1) return z; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; return (z < y ? x - 1 : x); } pair<u64, u32> divmod(u64 z) { if (m == 1) return {z, 0}; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; if (z < y) return {x - 1, z - y + m}; return {x, z - y}; } u32 mul(u32 a, u32 b) { return modulo(u64(a) * b); } }; struct Barrett_64 { u128 mod, mh, ml; explicit Barrett_64(u64 mod = 1) : mod(mod) { u128 m = u128(-1) / mod; if (m * mod + mod == u128(0)) ++m; mh = m >> 64; ml = m & u64(-1); } u64 umod() const { return mod; } u64 modulo(u128 x) { u128 z = (x & u64(-1)) * ml; z = (x & u64(-1)) * mh + (x >> 64) * ml + (z >> 64); z = (x >> 64) * mh + (z >> 64); x -= z * mod; return x < mod ? x : x - mod; } u64 mul(u64 a, u64 b) { return modulo(u128(a) * b); } }; #line 5 "mod/mod_pow.hpp" u32 mod_pow(int a, ll n, int mod) { assert(n >= 0); a = ((a %= mod) < 0 ? a + mod : a); if ((mod & 1) && (mod < (1 << 30))) { using mint = Mongomery_modint_32<202311021>; mint::set_mod(mod); return mint(a).pow(n).val(); } Barrett bt(mod); int r = 1; while (n) { if (n & 1) r = bt.mul(r, a); a = bt.mul(a, a), n >>= 1; } return r; } u64 mod_pow_64(ll a, ll n, u64 mod) { assert(n >= 0); a = ((a %= mod) < 0 ? a + mod : a); if ((mod & 1) && (mod < (u64(1) << 62))) { using mint = Mongomery_modint_64<202311021>; mint::set_mod(mod); return mint(a).pow(n).val(); } Barrett_64 bt(mod); ll r = 1; while (n) { if (n & 1) r = bt.mul(r, a); a = bt.mul(a, a), n >>= 1; } return r; } #line 6 "mod/primitive_root.hpp" // int int primitive_root(int p) { auto pf = factor(p - 1); auto is_ok = [&](int g) -> bool { for (auto&& [q, e]: pf) if (mod_pow(g, (p - 1) / q, p) == 1) return false; return true; }; while (1) { int x = RNG(1, p); if (is_ok(x)) return x; } return -1; } ll primitive_root_64(ll p) { auto pf = factor(p - 1); auto is_ok = [&](ll g) -> bool { for (auto&& [q, e]: pf) if (mod_pow_64(g, (p - 1) / q, p) == 1) return false; return true; }; while (1) { ll x = RNG(1, p); if (is_ok(x)) return x; } return -1; } #line 2 "mod/mod_inv.hpp" // long でも大丈夫 // (val * x - 1) が mod の倍数になるようにする // 特に mod=0 なら x=0 が満たす ll mod_inv(ll val, ll mod) { if (mod == 0) return 0; mod = abs(mod); val %= mod; if (val < 0) val += mod; ll a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } if (u < 0) u += mod; return u; } #line 2 "ds/hashmap.hpp" // u64 -> Val template <typename Val> struct HashMap { // n は入れたいものの個数で ok HashMap(u32 n = 0) { build(n); } void build(u32 n) { u32 k = 8; while (k < n * 2) k *= 2; cap = k / 2, mask = k - 1; key.resize(k), val.resize(k), used.assign(k, 0); } // size を保ったまま. size=0 にするときは build すること. void clear() { used.assign(len(used), 0); cap = (mask + 1) / 2; } int size() { return len(used) / 2 - cap; } int index(const u64& k) { int i = 0; for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {} return i; } Val& operator[](const u64& k) { if (cap == 0) extend(); int i = index(k); if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; } return val[i]; } Val get(const u64& k, Val default_value) { int i = index(k); return (used[i] ? val[i] : default_value); } bool count(const u64& k) { int i = index(k); return used[i] && key[i] == k; } // f(key, val) template <typename F> void enumerate_all(F f) { FOR(i, len(used)) if (used[i]) f(key[i], val[i]); } private: u32 cap, mask; vc<u64> key; vc<Val> val; vc<bool> used; u64 hash(u64 x) { static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count(); x += FIXED_RANDOM; x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9; x = (x ^ (x >> 27)) * 0x94d049bb133111eb; return (x ^ (x >> 31)) & mask; } void extend() { vc<pair<u64, Val>> dat; dat.reserve(len(used) / 2 - cap); FOR(i, len(used)) { if (used[i]) dat.eb(key[i], val[i]); } build(2 * len(dat)); for (auto& [a, b]: dat) (*this)[a] = b; } }; #line 5 "mod/mod_kth_root.hpp" // mod は int int mod_kth_root(ll k, ll a, int mod) { assert(primetest(mod) && 0 <= a && a < mod); if (k == 0) return (a == 1 ? 1 : -1); if (a == 0) return 0; if (mod == 2) return a; k %= mod - 1; Barrett bt(mod); ll g = gcd(k, mod - 1); if (mod_pow(a, (mod - 1) / g, mod) != 1) return -1; ll c = mod_inv(k / g, (mod - 1) / g); a = mod_pow(a, c, mod); k = (k * c) % (mod - 1); if (k == 0) return 1; g = primitive_root(mod); auto solve_pp = [&](ll p, int e, ll a) -> ll { int f = 0; ll pf = 1; while ((mod - 1) % (pf * p) == 0) ++f, pf *= p; ll m = (mod - 1) / pf; /* ・位数 Qm の巡回群 ・a の p^e 乗根をとりたい。持つことは分かっている ・a / x^{p^e} = b を維持する。まずは、b が p で割れる回数を増やしていく。 */ ll x = 1, b = a, c = f - e; // b ^ {mp^c} = 1 int pc = 1; FOR(c) pc *= p; int pe = 1; FOR(e) pe *= p; // 必要ならば原始 p 乗根に関する離散対数問題のセットアップ ll G = mod_pow(g, (mod - 1) / p, mod); int M = 0; HashMap<int> MP; ll GM_inv = -1; if (c) { while (M * M < p) ++M; MP.build(M); ll Gpow = 1; FOR(m, M) { MP[Gpow] = m; Gpow = bt.mul(Gpow, G); } GM_inv = mod_pow(Gpow, mod - 2, mod); } while (c) { /* b^{mp^c} = 1 が分かっている。(b/x^{p^e}})^{mp^{c-1}} = 1 にしたい。 x = g^{p^{f-c-e}*k} として探す。原始 p 乗根 B, G に対する B = G^k に帰着。 */ ll B = mod_pow(b, m * pc / p, mod); int k = [&](ll B) -> int { FOR(m, M + 1) { if (MP.count(B)) return m * M + MP[B]; B = bt.mul(B, GM_inv); } return -1; }(B); x = bt.mul(x, mod_pow(g, pf / pc / pe * k, mod)); ll exp = pf / pc * k % (mod - 1); b = bt.mul(b, mod_pow(g, mod - 1 - exp, mod)); --c; pc /= p; } int k = pe - mod_inv(m, pe); k = (k * m + 1) / pe; ll y = mod_pow(b, k, mod); x = bt.mul(x, y); return x; }; auto pf = factor(k); for (auto&& [p, e]: pf) a = solve_pp(p, e, a); return a; } ll mod_kth_root_64(ll k, ll a, ll mod) { assert(primetest(mod) && 0 <= a && a < mod); if (k == 0) return (a == 1 ? 1 : -1); if (a == 0) return 0; if (mod == 2) return a; k %= mod - 1; ll g = gcd(k, mod - 1); if (mod_pow_64(a, (mod - 1) / g, mod) != 1) return -1; ll c = mod_inv(k / g, (mod - 1) / g); a = mod_pow_64(a, c, mod); k = i128(k) * c % (mod - 1); if (k == 0) return 1; g = primitive_root_64(mod); auto solve_pp = [&](ll p, ll e, ll a) -> ll { ll f = 0; ll pf = 1; while (((mod - 1) / pf) % p == 0) ++f, pf *= p; ll m = (mod - 1) / pf; /* ・位数 Qm の巡回群 ・a の p^e 乗根をとりたい。持つことは分かっている ・a / x^{p^e} = b を維持する。まずは、b が p で割れる回数を増やしていく。 */ ll x = 1, b = a, c = f - e; // b ^ {mp^c} = 1 ll pc = 1; FOR(c) pc *= p; ll pe = 1; FOR(e) pe *= p; // 必要ならば原始 p 乗根に関する離散対数問題のセットアップ ll G = mod_pow_64(g, (mod - 1) / p, mod); ll M = 0; ll GM_inv = -1; HashMap<ll> MP; if (c) { while (M * M < p) ++M; MP.build(M); ll Gpow = 1; FOR(m, M) { MP[Gpow] = m; Gpow = i128(Gpow) * G % mod; } GM_inv = mod_pow_64(Gpow, mod - 2, mod); } while (c) { /* b^{mp^c} = 1 が分かっている。(b/x^{p^e}})^{mp^{c-1}} = 1 にしたい。 x = g^{p^{f-c-e}*k} として探す。原始 p 乗根 B, G に対する B = G^k に帰着。 */ ll B = mod_pow_64(b, pc / p * m, mod); ll k = [&](ll B) -> ll { FOR(m, M + 1) { if (MP.count(B)) return m * M + MP[B]; B = i128(B) * GM_inv % mod; } return -1; }(B); x = i128(x) * mod_pow_64(g, pf / pc / pe * k, mod) % mod; ll exp = pf / pc * i128(k) % (mod - 1); b = i128(b) * mod_pow_64(g, mod - 1 - exp, mod) % mod; --c; pc /= p; } ll k = pe - mod_inv(m, pe); k = (i128(k) * m + 1) / pe; ll y = mod_pow_64(b, k, mod); x = i128(x) * y % mod; return x; }; auto pf = factor(k); for (auto&& [p, e]: pf) a = solve_pp(p, e, a); return a; }