nt/four_square.hpp

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• Last update: 2023-11-02 05:38:57+09:00
• Include: #include "nt/four_square.hpp"

Depends on

• mod/barrett.hpp
• mod/mod_pow.hpp
• mod/mongomery_modint.hpp
• nt/factor.hpp
• nt/gaussian_integers.hpp
• nt/primetest.hpp
• nt/three_square.hpp
• random/base.hpp

Verified with

• test/mytest/four_square.test.cpp

Code

#include "nt/three_square.hpp"

tuple<ll, ll, ll, ll> four_square(ll N) {
  if (N == 0) return {0, 0, 0, 0};
  ll e = 0;
  while (N % 4 == 0) N /= 4, ++e;
  auto [a, b, c] = three_square(N);
  if (a != -1) return {a << e, b << e, c << e, 0};
  tie(a, b, c) = three_square(N - 1);
  return {a << e, b << e, c << e, 1LL << e};
}

#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 "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 3 "nt/gaussian_integers.hpp"

template <typename T>
struct Gaussian_Integer {
  T x, y;
  using G = Gaussian_Integer;

  Gaussian_Integer(T x = 0, T y = 0) : x(x), y(y) {}
  Gaussian_Integer(pair<T, T> p) : x(p.fi), y(p.se) {}

  T norm() const { return x * x + y * y; }
  G conjugate() const { return G(x, -y); }

  G &operator+=(const G &g) {
    x += g.x, y += g.y;
    return *this;
  }
  G &operator-=(const G &g) {
    x -= g.x, y -= g.y;
    return *this;
  }
  G &operator*=(const G &g) {
    tie(x, y) = mp(x * g.x - y * g.y, x * g.y + y * g.x);
    return *this;
  }
  G &operator/=(const G &g) {
    *this *= g.conjugate();
    T n = g.norm();
    x = floor(x + n / 2, n);
    y = floor(y + n / 2, n);
    return *this;
  }
  G &operator%=(const G &g) {
    auto q = G(*this) / g;
    q *= g;
    (*this) -= q;
    return *this;
  }
  G operator-() { return G(-x, -y); }
  G operator+(const G &g) { return G(*this) += g; }
  G operator-(const G &g) { return G(*this) -= g; }
  G operator*(const G &g) { return G(*this) *= g; }
  G operator/(const G &g) { return G(*this) /= g; }
  G operator%(const G &g) { return G(*this) %= g; }
  bool operator==(const G &g) { return (x == g.x && y == g.y); }

  static G gcd(G a, G b) {
    while (b.x != 0 || b.y != 0) {
      a %= b;
      swap(a, b);
    }
    return a;
  }

  // (g,x,y) s.t ax+by=g

  static tuple<G, G, G> extgcd(G a, G b) {
    if (b.x != 0 || b.y != 0) {
      G q = a / b;
      auto [g, x, y] = extgcd(b, a - q * b);
      return {g, y, x - q * y};
    }
    return {a, G{1, 0}, G{0, 0}};
  }
};

pair<ll, ll> solve_norm_equation_prime(ll p) {
  using G = Gaussian_Integer<i128>;
  assert(p == 2 || p % 4 == 1);
  if (p == 2) return {1, 1};
  ll x = [&]() -> ll {
    ll x = 1;
    while (1) {
      ++x;
      ll pow_x = 1;
      if (p < (1 << 30)) {
        pow_x = mod_pow(x, (p - 1) / 4, p);
        if (pow_x * pow_x % p == p - 1) return pow_x;
      } else {
        pow_x = mod_pow_64(x, (p - 1) / 4, p);
        if (i128(pow_x) * pow_x % p == p - 1) return pow_x;
      }
    }
    return -1;
  }();
  G a(p, 0), b(x, 1);
  a = G::gcd(a, b);
  assert(a.norm() == p);
  return {a.x, a.y};
}

template <typename T>
vc<Gaussian_Integer<T>> solve_norm_equation_factor(vc<pair<ll, int>> pfs) {
  using G = Gaussian_Integer<T>;
  vc<G> res;
  for (auto &&[p, e]: pfs) {
    if (p % 4 == 3 && e % 2 == 1) return {};
  }

  res.eb(G(1, 0));
  for (auto &&[p, e]: pfs) {
    if (p % 4 == 3) {
      T pp = 1;
      FOR(e / 2) pp *= p;
      for (auto &&g: res) {
        g.x *= pp;
        g.y *= pp;
      }
      continue;
    }
    G pi = solve_norm_equation_prime(p);
    vc<G> pows(e + 1);
    pows[0] = G(1, 0);
    FOR(i, e) pows[i + 1] = pows[i] * pi;
    if (p == 2) {
      for (auto &&g: res) g *= pows[e];
      continue;
    }
    vc<G> pis(e + 1);
    FOR(j, e + 1) { pis[j] = pows[j] * (pows[e - j].conjugate()); }
    vc<G> new_res;
    new_res.reserve(len(res) * (e + 1));
    for (auto &&g: res) {
      for (auto &&a: pis) { new_res.eb(g * a); }
    }
    swap(res, new_res);
  }

  for (auto &&g: res) {
    while (g.x <= 0 || g.y < 0) { g = G(-g.y, g.x); }
  }
  return res;
}

// i128 を使うと N <= 10^{18} もできる

// ノルムがとれるように、2 乗してもオーバーフローしない型を使おう

// 0 <= arg < 90 となるもののみ返す。

// 単数倍は作らないので、使うときに気を付ける。

template <typename T>
vc<Gaussian_Integer<T>> solve_norm_equation(T N) {
  using G = Gaussian_Integer<T>;
  vc<G> res;
  if (N < 0) return {};
  if (N == 0) {
    res.eb(G(0, 0));
    return res;
  }
  auto pfs = factor(N);
  return solve_norm_equation_factor<T>(pfs);
}
#line 3 "nt/three_square.hpp"

// https://math.stackexchange.com/questions/483101/rabin-and-shallit-algorithm
// ERH のもと O(log^2N) ？
tuple<ll, ll, ll> three_square(ll N) {
  if (N == 0) return {0, 0, 0};
  auto F = [&](ll n) -> tuple<ll, ll, ll> {
    if (N == 2) return {1, 1, 0};
    if (N == 3) return {1, 1, 1};
    if (N == 10) return {3, 1, 0};
    if (N == 34) return {5, 3, 0};
    if (N == 58) return {7, 3, 0};
    if (N == 85) return {9, 2, 0};
    if (N == 130) return {11, 3, 0};
    if (N == 214) return {14, 3, 3};
    if (N == 226) return {15, 1, 0};
    if (N == 370) return {19, 3, 0};
    if (N == 526) return {21, 9, 2};
    if (N == 706) return {25, 9, 0};
    if (N == 730) return {27, 1, 0};
    if (N == 1414) return {33, 18, 1};
    if (N == 1906) return {41, 15, 0};
    if (N == 2986) return {45, 31, 0};
    if (N == 9634) return {97, 15, 0};
    ll x = sqrtl(N);
    if (N == x * x) return {x, 0, 0};
    if (N % 4 != 1 && x % 2 == 0) --x;
    if (N % 4 == 1 && x % 2 == 1) --x;
    x += 2;
    while (1) {
      x -= 2;
      ll k = N - x * x;
      if (k < 0) break;
      if (k % 2 == 1 && primetest(k)) {
        auto [a, b] = solve_norm_equation_prime(k);
        a = abs(a), b = abs(b);
        return {a, b, x};
      }
      if (k % 2 == 0 && primetest(k / 2)) {
        auto [a, b] = solve_norm_equation_prime(k / 2);
        tie(a, b) = mp(a + b, a - b);
        a = abs(a), b = abs(b);
        return {a, b, x};
      }
    }
    return {-1, -1, -1};
    assert(0);
  };
  ll e = 0;
  while (N % 4 == 0) N /= 4, ++e;
  if (N % 8 == 7) return {-1, -1, -1};
  auto [a, b, c] = F(N);
  return {a << e, b << e, c << e};
}
#line 2 "nt/four_square.hpp"

tuple<ll, ll, ll, ll> four_square(ll N) {
  if (N == 0) return {0, 0, 0, 0};
  ll e = 0;
  while (N % 4 == 0) N /= 4, ++e;
  auto [a, b, c] = three_square(N);
  if (a != -1) return {a << e, b << e, c << e, 0};
  tie(a, b, c) = three_square(N - 1);
  return {a << e, b << e, c << e, 1LL << e};
}

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