CP-Algorithms Library
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cp-algo/number_theory/discrete_log.hpp
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- Last update: 2025-12-30 00:27:16+01:00
- Include:
#include "cp-algo/number_theory/discrete_log.hpp"
Depends on
- cp-algo/math/common.hpp
- cp-algo/number_theory/euler.hpp
- cp-algo/number_theory/factorize.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/number_theory/primality.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/big_alloc.hpp
Required by
cp-algo/number_theory.hppVerified with
Code
#ifndef CP_ALGO_NUMBER_THEORY_DISCRETE_LOG_HPP
#define CP_ALGO_NUMBER_THEORY_DISCRETE_LOG_HPP
#include "euler.hpp"
#include <optional>
namespace cp_algo::math {
// Find min non-negative x s.t. a*b^x = c (mod m)
template<typename _Int>
std::optional<_Int> discrete_log(_Int b, _Int c, _Int m, _Int a = 1) {
if(std::abs(a - c) % m == 0) {
return 0;
}
if(std::gcd(a, m) != std::gcd(int64_t(a) * b, int64_t(m))) {
auto res = discrete_log(b, c, m, _Int(int64_t(a) * b % m));
return res ? std::optional(*res + 1) : res;
}
// a * b^x is periodic here
using Int = std::make_signed_t<_Int>;
using base = dynamic_modint<Int>;
return base::with_mod(m, [&]() -> std::optional<_Int> {
int sqrtmod = std::max(1, (int)std::sqrt(m) / 2);
big_map<_Int, int> small;
base cur = a;
for(int i = 0; i < sqrtmod; i++) {
small[cur.getr()] = i;
cur *= b;
}
base step = bpow(base(b), sqrtmod);
cur = 1;
for(ptrdiff_t k = 0; k < m; k += sqrtmod) {
auto it = small.find((base(c) * cur).getr());
if(it != end(small)) {
auto cand = base::with_mod(period(base(b)), [&]() {
return base(it->second - k).getr();
});
if(base(a) * bpow(base(b), cand) == base(c)) {
return cand;
} else {
return std::nullopt;
}
}
cur *= step;
}
return std::nullopt;
});
}
}
#endif // CP_ALGO_NUMBER_THEORY_DISCRETE_LOG_HPP
#line 1 "cp-algo/number_theory/discrete_log.hpp"
#line 1 "cp-algo/number_theory/euler.hpp"
#line 1 "cp-algo/number_theory/factorize.hpp"
#line 1 "cp-algo/number_theory/primality.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
#include <bit>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
const int maxn = CP_ALGO_MAXN;
#else
const int maxn = 1 << 19;
#endif
const int magic = 64; // threshold for sizes to run the naive algo
auto bpow(auto const& x, auto n, auto const& one, auto op) {
if (n == 0) {
return one;
}
auto ans = x;
for(int j = std::bit_width<uint64_t>(n) - 2; ~j; j--) {
ans = op(ans, ans);
if((n >> j) & 1) {
ans = op(ans, x);
}
}
return ans;
}
auto bpow(auto x, auto n, auto ans) {
return bpow(x, n, ans, std::multiplies{});
}
template<typename T>
T bpow(T const& x, auto n) {
return bpow(x, n, T(1));
}
inline constexpr auto inv2(auto x) {
assert(x % 2);
std::make_unsigned_t<decltype(x)> y = 1;
while(y * x != 1) {
y *= 2 - x * y;
}
return y;
}
}
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {
template<typename modint, typename _Int>
struct modint_base {
using Int = _Int;
using UInt = std::make_unsigned_t<Int>;
static constexpr size_t bits = sizeof(Int) * 8;
using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
constexpr static Int mod() {
return modint::mod();
}
constexpr static Int remod() {
return modint::remod();
}
constexpr static UInt2 modmod() {
return UInt2(mod()) * mod();
}
constexpr modint_base() = default;
constexpr modint_base(Int2 rr) {
to_modint().setr(UInt((rr + modmod()) % mod()));
}
modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator - () const {
modint neg;
neg.r = std::min(-r, remod() - r);
return neg;
}
modint& operator /= (const modint &t) {
return to_modint() *= t.inv();
}
modint& operator *= (const modint &t) {
r = UInt(UInt2(r) * t.r % mod());
return to_modint();
}
modint& operator += (const modint &t) {
r += t.r; r = std::min(r, r - remod());
return to_modint();
}
modint& operator -= (const modint &t) {
r -= t.r; r = std::min(r, r + remod());
return to_modint();
}
modint operator + (const modint &t) const {return modint(to_modint()) += t;}
modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
// Why <=> doesn't work?..
auto operator == (const modint &t) const {return to_modint().getr() == t.getr();}
auto operator != (const modint &t) const {return to_modint().getr() != t.getr();}
auto operator <= (const modint &t) const {return to_modint().getr() <= t.getr();}
auto operator >= (const modint &t) const {return to_modint().getr() >= t.getr();}
auto operator < (const modint &t) const {return to_modint().getr() < t.getr();}
auto operator > (const modint &t) const {return to_modint().getr() > t.getr();}
Int rem() const {
UInt R = to_modint().getr();
return R - (R > (UInt)mod() / 2) * mod();
}
constexpr void setr(UInt rr) {
r = rr;
}
constexpr UInt getr() const {
return r;
}
// Only use these if you really know what you're doing!
static uint64_t modmod8() {return uint64_t(8 * modmod());}
void add_unsafe(UInt t) {r += t;}
void pseudonormalize() {r = std::min(r, r - modmod8());}
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr_direct(UInt rr) {r = rr;}
UInt getr_direct() const {return r;}
protected:
UInt r;
private:
constexpr modint& to_modint() {return static_cast<modint&>(*this);}
constexpr modint const& to_modint() const {return static_cast<modint const&>(*this);}
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
decltype(std::cin)& operator >> (decltype(std::cin) &in, modint &x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
decltype(std::cout)& operator << (decltype(std::cout) &out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::Int mod() {return m;}
static constexpr Base::UInt remod() {return m;}
auto getr() const {return Base::r;}
};
template<typename Int = int>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
static Base::UInt m_reduce(Base::UInt2 ab) {
if(mod() % 2 == 0) [[unlikely]] {
return typename Base::UInt(ab % mod());
} else {
typename Base::UInt2 m = typename Base::UInt(ab) * imod();
return typename Base::UInt((ab + m * mod()) >> Base::bits);
}
}
static Base::UInt m_transform(Base::UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
dynamic_modint& operator *= (const dynamic_modint &t) {
Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
return *this;
}
void setr(Base::UInt rr) {
Base::r = m_transform(rr);
}
Base::UInt getr() const {
typename Base::UInt res = m_reduce(Base::r);
return std::min(res, res - mod());
}
static Int mod() {return m;}
static Int remod() {return 2 * m;}
static Base::UInt imod() {return im;}
static Base::UInt2 pw128() {return r2;}
static void switch_mod(Int nm) {
m = nm;
im = m % 2 ? inv2(-m) : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() {switch_mod(prev);}
} _;
switch_mod(tmp);
return callback();
}
private:
static thread_local Int m;
static thread_local Base::UInt im, r2;
};
template<typename Int>
Int thread_local dynamic_modint<Int>::m = 1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}
#line 4 "cp-algo/number_theory/primality.hpp"
#include <algorithm>
#line 6 "cp-algo/number_theory/primality.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
template<typename _Int>
bool is_prime(_Int m) {
using Int = std::make_signed_t<_Int>;
using UInt = std::make_unsigned_t<Int>;
if(m == 1 || m % 2 == 0) {
return m == 2;
}
// m - 1 = 2^s * d
int s = std::countr_zero(UInt(m - 1));
auto d = (m - 1) >> s;
using base = dynamic_modint<Int>;
auto test = [&](base x) {
x = bpow(x, d);
if(std::abs(x.rem()) <= 1) {
return true;
}
for(int i = 1; i < s && x != -1; i++) {
x *= x;
}
return x == -1;
};
return base::with_mod(m, [&]() {
#ifdef CP_ALGO_NUMBER_THEORY_PRIMALITY_BASES_HPP
uint16_t base2 = 7, base3 = 61;
if (m != uint32_t(m)) {
base2 = base_table1[uint32_t(m * 0xAD625B89) >> 18];
base3 = base_table2[base2 >> 13];
}
return test(2) && test(base2) && test(base3);
#else
return std::ranges::all_of(std::array{2, 325, 9375, 28178, 450775, 9780504, 1795265022}, test);
#endif
});
}
}
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>
// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
# define CP_ALGO_USE_MMAP 1
# include <sys/mman.h>
#else
# define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo {
template <typename T, size_t Align = 32>
class big_alloc {
static_assert( Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::popcount(Align) == 1, "Align must be a power of two");
public:
using value_type = T;
template <class U> struct rebind { using other = big_alloc<U, Align>; };
constexpr bool operator==(const big_alloc&) const = default;
constexpr bool operator!=(const big_alloc&) const = default;
big_alloc() noexcept = default;
template <typename U, std::size_t A>
big_alloc(const big_alloc<U, A>&) noexcept {}
[[nodiscard]] T* allocate(std::size_t n) {
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if (padded >= MEGABYTE) {
void* raw = mmap(nullptr, padded,
PROT_READ | PROT_WRITE,
MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
madvise(raw, padded, MADV_HUGEPAGE);
madvise(raw, padded, MADV_POPULATE_WRITE);
return static_cast<T*>(raw);
}
#endif
return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
}
void deallocate(T* p, std::size_t n) noexcept {
if (!p) return;
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if (padded >= MEGABYTE) { munmap(p, padded); return; }
#endif
::operator delete(p, padded, std::align_val_t(align));
}
private:
static constexpr std::size_t MEGABYTE = 1 << 20;
static constexpr std::size_t round_up(std::size_t x) noexcept {
return (x + Align - 1) / Align * Align;
}
};
template<typename T> using big_vector = std::vector<T, big_alloc<T>>;
template<typename T> using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
template<typename T> using big_deque = std::deque<T, big_alloc<T>>;
template<typename T> using big_stack = std::stack<T, big_deque<T>>;
template<typename T> using big_queue = std::queue<T, big_deque<T>>;
template<typename T> using big_priority_queue = std::priority_queue<T, big_vector<T>>;
template<typename T> using big_forward_list = std::forward_list<T, big_alloc<T>>;
using big_string = big_basic_string<char>;
template<typename Key, typename Value, typename Compare = std::less<Key>>
using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
template<typename T, typename Compare = std::less<T>>
using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
template<typename T, typename Compare = std::less<T>>
using big_set = std::set<T, Compare, big_alloc<T>>;
template<typename Ref, typename V = void>
using big_generator = std::generator<Ref, V, big_alloc<std::byte>>;
}
// Deduction guide to make elements_of with big_generator default to big_alloc
namespace std::ranges {
template<typename Ref, typename V>
elements_of(cp_algo::big_generator<Ref, V>&&) -> elements_of<cp_algo::big_generator<Ref, V>&&, cp_algo::big_alloc<std::byte>>;
}
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random {
std::mt19937_64 gen(
std::chrono::steady_clock::now().time_since_epoch().count()
);
uint64_t rng() {
return gen();
}
}
#line 7 "cp-algo/number_theory/factorize.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
template<typename _Int>
auto proper_divisor(_Int m) {
using Int = std::make_signed_t<_Int>;
using base = dynamic_modint<Int>;
return m % 2 == 0 ? 2 : base::with_mod(m, [&]() {
base t = random::rng();
auto f = [&](auto x) {
return x * x + t;
};
base x = 0, y = 0;
base g = 1;
while(g == 1) {
for(int i = 0; i < 64; i++) {
x = f(x);
y = f(f(y));
if(x == y) [[unlikely]] {
t = random::rng();
x = y = 0;
} else {
base t = g * (x - y);
g = t == 0 ? g : t;
}
}
g = std::gcd(g.getr(), m);
}
return g.getr();
});
}
template<typename Int>
big_generator<Int> factorize(Int m) {
if(is_prime(m)) {
co_yield m;
} else if(m > 1) {
auto g = proper_divisor(m);
co_yield std::ranges::elements_of(factorize(g));
co_yield std::ranges::elements_of(factorize(m / g));
}
}
template<typename Int>
big_generator<Int> divisors_sqrt(Int m) {
for(Int i = 1; i * i <= m; i++) {
if(m % i == 0) {
co_yield i;
if(i * i != m) {
co_yield m / i;
}
}
}
}
}
#line 5 "cp-algo/number_theory/euler.hpp"
namespace cp_algo::math {
auto euler_phi(auto m) {
using T = std::decay_t<decltype(m)>;
auto primes = to<big_vector<T>>(factorize(m));
std::ranges::sort(primes);
auto [from, to] = std::ranges::unique(primes);
primes.erase(from, to);
auto ans = m;
for(auto it: primes) {
ans -= ans / it;
}
return ans;
}
template<modint_type base>
auto period(base x) {
auto ans = euler_phi(base::mod());
base x0 = bpow(x, ans);
for(auto t: factorize(ans)) {
while(ans % t == 0 && x0 * bpow(x, ans / t) == x0) {
ans /= t;
}
}
return ans;
}
template<typename _Int>
_Int primitive_root(_Int p) {
using Int = std::make_signed_t<_Int>;
using base = dynamic_modint<Int>;
return base::with_mod(p, [p](){
base t = 1;
while(period(t) != p - 1) {
t = random::rng();
}
return t.getr();
});
}
}
#line 4 "cp-algo/number_theory/discrete_log.hpp"
#include <optional>
namespace cp_algo::math {
// Find min non-negative x s.t. a*b^x = c (mod m)
template<typename _Int>
std::optional<_Int> discrete_log(_Int b, _Int c, _Int m, _Int a = 1) {
if(std::abs(a - c) % m == 0) {
return 0;
}
if(std::gcd(a, m) != std::gcd(int64_t(a) * b, int64_t(m))) {
auto res = discrete_log(b, c, m, _Int(int64_t(a) * b % m));
return res ? std::optional(*res + 1) : res;
}
// a * b^x is periodic here
using Int = std::make_signed_t<_Int>;
using base = dynamic_modint<Int>;
return base::with_mod(m, [&]() -> std::optional<_Int> {
int sqrtmod = std::max(1, (int)std::sqrt(m) / 2);
big_map<_Int, int> small;
base cur = a;
for(int i = 0; i < sqrtmod; i++) {
small[cur.getr()] = i;
cur *= b;
}
base step = bpow(base(b), sqrtmod);
cur = 1;
for(ptrdiff_t k = 0; k < m; k += sqrtmod) {
auto it = small.find((base(c) * cur).getr());
if(it != end(small)) {
auto cand = base::with_mod(period(base(b)), [&]() {
return base(it->second - k).getr();
});
if(base(a) * bpow(base(b), cand) == base(c)) {
return cand;
} else {
return std::nullopt;
}
}
cur *= step;
}
return std::nullopt;
});
}
}
#ifndef CP_ALGO_NUMBER_THEORY_DISCRETE_LOG_HPP
#define CP_ALGO_NUMBER_THEORY_DISCRETE_LOG_HPP
#include "euler.hpp"
#include <optional>
namespace cp_algo::math{template<typename _Int>std::optional<_Int>discrete_log(_Int b,_Int c,_Int m,_Int a=1){if(std::abs(a-c)%m==0){return 0;}if(std::gcd(a,m)!=std::gcd(int64_t(a)*b,int64_t(m))){auto res=discrete_log(b,c,m,_Int(int64_t(a)*b%m));return res?std::optional(*res+1):res;}using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return base::with_mod(m,[&]()->std::optional<_Int>{int sqrtmod=std::max(1,(int)std::sqrt(m)/2);big_map<_Int,int>small;base cur=a;for(int i=0;i<sqrtmod;i++){small[cur.getr()]=i;cur*=b;}base step=bpow(base(b),sqrtmod);cur=1;for(ptrdiff_t k=0;k<m;k+=sqrtmod){auto it=small.find((base(c)*cur).getr());if(it!=end(small)){auto cand=base::with_mod(period(base(b)),[&](){return base(it->second-k).getr();});if(base(a)*bpow(base(b),cand)==base(c)){return cand;}else{return std::nullopt;}}cur*=step;}return std::nullopt;});}}
#endif
#line 1 "cp-algo/number_theory/discrete_log.hpp"
#line 1 "cp-algo/number_theory/euler.hpp"
#line 1 "cp-algo/number_theory/factorize.hpp"
#line 1 "cp-algo/number_theory/primality.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
#include <bit>
namespace cp_algo::math{
#ifdef CP_ALGO_MAXN
const int maxn=CP_ALGO_MAXN;
#else
const int maxn=1<<19;
#endif
const int magic=64;auto bpow(auto const&x,auto n,auto const&one,auto op){if(n==0){return one;}auto ans=x;for(int j=std::bit_width<uint64_t>(n)-2;~j;j--){ans=op(ans,ans);if((n>>j)&1){ans=op(ans,x);}}return ans;}auto bpow(auto x,auto n,auto ans){return bpow(x,n,ans,std::multiplies{});}template<typename T>T bpow(T const&x,auto n){return bpow(x,n,T(1));}inline constexpr auto inv2(auto x){assert(x%2);std::make_unsigned_t<decltype(x)>y=1;while(y*x!=1){y*=2-x*y;}return y;}}
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math{template<typename modint,typename _Int>struct modint_base{using Int=_Int;using UInt=std::make_unsigned_t<Int>;static constexpr size_t bits=sizeof(Int)*8;using Int2=std::conditional_t<bits<=32,int64_t,__int128_t>;using UInt2=std::conditional_t<bits<=32,uint64_t,__uint128_t>;constexpr static Int mod(){return modint::mod();}constexpr static Int remod(){return modint::remod();}constexpr static UInt2 modmod(){return UInt2(mod())*mod();}constexpr modint_base()=default;constexpr modint_base(Int2 rr){to_modint().setr(UInt((rr+modmod())%mod()));}modint inv()const{return bpow(to_modint(),mod()-2);}modint operator-()const{modint neg;neg.r=std::min(-r,remod()-r);return neg;}modint&operator/=(const modint&t){return to_modint()*=t.inv();}modint&operator*=(const modint&t){r=UInt(UInt2(r)*t.r%mod());return to_modint();}modint&operator+=(const modint&t){r+=t.r;r=std::min(r,r-remod());return to_modint();}modint&operator-=(const modint&t){r-=t.r;r=std::min(r,r+remod());return to_modint();}modint operator+(const modint&t)const{return modint(to_modint())+=t;}modint operator-(const modint&t)const{return modint(to_modint())-=t;}modint operator*(const modint&t)const{return modint(to_modint())*=t;}modint operator/(const modint&t)const{return modint(to_modint())/=t;}auto operator==(const modint&t)const{return to_modint().getr()==t.getr();}auto operator!=(const modint&t)const{return to_modint().getr()!=t.getr();}auto operator<=(const modint&t)const{return to_modint().getr()<=t.getr();}auto operator>=(const modint&t)const{return to_modint().getr()>=t.getr();}auto operator<(const modint&t)const{return to_modint().getr()<t.getr();}auto operator>(const modint&t)const{return to_modint().getr()>t.getr();}Int rem()const{UInt R=to_modint().getr();return R-(R>(UInt)mod()/2)*mod();}constexpr void setr(UInt rr){r=rr;}constexpr UInt getr()const{return r;}static uint64_t modmod8(){return uint64_t(8*modmod());}void add_unsafe(UInt t){r+=t;}void pseudonormalize(){r=std::min(r,r-modmod8());}modint const&normalize(){if(r>=(UInt)mod()){r%=mod();}return to_modint();}void setr_direct(UInt rr){r=rr;}UInt getr_direct()const{return r;}protected:UInt r;private:constexpr modint&to_modint(){return static_cast<modint&>(*this);}constexpr modint const&to_modint()const{return static_cast<modint const&>(*this);}};template<typename modint>concept modint_type=std::is_base_of_v<modint_base<modint,typename modint::Int>,modint>;template<modint_type modint>decltype(std::cin)&operator>>(decltype(std::cin)&in,modint&x){typename modint::UInt r;auto&res=in>>r;x.setr(r);return res;}template<modint_type modint>decltype(std::cout)&operator<<(decltype(std::cout)&out,modint const&x){return out<<x.getr();}template<auto m>struct modint:modint_base<modint<m>,decltype(m)>{using Base=modint_base<modint<m>,decltype(m)>;using Base::Base;static constexpr Base::Int mod(){return m;}static constexpr Base::UInt remod(){return m;}auto getr()const{return Base::r;}};template<typename Int=int>struct dynamic_modint:modint_base<dynamic_modint<Int>,Int>{using Base=modint_base<dynamic_modint<Int>,Int>;using Base::Base;static Base::UInt m_reduce(Base::UInt2 ab){if(mod()%2==0)[[unlikely]]{return typename Base::UInt(ab%mod());}else{typename Base::UInt2 m=typename Base::UInt(ab)*imod();return typename Base::UInt((ab+m*mod())>>Base::bits);}}static Base::UInt m_transform(Base::UInt a){if(mod()%2==0)[[unlikely]]{return a;}else{return m_reduce(a*pw128());}}dynamic_modint&operator*=(const dynamic_modint&t){Base::r=m_reduce(typename Base::UInt2(Base::r)*t.r);return*this;}void setr(Base::UInt rr){Base::r=m_transform(rr);}Base::UInt getr()const{typename Base::UInt res=m_reduce(Base::r);return std::min(res,res-mod());}static Int mod(){return m;}static Int remod(){return 2*m;}static Base::UInt imod(){return im;}static Base::UInt2 pw128(){return r2;}static void switch_mod(Int nm){m=nm;im=m%2?inv2(-m):0;r2=static_cast<Base::UInt>(static_cast<Base::UInt2>(-1)%m+1);}auto static with_mod(Int tmp,auto callback){struct scoped{Int prev=mod();~scoped(){switch_mod(prev);}}_;switch_mod(tmp);return callback();}private:static thread_local Int m;static thread_local Base::UInt im,r2;};template<typename Int>Int thread_local dynamic_modint<Int>::m=1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im=-1;template<typename Int>dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2=0;}
#line 4 "cp-algo/number_theory/primality.hpp"
#include <algorithm>
#line 6 "cp-algo/number_theory/primality.hpp"
namespace cp_algo::math{template<typename _Int>bool is_prime(_Int m){using Int=std::make_signed_t<_Int>;using UInt=std::make_unsigned_t<Int>;if(m==1||m%2==0){return m==2;}int s=std::countr_zero(UInt(m-1));auto d=(m-1)>>s;using base=dynamic_modint<Int>;auto test=[&](base x){x=bpow(x,d);if(std::abs(x.rem())<=1){return true;}for(int i=1;i<s&&x!=-1;i++){x*=x;}return x==-1;};return base::with_mod(m,[&](){
#ifdef CP_ALGO_NUMBER_THEORY_PRIMALITY_BASES_HPP
uint16_t base2=7,base3=61;if(m!=uint32_t(m)){base2=base_table1[uint32_t(m*0xAD625B89)>>18];base3=base_table2[base2>>13];}return test(2)&&test(base2)&&test(base3);
#else
return std::ranges::all_of(std::array{2,325,9375,28178,450775,9780504,1795265022},test);
#endif
});}}
#line 1 "cp-algo/util/big_alloc.hpp"
#include <set>
#include <map>
#include <deque>
#include <stack>
#include <queue>
#include <vector>
#include <string>
#include <cstddef>
#line 13 "cp-algo/util/big_alloc.hpp"
#include <generator>
#include <forward_list>
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
# define CP_ALGO_USE_MMAP 1
# include <sys/mman.h>
#else
# define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo{template<typename T,size_t Align=32>class big_alloc{static_assert(Align>=alignof(void*),"Align must be at least pointer-size");static_assert(std::popcount(Align)==1,"Align must be a power of two");public:using value_type=T;template<class U>struct rebind{using other=big_alloc<U,Align>;};constexpr bool operator==(const big_alloc&)const=default;constexpr bool operator!=(const big_alloc&)const=default;big_alloc()noexcept=default;template<typename U,std::size_t A>big_alloc(const big_alloc<U,A>&)noexcept{}[[nodiscard]]T*allocate(std::size_t n){std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0);madvise(raw,padded,MADV_HUGEPAGE);madvise(raw,padded,MADV_POPULATE_WRITE);return static_cast<T*>(raw);}
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align)));}void deallocate(T*p,std::size_t n)noexcept{if(!p)return;std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){munmap(p,padded);return;}
#endif
::operator delete(p,padded,std::align_val_t(align));}private:static constexpr std::size_t MEGABYTE=1<<20;static constexpr std::size_t round_up(std::size_t x)noexcept{return(x+Align-1)/Align*Align;}};template<typename T>using big_vector=std::vector<T,big_alloc<T>>;template<typename T>using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>;template<typename T>using big_deque=std::deque<T,big_alloc<T>>;template<typename T>using big_stack=std::stack<T,big_deque<T>>;template<typename T>using big_queue=std::queue<T,big_deque<T>>;template<typename T>using big_priority_queue=std::priority_queue<T,big_vector<T>>;template<typename T>using big_forward_list=std::forward_list<T,big_alloc<T>>;using big_string=big_basic_string<char>;template<typename Key,typename Value,typename Compare=std::less<Key>>using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>;template<typename T,typename Compare=std::less<T>>using big_multiset=std::multiset<T,Compare,big_alloc<T>>;template<typename T,typename Compare=std::less<T>>using big_set=std::set<T,Compare,big_alloc<T>>;template<typename Ref,typename V=void>using big_generator=std::generator<Ref,V,big_alloc<std::byte>>;}namespace std::ranges{template<typename Ref,typename V>elements_of(cp_algo::big_generator<Ref,V>&&)->elements_of<cp_algo::big_generator<Ref,V>&&,cp_algo::big_alloc<std::byte>>;}
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random{std::mt19937_64 gen(std::chrono::steady_clock::now().time_since_epoch().count());uint64_t rng(){return gen();}}
#line 7 "cp-algo/number_theory/factorize.hpp"
namespace cp_algo::math{template<typename _Int>auto proper_divisor(_Int m){using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return m%2==0?2:base::with_mod(m,[&](){base t=random::rng();auto f=[&](auto x){return x*x+t;};base x=0,y=0;base g=1;while(g==1){for(int i=0;i<64;i++){x=f(x);y=f(f(y));if(x==y)[[unlikely]]{t=random::rng();x=y=0;}else{base t=g*(x-y);g=t==0?g:t;}}g=std::gcd(g.getr(),m);}return g.getr();});}template<typename Int>big_generator<Int>factorize(Int m){if(is_prime(m)){co_yield m;}else if(m>1){auto g=proper_divisor(m);co_yield std::ranges::elements_of(factorize(g));co_yield std::ranges::elements_of(factorize(m/g));}}template<typename Int>big_generator<Int>divisors_sqrt(Int m){for(Int i=1;i*i<=m;i++){if(m%i==0){co_yield i;if(i*i!=m){co_yield m/i;}}}}}
#line 5 "cp-algo/number_theory/euler.hpp"
namespace cp_algo::math{auto euler_phi(auto m){using T=std::decay_t<decltype(m)>;auto primes=to<big_vector<T>>(factorize(m));std::ranges::sort(primes);auto[from,to]=std::ranges::unique(primes);primes.erase(from,to);auto ans=m;for(auto it:primes){ans-=ans/it;}return ans;}template<modint_type base>auto period(base x){auto ans=euler_phi(base::mod());base x0=bpow(x,ans);for(auto t:factorize(ans)){while(ans%t==0&&x0*bpow(x,ans/t)==x0){ans/=t;}}return ans;}template<typename _Int>_Int primitive_root(_Int p){using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return base::with_mod(p,[p](){base t=1;while(period(t)!=p-1){t=random::rng();}return t.getr();});}}
#line 4 "cp-algo/number_theory/discrete_log.hpp"
#include <optional>
namespace cp_algo::math{template<typename _Int>std::optional<_Int>discrete_log(_Int b,_Int c,_Int m,_Int a=1){if(std::abs(a-c)%m==0){return 0;}if(std::gcd(a,m)!=std::gcd(int64_t(a)*b,int64_t(m))){auto res=discrete_log(b,c,m,_Int(int64_t(a)*b%m));return res?std::optional(*res+1):res;}using Int=std::make_signed_t<_Int>;using base=dynamic_modint<Int>;return base::with_mod(m,[&]()->std::optional<_Int>{int sqrtmod=std::max(1,(int)std::sqrt(m)/2);big_map<_Int,int>small;base cur=a;for(int i=0;i<sqrtmod;i++){small[cur.getr()]=i;cur*=b;}base step=bpow(base(b),sqrtmod);cur=1;for(ptrdiff_t k=0;k<m;k+=sqrtmod){auto it=small.find((base(c)*cur).getr());if(it!=end(small)){auto cand=base::with_mod(period(base(b)),[&](){return base(it->second-k).getr();});if(base(a)*bpow(base(b),cand)==base(c)){return cand;}else{return std::nullopt;}}cur*=step;}return std::nullopt;});}}