bigint/binary.hpp

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• Last update: 2024-02-11 04:08:39+09:00
• Include: #include "bigint/binary.hpp"
• Link: https://codeforces.com/contest/477/problem/D

Depends on

• mod/crt3.hpp
• mod/mod_inv.hpp
• mod/modint.hpp
• mod/modint_common.hpp
• poly/convolution.hpp
• poly/convolution_karatsuba.hpp
• poly/convolution_naive.hpp
• poly/fft.hpp
• poly/ntt.hpp

Verified with

• test/mytest/bigint.test.cpp

Code

#include "poly/convolution.hpp"

struct BigInteger_Binary {
  static constexpr int LOG = 30;
  static constexpr int MOD = 1 << LOG;
  using bint = BigInteger_Binary;
  int sgn;
  vc<int> dat;

  BigInteger_Binary() : sgn(0) {}
  BigInteger_Binary(i128 val) {
    if (val == 0) {
      sgn = 0;
      return;
    }
    sgn = 1;
    if (val < 0) sgn = -1, val = -val;
    while (val > 0) {
      dat.eb(val % MOD);
      val /= MOD;
    }
  }
  BigInteger_Binary(string s) {
    assert(!s.empty());
    sgn = 1;
    if (s[0] == '-') {
      sgn = -1;
      s.erase(s.begin());
      assert(!s.empty());
    }
    if (s[0] == '0') {
      sgn = 0;
      return;
    }
    reverse(all(s));
    int n = len(s);
    int m = ceil(n, LOG);
    dat.assign(m, 0);
    FOR(i, n) { dat[i / LOG] += ((s[i] - '0') << (i % LOG)); }
  }
  bint &operator=(const bint &p) {
    sgn = p.sgn;
    dat = p.dat;
    return *this;
  }
  bool operator<(const bint &p) const {
    if (sgn != p.sgn) return sgn < p.sgn;
    if (sgn == 0) return false;
    if (len(dat) != len(p.dat)) {
      if (sgn == 1) return len(dat) < len(p.dat);
      if (sgn == -1) return len(dat) > len(p.dat);
    }
    FOR_R(i, len(dat)) {
      if (dat[i] == p.dat[i]) continue;
      if (sgn == 1) return dat[i] < p.dat[i];
      if (sgn == -1) return dat[i] > p.dat[i];
    }
    return false;
  }
  bool operator>(const bint &p) const { return p < *this; }
  bool operator<=(const bint &p) const { return !(*this > p); }
  bool operator>=(const bint &p) const { return !(*this < p); }
  bint &operator+=(const bint p) {
    if (sgn == 0) { return *this = p; }
    if (p.sgn == 0) { return *this; }
    if (sgn != p.sgn) {
      *this -= (-p);
      return *this;
    }
    int n = max(len(dat), len(p.dat));
    dat.resize(n + 1);
    FOR(i, n) {
      if (i < len(p.dat)) dat[i] += p.dat[i];
      if (dat[i] >= MOD) dat[i] -= MOD, dat[i + 1] += 1;
    }
    while (len(dat) && dat.back() == 0) dat.pop_back();
    return *this;
  }
  bint &operator-=(const bint p) {
    if (sgn == 0) return *this = (-p);
    if (p.sgn == 0) return *this;
    if (sgn != p.sgn) {
      *this += (-p);
      return *this;
    }
    if ((sgn == 1 && *this < p) || (sgn == -1 && *this > p)) {
      *this = p - *this;
      sgn = -sgn;
      return *this;
    }
    FOR(i, len(p.dat)) { dat[i] -= p.dat[i]; }
    FOR(i, len(dat) - 1) {
      if (dat[i] < 0) dat[i] += MOD, dat[i + 1] -= 1;
    }
    while (len(dat) && dat.back() == 0) { dat.pop_back(); }
    if (dat.empty()) sgn = 0;
    return *this;
  }
  bint &operator*=(const bint &p) {
    sgn *= p.sgn;
    dat = convolve(dat, p.dat);
    return *this;
  }
  // bint &operator/=(const bint &p) { return *this; }
  bint operator-() const {
    bint p = *this;
    p.sgn *= -1;
    return p;
  }
  bint operator+(const bint &p) const { return bint(*this) += p; }
  bint operator-(const bint &p) const { return bint(*this) -= p; }
  bint operator*(const bint &p) const { return bint(*this) *= p; }
  // bint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const bint &p) const {
    return (sgn == p.sgn && dat == p.dat);
  }
  bool operator!=(const bint &p) const {
    return (sgn != p.sgn || dat != p.dat);
  }

  vc<int> convolve(const vc<int> &a, const vc<int> &b) {
    int n = len(a), m = len(b);
    if (!n || !m) return {};
    if (min(n, m) <= 500) {
      vc<int> c(n + m - 1);
      u128 x = 0;
      FOR(k, n + m - 1) {
        int s = max<int>(0, k + 1 - m), t = min<int>(k, n - 1);
        FOR(i, s, t + 1) { x += u64(a[i]) * b[k - i]; }
        c[k] = x % MOD, x = x / MOD;
      }
      while (x > 0) { c.eb(x % MOD), x = x / MOD; }
      return c;
    }
    static constexpr int p0 = 167772161;
    static constexpr int p1 = 469762049;
    static constexpr int p2 = 754974721;
    using mint0 = modint<p0>;
    using mint1 = modint<p1>;
    using mint2 = modint<p2>;
    vc<mint0> a0(all(a)), b0(all(b));
    vc<mint1> a1(all(a)), b1(all(b));
    vc<mint2> a2(all(a)), b2(all(b));
    auto c0 = convolution_ntt<mint0>(a0, b0);
    auto c1 = convolution_ntt<mint1>(a1, b1);
    auto c2 = convolution_ntt<mint2>(a2, b2);
    vc<int> c(len(c0));
    u128 x = 0;
    FOR(i, n + m - 1) {
      x += CRT3<u128, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val);
      c[i] = x % MOD, x = x / MOD;
    }
    while (x) { c.eb(x % MOD), x = x / MOD; }
    return c;
  }

  string to_string() {
    if (dat.empty()) return "0";
    string s;
    for (int x: dat) {
      FOR(LOG) {
        s += '0' + (x & 1);
        x /= 2;
      }
    }
    while (s.back() == '0') s.pop_back();
    if (sgn == -1) s += '-';
    reverse(all(s));
    return s;
  }

  string to_decimal_string() {
    assert(0);
    return "";
  }

  // https://codeforces.com/contest/477/problem/D
  pair<bint, int> divmod(int p) {
    assert(dat.empty() || sgn == 1);
    vc<int> after;
    ll rm = 0;
    FOR_R(i, len(dat)) {
      rm = rm * MOD + dat[i];
      after.eb(rm / p);
      rm = rm % p;
    }
    reverse(all(after));
    while (len(after) && after.back() == 0) POP(after);
    bint q;
    q.sgn = 1;
    q.dat = after;
    return {q, rm};
  }

  vc<int> base_p_representation(int p) {
    vc<u32> A(all(dat));
    vc<int> res;
    while (1) {
      while (len(A) && A.back() == u32(0)) POP(A);
      if (A.empty()) break;
      u64 rm = 0;
      FOR_R(i, len(A)) {
        rm = rm * MOD + A[i];
        A[i] = rm / p;
        rm %= p;
      }
      res.eb(rm);
    }
    reverse(all(res));
    return res;
  }

  void add_power_of_2(int k) {
    int q = k / LOG, r = k % LOG;
    if (sgn == 0) sgn = 1;
    if (q >= len(dat)) dat.resize(q + 1);
    if (sgn == 1) {
      dat[q] += 1 << r;
      while (dat[q] >= MOD) {
        dat[q] -= MOD;
        if (q + 1 >= len(dat)) dat.resize(q + 2);
        dat[q + 1] += 1;
        q += 1;
      }
    } else {
      dat[q] += 1 << r;
      while (dat[q] >= MOD) {
        dat[q] -= MOD;
        if (q + 1 >= len(dat)) dat.resize(q + 2);
        dat[q + 1] += 1;
        q += 1;
      }
    }
  }

  void substract_power_of_2(int k) {}
};

#ifdef FASTIO
void wt(BigInteger_Binary x) { fastio::wt(x.to_string()); }
#endif

#line 2 "mod/modint_common.hpp"

struct has_mod_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};

template <typename mint>
mint inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {0, 1};
  assert(0 <= n);
  if (n >= mod) n %= mod;
  while (len(dat) <= n) {
    int k = len(dat);
    int q = (mod + k - 1) / k;
    dat.eb(dat[k * q - mod] * mint::raw(q));
  }
  return dat[n];
}

template <typename mint>
mint fact(int n) {
  static const int mod = mint::get_mod();
  assert(0 <= n && n < mod);
  static vector<mint> dat = {1, 1};
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat)));
  return dat[n];
}

template <typename mint>
mint fact_inv(int n) {
  static vector<mint> dat = {1, 1};
  if (n < 0) return mint(0);
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
  return dat[n];
}

template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
  return (mint(1) * ... * fact_inv<mint>(xs));
}

template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
  return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}

template <typename mint>
mint C_dense(int n, int k) {
  static vvc<mint> C;
  static int H = 0, W = 0;
  auto calc = [&](int i, int j) -> mint {
    if (i == 0) return (j == 0 ? mint(1) : mint(0));
    return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
  };
  if (W <= k) {
    FOR(i, H) {
      C[i].resize(k + 1);
      FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
    }
    W = k + 1;
  }
  if (H <= n) {
    C.resize(n + 1);
    FOR(i, H, n + 1) {
      C[i].resize(W);
      FOR(j, W) { C[i][j] = calc(i, j); }
    }
    H = n + 1;
  }
  return C[n][k];
}

template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
  assert(n >= 0);
  if (k < 0 || n < k) return 0;
  if constexpr (dense) return C_dense<mint>(n, k);
  if constexpr (!large) return multinomial<mint>(n, k, n - k);
  k = min(k, n - k);
  mint x(1);
  FOR(i, k) x *= mint(n - i);
  return x * fact_inv<mint>(k);
}

template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
  assert(n >= 0);
  assert(0 <= k && k <= n);
  if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
  return mint(1) / C<mint, 1>(n, k);
}

// [x^d](1-x)^{-n}
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
  assert(n >= 0);
  if (d < 0) return mint(0);
  if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
  return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "mod/modint.hpp"

template <int mod>
struct modint {
  static constexpr u32 umod = u32(mod);
  static_assert(umod < u32(1) << 31);
  u32 val;

  static modint raw(u32 v) {
    modint x;
    x.val = v;
    return x;
  }
  constexpr modint() : val(0) {}
  constexpr modint(u32 x) : val(x % umod) {}
  constexpr modint(u64 x) : val(x % umod) {}
  constexpr modint(u128 x) : val(x % umod) {}
  constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
  bool operator<(const modint &other) const { return val < other.val; }
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += umod - p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = u64(val) * p.val % umod;
    return *this;
  }
  modint &operator/=(const modint &p) {
    *this *= p.inverse();
    return *this;
  }
  modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
  modint operator+(const modint &p) const { return modint(*this) += p; }
  modint operator-(const modint &p) const { return modint(*this) -= p; }
  modint operator*(const modint &p) const { return modint(*this) *= p; }
  modint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const modint &p) const { return val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }
  modint inverse() const {
    int 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);
    }
    return modint(u);
  }
  modint pow(ll n) const {
    assert(n >= 0);
    modint ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  static constexpr int get_mod() { return mod; }
  // (n, r), r は 1 の 2^n 乗根
  static constexpr pair<int, int> ntt_info() {
    if (mod == 120586241) return {20, 74066978};
    if (mod == 167772161) return {25, 17};
    if (mod == 469762049) return {26, 30};
    if (mod == 754974721) return {24, 362};
    if (mod == 880803841) return {23, 211};
    if (mod == 943718401) return {22, 663003469};
    if (mod == 998244353) return {23, 31};
    if (mod == 1045430273) return {20, 363};
    if (mod == 1051721729) return {20, 330};
    if (mod == 1053818881) return {20, 2789};
    return {-1, -1};
  }
  static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};

#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
  fastio::rd(x.val);
  x.val %= mod;
  // assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
  fastio::wt(x.val);
}
#endif

using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#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 1 "mod/crt3.hpp"

constexpr u32 mod_pow_constexpr(u64 a, u64 n, u32 mod) {
  a %= mod;
  u64 res = 1;
  FOR(32) {
    if (n & 1) res = res * a % mod;
    a = a * a % mod, n /= 2;
  }
  return res;
}

template <typename T, u32 p0, u32 p1, u32 p2>
T CRT3(u64 a0, u64 a1, u64 a2) {
  static_assert(p0 < p1 && p1 < p2);
  static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1);
  static constexpr u64 x01_2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
  u64 c = (a1 - a0 + p1) * x0_1 % p1;
  u64 a = a0 + c * p0;
  c = (a2 - a % p2 + p2) * x01_2 % p2;
  return T(a) + T(c) * T(p0) * T(p1);
}
#line 2 "poly/convolution_naive.hpp"

template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
  int n = int(a.size()), m = int(b.size());
  if (n > m) return convolution_naive<T>(b, a);
  if (n == 0) return {};
  vector<T> ans(n + m - 1);
  FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j];
  return ans;
}

template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
  int n = int(a.size()), m = int(b.size());
  if (n > m) return convolution_naive<T>(b, a);
  if (n == 0) return {};
  vc<T> ans(n + m - 1);
  if (n <= 16 && (T::get_mod() < (1 << 30))) {
    for (int k = 0; k < n + m - 1; ++k) {
      int s = max(0, k - m + 1);
      int t = min(n, k + 1);
      u64 sm = 0;
      for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
      ans[k] = sm;
    }
  } else {
    for (int k = 0; k < n + m - 1; ++k) {
      int s = max(0, k - m + 1);
      int t = min(n, k + 1);
      u128 sm = 0;
      for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
      ans[k] = T::raw(sm % T::get_mod());
    }
  }
  return ans;
}
#line 2 "poly/convolution_karatsuba.hpp"

// 任意の環でできる
template <typename T>
vc<T> convolution_karatsuba(const vc<T>& f, const vc<T>& g) {
  const int thresh = 30;
  if (min(len(f), len(g)) <= thresh) return convolution_naive(f, g);
  int n = max(len(f), len(g));
  int m = ceil(n, 2);
  vc<T> f1, f2, g1, g2;
  if (len(f) < m) f1 = f;
  if (len(f) >= m) f1 = {f.begin(), f.begin() + m};
  if (len(f) >= m) f2 = {f.begin() + m, f.end()};
  if (len(g) < m) g1 = g;
  if (len(g) >= m) g1 = {g.begin(), g.begin() + m};
  if (len(g) >= m) g2 = {g.begin() + m, g.end()};
  vc<T> a = convolution_karatsuba(f1, g1);
  vc<T> b = convolution_karatsuba(f2, g2);
  FOR(i, len(f2)) f1[i] += f2[i];
  FOR(i, len(g2)) g1[i] += g2[i];
  vc<T> c = convolution_karatsuba(f1, g1);
  vc<T> F(len(f) + len(g) - 1);
  assert(2 * m + len(b) <= len(F));
  FOR(i, len(a)) F[i] += a[i], c[i] -= a[i];
  FOR(i, len(b)) F[2 * m + i] += b[i], c[i] -= b[i];
  if (c.back() == T(0)) c.pop_back();
  FOR(i, len(c)) if (c[i] != T(0)) F[m + i] += c[i];
  return F;
}
#line 2 "poly/ntt.hpp"

template <class mint>
void ntt(vector<mint>& a, bool inverse) {
  assert(mint::can_ntt());
  const int rank2 = mint::ntt_info().fi;
  const int mod = mint::get_mod();
  static array<mint, 30> root, iroot;
  static array<mint, 30> rate2, irate2;
  static array<mint, 30> rate3, irate3;

  assert(rank2 != -1 && len(a) <= (1 << max(0, rank2)));

  static bool prepared = 0;
  if (!prepared) {
    prepared = 1;
    root[rank2] = mint::ntt_info().se;
    iroot[rank2] = mint(1) / root[rank2];
    FOR_R(i, rank2) {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }
    mint prod = 1, iprod = 1;
    for (int i = 0; i <= rank2 - 2; i++) {
      rate2[i] = root[i + 2] * prod;
      irate2[i] = iroot[i + 2] * iprod;
      prod *= iroot[i + 2];
      iprod *= root[i + 2];
    }
    prod = 1, iprod = 1;
    for (int i = 0; i <= rank2 - 3; i++) {
      rate3[i] = root[i + 3] * prod;
      irate3[i] = iroot[i + 3] * iprod;
      prod *= iroot[i + 3];
      iprod *= root[i + 3];
    }
  }

  int n = int(a.size());
  int h = topbit(n);
  assert(n == 1 << h);
  if (!inverse) {
    int len = 0;
    while (len < h) {
      if (h - len == 1) {
        int p = 1 << (h - len - 1);
        mint rot = 1;
        FOR(s, 1 << len) {
          int offset = s << (h - len);
          FOR(i, p) {
            auto l = a[i + offset];
            auto r = a[i + offset + p] * rot;
            a[i + offset] = l + r;
            a[i + offset + p] = l - r;
          }
          rot *= rate2[topbit(~s & -~s)];
        }
        len++;
      } else {
        int p = 1 << (h - len - 2);
        mint rot = 1, imag = root[2];
        for (int s = 0; s < (1 << len); s++) {
          mint rot2 = rot * rot;
          mint rot3 = rot2 * rot;
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            u64 mod2 = u64(mod) * mod;
            u64 a0 = a[i + offset].val;
            u64 a1 = u64(a[i + offset + p].val) * rot.val;
            u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val;
            u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val;
            u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val;
            u64 na2 = mod2 - a2;
            a[i + offset] = a0 + a2 + a1 + a3;
            a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
            a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
            a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
          }
          rot *= rate3[topbit(~s & -~s)];
        }
        len += 2;
      }
    }
  } else {
    mint coef = mint(1) / mint(len(a));
    FOR(i, len(a)) a[i] *= coef;
    int len = h;
    while (len) {
      if (len == 1) {
        int p = 1 << (h - len);
        mint irot = 1;
        FOR(s, 1 << (len - 1)) {
          int offset = s << (h - len + 1);
          FOR(i, p) {
            u64 l = a[i + offset].val;
            u64 r = a[i + offset + p].val;
            a[i + offset] = l + r;
            a[i + offset + p] = (mod + l - r) * irot.val;
          }
          irot *= irate2[topbit(~s & -~s)];
        }
        len--;
      } else {
        int p = 1 << (h - len);
        mint irot = 1, iimag = iroot[2];
        FOR(s, (1 << (len - 2))) {
          mint irot2 = irot * irot;
          mint irot3 = irot2 * irot;
          int offset = s << (h - len + 2);
          for (int i = 0; i < p; i++) {
            u64 a0 = a[i + offset + 0 * p].val;
            u64 a1 = a[i + offset + 1 * p].val;
            u64 a2 = a[i + offset + 2 * p].val;
            u64 a3 = a[i + offset + 3 * p].val;
            u64 x = (mod + a2 - a3) * iimag.val % mod;
            a[i + offset] = a0 + a1 + a2 + a3;
            a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val;
            a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val;
            a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val;
          }
          irot *= irate3[topbit(~s & -~s)];
        }
        len -= 2;
      }
    }
  }
}
#line 1 "poly/fft.hpp"
namespace CFFT {
using real = double;

struct C {
  real x, y;

  C() : x(0), y(0) {}

  C(real x, real y) : x(x), y(y) {}
  inline C operator+(const C& c) const { return C(x + c.x, y + c.y); }
  inline C operator-(const C& c) const { return C(x - c.x, y - c.y); }
  inline C operator*(const C& c) const {
    return C(x * c.x - y * c.y, x * c.y + y * c.x);
  }

  inline C conj() const { return C(x, -y); }
};

const real PI = acosl(-1);
int base = 1;
vector<C> rts = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};

void ensure_base(int nbase) {
  if (nbase <= base) return;
  rev.resize(1 << nbase);
  rts.resize(1 << nbase);
  for (int i = 0; i < (1 << nbase); i++) {
    rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
  }
  while (base < nbase) {
    real angle = PI * 2.0 / (1 << (base + 1));
    for (int i = 1 << (base - 1); i < (1 << base); i++) {
      rts[i << 1] = rts[i];
      real angle_i = angle * (2 * i + 1 - (1 << base));
      rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
    }
    ++base;
  }
}

void fft(vector<C>& a, int n) {
  assert((n & (n - 1)) == 0);
  int zeros = __builtin_ctz(n);
  ensure_base(zeros);
  int shift = base - zeros;
  for (int i = 0; i < n; i++) {
    if (i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); }
  }
  for (int k = 1; k < n; k <<= 1) {
    for (int i = 0; i < n; i += 2 * k) {
      for (int j = 0; j < k; j++) {
        C z = a[i + j + k] * rts[j + k];
        a[i + j + k] = a[i + j] - z;
        a[i + j] = a[i + j] + z;
      }
    }
  }
}
} // namespace CFFT
#line 9 "poly/convolution.hpp"

template <class mint>
vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
  if (a.empty() || b.empty()) return {};
  int n = int(a.size()), m = int(b.size());
  int sz = 1;
  while (sz < n + m - 1) sz *= 2;

  // sz = 2^k のときの高速化。分割統治的なやつで損しまくるので。

  if ((n + m - 3) <= sz / 2) {
    auto a_last = a.back(), b_last = b.back();
    a.pop_back(), b.pop_back();
    auto c = convolution(a, b);
    c.resize(n + m - 1);
    c[n + m - 2] = a_last * b_last;
    FOR(i, len(a)) c[i + len(b)] += a[i] * b_last;
    FOR(i, len(b)) c[i + len(a)] += b[i] * a_last;
    return c;
  }

  a.resize(sz), b.resize(sz);
  bool same = a == b;
  ntt(a, 0);
  if (same) {
    b = a;
  } else {
    ntt(b, 0);
  }
  FOR(i, sz) a[i] *= b[i];
  ntt(a, 1);
  a.resize(n + m - 1);
  return a;
}

template <typename mint>
vector<mint> convolution_garner(const vector<mint>& a, const vector<mint>& b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  static constexpr int p0 = 167772161;
  static constexpr int p1 = 469762049;
  static constexpr int p2 = 754974721;
  using mint0 = modint<p0>;
  using mint1 = modint<p1>;
  using mint2 = modint<p2>;
  vc<mint0> a0(n), b0(m);
  vc<mint1> a1(n), b1(m);
  vc<mint2> a2(n), b2(m);
  FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
  FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
  auto c0 = convolution_ntt<mint0>(a0, b0);
  auto c1 = convolution_ntt<mint1>(a1, b1);
  auto c2 = convolution_ntt<mint2>(a2, b2);
  vc<mint> c(len(c0));
  FOR(i, n + m - 1) {
    c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val);
  }
  return c;
}

template <typename R>
vc<double> convolution_fft(const vc<R>& a, const vc<R>& b) {
  using C = CFFT::C;
  int need = (int)a.size() + (int)b.size() - 1;
  int nbase = 1;
  while ((1 << nbase) < need) nbase++;
  CFFT::ensure_base(nbase);
  int sz = 1 << nbase;
  vector<C> fa(sz);
  for (int i = 0; i < sz; i++) {
    double x = (i < (int)a.size() ? a[i] : 0);
    double y = (i < (int)b.size() ? b[i] : 0);
    fa[i] = C(x, y);
  }
  CFFT::fft(fa, sz);
  C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
  for (int i = 0; i <= (sz >> 1); i++) {
    int j = (sz - i) & (sz - 1);
    C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
    fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
    fa[i] = z;
  }
  for (int i = 0; i < (sz >> 1); i++) {
    C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
    C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * CFFT::rts[(sz >> 1) + i];
    fa[i] = A0 + A1 * s;
  }
  CFFT::fft(fa, sz >> 1);
  vector<double> ret(need);
  for (int i = 0; i < need; i++) {
    ret[i] = (i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
  }
  return ret;
}

vector<ll> convolution(const vector<ll>& a, const vector<ll>& b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  if (min(n, m) <= 2500) return convolution_naive(a, b);
  ll abs_sum_a = 0, abs_sum_b = 0;
  ll LIM = 1e15;
  FOR(i, n) abs_sum_a = min(LIM, abs_sum_a + abs(a[i]));
  FOR(i, m) abs_sum_b = min(LIM, abs_sum_b + abs(b[i]));
  if (i128(abs_sum_a) * abs_sum_b < 1e15) {
    vc<double> c = convolution_fft<ll>(a, b);
    vc<ll> res(len(c));
    FOR(i, len(c)) res[i] = ll(floor(c[i] + .5));
    return res;
  }

  static constexpr unsigned long long MOD1 = 754974721; // 2^24

  static constexpr unsigned long long MOD2 = 167772161; // 2^25

  static constexpr unsigned long long MOD3 = 469762049; // 2^26

  static constexpr unsigned long long M2M3 = MOD2 * MOD3;
  static constexpr unsigned long long M1M3 = MOD1 * MOD3;
  static constexpr unsigned long long M1M2 = MOD1 * MOD2;
  static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

  static const unsigned long long i1 = mod_inv(MOD2 * MOD3, MOD1);
  static const unsigned long long i2 = mod_inv(MOD1 * MOD3, MOD2);
  static const unsigned long long i3 = mod_inv(MOD1 * MOD2, MOD3);

  using mint1 = modint<MOD1>;
  using mint2 = modint<MOD2>;
  using mint3 = modint<MOD3>;

  vc<mint1> a1(n), b1(m);
  vc<mint2> a2(n), b2(m);
  vc<mint3> a3(n), b3(m);
  FOR(i, n) a1[i] = a[i], a2[i] = a[i], a3[i] = a[i];
  FOR(i, m) b1[i] = b[i], b2[i] = b[i], b3[i] = b[i];

  auto c1 = convolution_ntt<mint1>(a1, b1);
  auto c2 = convolution_ntt<mint2>(a2, b2);
  auto c3 = convolution_ntt<mint3>(a3, b3);

  vc<ll> c(n + m - 1);
  FOR(i, n + m - 1) {
    u64 x = 0;
    x += (c1[i].val * i1) % MOD1 * M2M3;
    x += (c2[i].val * i2) % MOD2 * M1M3;
    x += (c3[i].val * i3) % MOD3 * M1M2;
    ll diff = c1[i].val - ((long long)(x) % (long long)(MOD1));
    if (diff < 0) diff += MOD1;
    static constexpr unsigned long long offset[5]
        = {0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
    x -= offset[diff % 5];
    c[i] = x;
  }
  return c;
}

template <typename mint>
vc<mint> convolution(const vc<mint>& a, const vc<mint>& b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  if (mint::can_ntt()) {
    if (min(n, m) <= 50) return convolution_karatsuba<mint>(a, b);
    return convolution_ntt(a, b);
  }
  if (min(n, m) <= 200) return convolution_karatsuba<mint>(a, b);
  return convolution_garner(a, b);
}
#line 2 "bigint/binary.hpp"

struct BigInteger_Binary {
  static constexpr int LOG = 30;
  static constexpr int MOD = 1 << LOG;
  using bint = BigInteger_Binary;
  int sgn;
  vc<int> dat;

  BigInteger_Binary() : sgn(0) {}
  BigInteger_Binary(i128 val) {
    if (val == 0) {
      sgn = 0;
      return;
    }
    sgn = 1;
    if (val < 0) sgn = -1, val = -val;
    while (val > 0) {
      dat.eb(val % MOD);
      val /= MOD;
    }
  }
  BigInteger_Binary(string s) {
    assert(!s.empty());
    sgn = 1;
    if (s[0] == '-') {
      sgn = -1;
      s.erase(s.begin());
      assert(!s.empty());
    }
    if (s[0] == '0') {
      sgn = 0;
      return;
    }
    reverse(all(s));
    int n = len(s);
    int m = ceil(n, LOG);
    dat.assign(m, 0);
    FOR(i, n) { dat[i / LOG] += ((s[i] - '0') << (i % LOG)); }
  }
  bint &operator=(const bint &p) {
    sgn = p.sgn;
    dat = p.dat;
    return *this;
  }
  bool operator<(const bint &p) const {
    if (sgn != p.sgn) return sgn < p.sgn;
    if (sgn == 0) return false;
    if (len(dat) != len(p.dat)) {
      if (sgn == 1) return len(dat) < len(p.dat);
      if (sgn == -1) return len(dat) > len(p.dat);
    }
    FOR_R(i, len(dat)) {
      if (dat[i] == p.dat[i]) continue;
      if (sgn == 1) return dat[i] < p.dat[i];
      if (sgn == -1) return dat[i] > p.dat[i];
    }
    return false;
  }
  bool operator>(const bint &p) const { return p < *this; }
  bool operator<=(const bint &p) const { return !(*this > p); }
  bool operator>=(const bint &p) const { return !(*this < p); }
  bint &operator+=(const bint p) {
    if (sgn == 0) { return *this = p; }
    if (p.sgn == 0) { return *this; }
    if (sgn != p.sgn) {
      *this -= (-p);
      return *this;
    }
    int n = max(len(dat), len(p.dat));
    dat.resize(n + 1);
    FOR(i, n) {
      if (i < len(p.dat)) dat[i] += p.dat[i];
      if (dat[i] >= MOD) dat[i] -= MOD, dat[i + 1] += 1;
    }
    while (len(dat) && dat.back() == 0) dat.pop_back();
    return *this;
  }
  bint &operator-=(const bint p) {
    if (sgn == 0) return *this = (-p);
    if (p.sgn == 0) return *this;
    if (sgn != p.sgn) {
      *this += (-p);
      return *this;
    }
    if ((sgn == 1 && *this < p) || (sgn == -1 && *this > p)) {
      *this = p - *this;
      sgn = -sgn;
      return *this;
    }
    FOR(i, len(p.dat)) { dat[i] -= p.dat[i]; }
    FOR(i, len(dat) - 1) {
      if (dat[i] < 0) dat[i] += MOD, dat[i + 1] -= 1;
    }
    while (len(dat) && dat.back() == 0) { dat.pop_back(); }
    if (dat.empty()) sgn = 0;
    return *this;
  }
  bint &operator*=(const bint &p) {
    sgn *= p.sgn;
    dat = convolve(dat, p.dat);
    return *this;
  }
  // bint &operator/=(const bint &p) { return *this; }
  bint operator-() const {
    bint p = *this;
    p.sgn *= -1;
    return p;
  }
  bint operator+(const bint &p) const { return bint(*this) += p; }
  bint operator-(const bint &p) const { return bint(*this) -= p; }
  bint operator*(const bint &p) const { return bint(*this) *= p; }
  // bint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const bint &p) const {
    return (sgn == p.sgn && dat == p.dat);
  }
  bool operator!=(const bint &p) const {
    return (sgn != p.sgn || dat != p.dat);
  }

  vc<int> convolve(const vc<int> &a, const vc<int> &b) {
    int n = len(a), m = len(b);
    if (!n || !m) return {};
    if (min(n, m) <= 500) {
      vc<int> c(n + m - 1);
      u128 x = 0;
      FOR(k, n + m - 1) {
        int s = max<int>(0, k + 1 - m), t = min<int>(k, n - 1);
        FOR(i, s, t + 1) { x += u64(a[i]) * b[k - i]; }
        c[k] = x % MOD, x = x / MOD;
      }
      while (x > 0) { c.eb(x % MOD), x = x / MOD; }
      return c;
    }
    static constexpr int p0 = 167772161;
    static constexpr int p1 = 469762049;
    static constexpr int p2 = 754974721;
    using mint0 = modint<p0>;
    using mint1 = modint<p1>;
    using mint2 = modint<p2>;
    vc<mint0> a0(all(a)), b0(all(b));
    vc<mint1> a1(all(a)), b1(all(b));
    vc<mint2> a2(all(a)), b2(all(b));
    auto c0 = convolution_ntt<mint0>(a0, b0);
    auto c1 = convolution_ntt<mint1>(a1, b1);
    auto c2 = convolution_ntt<mint2>(a2, b2);
    vc<int> c(len(c0));
    u128 x = 0;
    FOR(i, n + m - 1) {
      x += CRT3<u128, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val);
      c[i] = x % MOD, x = x / MOD;
    }
    while (x) { c.eb(x % MOD), x = x / MOD; }
    return c;
  }

  string to_string() {
    if (dat.empty()) return "0";
    string s;
    for (int x: dat) {
      FOR(LOG) {
        s += '0' + (x & 1);
        x /= 2;
      }
    }
    while (s.back() == '0') s.pop_back();
    if (sgn == -1) s += '-';
    reverse(all(s));
    return s;
  }

  string to_decimal_string() {
    assert(0);
    return "";
  }

  // https://codeforces.com/contest/477/problem/D
  pair<bint, int> divmod(int p) {
    assert(dat.empty() || sgn == 1);
    vc<int> after;
    ll rm = 0;
    FOR_R(i, len(dat)) {
      rm = rm * MOD + dat[i];
      after.eb(rm / p);
      rm = rm % p;
    }
    reverse(all(after));
    while (len(after) && after.back() == 0) POP(after);
    bint q;
    q.sgn = 1;
    q.dat = after;
    return {q, rm};
  }

  vc<int> base_p_representation(int p) {
    vc<u32> A(all(dat));
    vc<int> res;
    while (1) {
      while (len(A) && A.back() == u32(0)) POP(A);
      if (A.empty()) break;
      u64 rm = 0;
      FOR_R(i, len(A)) {
        rm = rm * MOD + A[i];
        A[i] = rm / p;
        rm %= p;
      }
      res.eb(rm);
    }
    reverse(all(res));
    return res;
  }

  void add_power_of_2(int k) {
    int q = k / LOG, r = k % LOG;
    if (sgn == 0) sgn = 1;
    if (q >= len(dat)) dat.resize(q + 1);
    if (sgn == 1) {
      dat[q] += 1 << r;
      while (dat[q] >= MOD) {
        dat[q] -= MOD;
        if (q + 1 >= len(dat)) dat.resize(q + 2);
        dat[q + 1] += 1;
        q += 1;
      }
    } else {
      dat[q] += 1 << r;
      while (dat[q] >= MOD) {
        dat[q] -= MOD;
        if (q + 1 >= len(dat)) dat.resize(q + 2);
        dat[q + 1] += 1;
        q += 1;
      }
    }
  }

  void substract_power_of_2(int k) {}
};

#ifdef FASTIO
void wt(BigInteger_Binary x) { fastio::wt(x.to_string()); }
#endif

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