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Multivarate Convolution Cyclic
(Convolution/multivariatecyclic.hpp)

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• Last update: 2025-05-25 16:11:40+09:00
• Include: #include "Convolution/multivariatecyclic.hpp"

Depends on

• Multipoint Evaluation on Geometric Sequence (FPS/multievalgeom.hpp)
• Fast Division (Math/fastdiv.hpp)
• Miller-Rabin (Math/miller.hpp)
• Pollard-Rho (Math/pollard.hpp)
• Primitive Function (Math/primitive.hpp)
• Random (Utility/random.hpp)

Verified with

• Verify/LC_multivariate_convolution_cyclic.test.cpp

Code

#pragma once
#include "Math/primitive.hpp"
#include "FPS/multievalgeom.hpp"

template<typename T>vector<T> MultivariateCyclic
    (vector<T> f,vector<T> g,vector<int>& a){
    int MO=T::get_mod();
    int k=a.size(),n=1;
    for(auto& x:a)n*=x;
    T pr=getPrimitiveRoot(MO),ipr=T(pr).inv();

    int offset=1;
    rep(x,0,k){
        assert((MO-1)%a[x]==0);
        T w=pr.pow((MO-1)/a[x]);
        rep(i,0,n)if(i%(offset*a[x])<offset){
            vector<T> na(a[x]),nb(a[x]);
            rep(j,0,a[x]){
                na[j]=f[i+offset*j];
                nb[j]=g[i+offset*j];
            }
            na=MultievalGeomSeq(na,T(1),w,a[x]);
            nb=MultievalGeomSeq(nb,T(1),w,a[x]);
            rep(j,0,a[x]){
                f[i+offset*j]=na[j];
                g[i+offset*j]=nb[j];
            }
        }
        offset*=a[x];
    }

    rep(i,0,n)f[i]*=g[i];
    
    offset=1;
    rep(x,0,k){
        T iw=ipr.pow((MO-1)/a[x]);
        rep(i,0,n)if(i%(offset*a[x])<offset){
            vector<T> na(a[x]);
            rep(j,0,a[x])na[j]=f[i+offset*j];
            na=MultievalGeomSeq(na,T(1),iw,a[x]);
            rep(j,0,a[x])f[i+offset*j]=na[j];
        }
        offset*=a[x];
    }
    T ninv=T(n).inv();
    rep(i,0,n)f[i]*=ninv;
    return f;
}

/**
 * @brief Multivarate Convolution Cyclic
*/

#line 2 "Math/fastdiv.hpp"

struct FastDiv {
    using u64 = uint64_t;
    using u128 = __uint128_t;
    constexpr FastDiv() : m(), s(), x() {}
    constexpr FastDiv(int _m)
        : m(_m), s(__lg(m - 1)), x(((u128(1) << (s + 64)) + m - 1) / m) {}
    constexpr int get() {
        return m;
    }
    constexpr friend u64 operator/(u64 n, const FastDiv &d) {
        return (u128(n) * d.x >> d.s) >> 64;
    }
    constexpr friend int operator%(u64 n, const FastDiv &d) {
        return n - n / d * d.m;
    }
    constexpr pair<u64, int> divmod(u64 n) const {
        u64 q = n / (*this);
        return {q, n - q * m};
    }
    int m, s;
    u64 x;
};

struct FastDiv64 {
    using u64 = uint64_t;
    using u128 = __uint128_t;
    u128 mod, mh, ml;
    explicit FastDiv64(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);
    }
};

/**
 * @brief Fast Division
 */
#line 2 "Math/miller.hpp"

struct m64 {
    using i64 = int64_t;
    using u64 = uint64_t;
    using u128 = __uint128_t;

    static u64 mod;
    static u64 r;
    static u64 n2;

    static u64 get_r() {
        u64 ret = mod;
        rep(_,0,5) ret *= 2 - mod * ret;
        return ret;
    }

    static void set_mod(u64 m) {
        assert(m < (1LL << 62));
        assert((m & 1) == 1);
        mod = m;
        n2 = -u128(m) % m;
        r = get_r();
        assert(r * mod == 1);
    }
    static u64 get_mod() { return mod; }

    u64 a;
    m64() : a(0) {}
    m64(const int64_t &b) : a(reduce((u128(b) + mod) * n2)){};

    static u64 reduce(const u128 &b) {
        return (b + u128(u64(b) * u64(-r)) * mod) >> 64;
    }
    u64 get() const {
        u64 ret = reduce(a);
        return ret >= mod ? ret - mod : ret;
    }
    m64 &operator*=(const m64 &b) {
        a = reduce(u128(a) * b.a);
        return *this;
    }
    m64 operator*(const m64 &b) const { return m64(*this) *= b; }
    bool operator==(const m64 &b) const {
        return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
    }
    bool operator!=(const m64 &b) const {
        return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
    }
    m64 pow(u128 n) const {
        m64 ret(1), mul(*this);
        while (n > 0) {
        if (n & 1) ret *= mul;
        mul *= mul;
        n >>= 1;
        }
        return ret;
    }
};
typename m64::u64 m64::mod, m64::r, m64::n2;

bool Miller(ll n){
    if(n<2 or (n&1)==0)return (n==2);
    m64::set_mod(n);
    ll d=n-1; while((d&1)==0)d>>=1;
    vector<ll> seeds;
    if(n<(1<<30))seeds={2, 7, 61};
    else seeds={2, 325, 9375, 28178, 450775, 9780504};
    for(auto& x:seeds){
        if(n<=x)break;
        ll t=d;
        m64 y=m64(x).pow(t);
        while(t!=n-1 and y!=1 and y!=n-1){
            y*=y;
            t<<=1;
        }
        if(y!=n-1 and (t&1)==0)return 0;
    } return 1;
}

/**
 * @brief Miller-Rabin
 */
#line 2 "Utility/random.hpp"

namespace Random {
mt19937_64 randgen(chrono::steady_clock::now().time_since_epoch().count());
using u64 = unsigned long long;
u64 get() {
    return randgen();
}
template <typename T> T get(T L) { // [0,L]

    return get() % (L + 1);
}
template <typename T> T get(T L, T R) { // [L,R]

    return get(R - L) + L;
}
double uniform() {
    return double(get(1000000000)) / 1000000000;
}
string str(int n) {
    string ret;
    rep(i, 0, n) ret += get('a', 'z');
    return ret;
}
template <typename Iter> void shuffle(Iter first, Iter last) {
    if (first == last)
        return;
    int len = 1;
    for (auto it = first + 1; it != last; it++) {
        len++;
        int j = get(0, len - 1);
        if (j != len - 1)
            iter_swap(it, first + j);
    }
}
template <typename T> vector<T> select(int n, T L, T R) { // [L,R]

    if (n * 2 >= R - L + 1) {
        vector<T> ret(R - L + 1);
        iota(ALL(ret), L);
        shuffle(ALL(ret));
        ret.resize(n);
        return ret;
    } else {
        unordered_set<T> used;
        vector<T> ret;
        while (SZ(used) < n) {
            T x = get(L, R);
            if (!used.count(x)) {
                used.insert(x);
                ret.push_back(x);
            }
        }
        return ret;
    }
}

void relabel(int n, vector<pair<int, int>> &es) {
    shuffle(ALL(es));
    vector<int> ord(n);
    iota(ALL(ord), 0);
    shuffle(ALL(ord));
    for (auto &[u, v] : es)
        u = ord[u], v = ord[v];
}
template <bool directed, bool multi, bool self>
vector<pair<int, int>> genGraph(int n, int m) {
    vector<pair<int, int>> cand, es;
    rep(u, 0, n) rep(v, 0, n) {
        if (!self and u == v)
            continue;
        if (!directed and u > v)
            continue;
        cand.push_back({u, v});
    }
    if (m == -1)
        m = get(SZ(cand));
    // chmin(m, SZ(cand));

    vector<int> ord;
    if (multi)
        rep(_, 0, m) ord.push_back(get(SZ(cand) - 1));
    else {
        ord = select(m, 0, SZ(cand) - 1);
    }
    for (auto &i : ord)
        es.push_back(cand[i]);
    relabel(n, es);
    return es;
}
vector<pair<int, int>> genTree(int n) {
    vector<pair<int, int>> es;
    rep(i, 1, n) es.push_back({get(i - 1), i});
    relabel(n, es);
    return es;
}
}; // namespace Random


/**
 * @brief Random
 */
#line 4 "Math/pollard.hpp"

vector<ll> Pollard(ll n) {
    if (n <= 1)
        return {};
    if (Miller(n))
        return {n};
    if ((n & 1) == 0) {
        vector<ll> v = Pollard(n >> 1);
        v.push_back(2);
        return v;
    }
    for (ll x = 2, y = 2, d;;) {
        ll c = Random::get(2LL, n - 1);
        do {
            x = (__int128_t(x) * x + c) % n;
            y = (__int128_t(y) * y + c) % n;
            y = (__int128_t(y) * y + c) % n;
            d = __gcd(x - y + n, n);
        } while (d == 1);
        if (d < n) {
            vector<ll> lb = Pollard(d), rb = Pollard(n / d);
            lb.insert(lb.end(), ALL(rb));
            return lb;
        }
    }
}

vector<pair<ll, int>> Pollard2(ll n) {
    auto ps = Pollard(n);
    sort(ALL(ps));
    using P = pair<ll, int>;
    vector<P> pes;
    for (auto &p : ps) {
        if (pes.empty() or pes.back().first != p) {
            pes.push_back({p, 1});
        } else {
            pes.back().second++;
        }
    }
    return pes;
}

vector<ll> EnumDivisors(ll n) {
    auto pes = Pollard2(n);
    vector<ll> ret;
    auto rec = [&](auto &rec, int id, ll d) -> void {
        if (id == SZ(pes)) {
            ret.push_back(d);
            return;
        }
        rec(rec, id + 1, d);
        rep(e, 0, pes[id].second) {
            d *= pes[id].first;
            rec(rec, id + 1, d);
        }
    };
    rec(rec, 0, 1);
    sort(ALL(ret));
    return ret;
}

/**
 * @brief Pollard-Rho
 */
#line 4 "Math/primitive.hpp"

ll mpow(ll a, ll t, ll m) {
    ll res = 1;
    FastDiv64 im(m);
    while (t) {
        if (t & 1)
            res = im.modulo(__int128_t(res) * a);
        a = im.modulo(__int128_t(a) * a);
        t >>= 1;
    }
    return res;
}
ll minv(ll a, ll m) {
    ll b = m, u = 1, v = 0;
    while (b) {
        ll t = a / b;
        a -= t * b;
        swap(a, b);
        u -= t * v;
        swap(u, v);
    }
    u = (u % m + m) % m;
    return u;
}
ll getPrimitiveRoot(ll p) {
    vector<ll> ps = Pollard(p - 1);
    sort(ALL(ps));
    rep(x, 1, inf) {
        for (auto &q : ps) {
            if (mpow(x, (p - 1) / q, p) == 1)
                goto fail;
        }
        return x;
    fail:;
    }
    assert(0);
}
template <typename T> T extgcd(T a, T b, T &p, T &q) {
    if (b == 0) {
        p = 1;
        q = 0;
        return a;
    }
    T d = extgcd(b, a % b, q, p);
    q -= a / b * p;
    return d;
}
template <typename T> pair<T, T> crt(vector<T> vs, vector<T> ms) {
    T V = vs[0], M = ms[0];
    rep(i, 1, vs.size()) {
        T p, q, v = vs[i], m = ms[i];
        if (M < m)
            swap(M, m), swap(V, v);
        T d = extgcd(M, m, p, q);
        if ((v - V) % d != 0)
            return {0, -1};
        T md = m / d, tmp = (v - V) / d % md * p % md;
        V += M * tmp;
        M *= md;
    }
    V = (V % M + M) % M;
    return {V, M};
}
ll garner(vector<ll> vs, vector<ll> p, int mod) {
    int sz = SZ(vs);
    vector<ll> kp(sz + 1), rmul(sz + 1, 1);
    p.push_back(mod);
    rep(i, 0, sz) {
        ll x = (vs[i] - kp[i]) * minv(rmul[i], p[i]) % p[i];
        if (x < 0)
            x += p[i];
        rep(j, i + 1, sz + 1) {
            kp[j] += rmul[j] * x;
            kp[j] %= p[j];
            rmul[j] *= p[i];
            rmul[j] %= p[j];
        }
    }
    return kp.back();
}

ll ModLog(ll a, ll b, ll p) {
    ll g = 1;
    for (ll t = p; t; t >>= 1)
        g = g * a % p;
    g = __gcd(g, p);
    ll t = 1, c = 0;
    for (; t % g; c++) {
        if (t == b)
            return c;
        t = t * a % p;
    }
    if (b % g)
        return -1;
    t /= g, b /= g;
    ll n = p / g, h = 0, gs = 1;
    for (; h * h < n; h++)
        gs = gs * a % n;
    unordered_map<ll, ll> bs;
    for (ll s = 0, e = b; s < h; bs[e] = ++s)
        e = e * a % n;
    for (ll s = 0, e = t; s < n;) {
        e = e * gs % n, s += h;
        if (bs.count(e)) {
            return c + s - bs[e];
        }
    }
    return -1;
}

ll TonelliShanks(ll a, ll p) {
    a %= p;
    if (a == 0)
        return 0;
    if (p == 2)
        return a;
    if (mpow(a, (p - 1) >> 1, p) != 1)
        return -1;
    ll b = 1;
    while (mpow(b, (p - 1) >> 1, p) == 1)
        b = Random::get(1LL, p - 1);

    ll q = p - 1, k = 0;
    while (q % 2 == 0) {
        q >>= 1;
        k++;
    }
    ll x = mpow(a, (q + 1) >> 1, p);
    b = mpow(b, q, p);
    k -= 2;
    while (mpow(x, 2, p) != a) {
        ll err = minv(a, p) * mpow(x, 2, p) % p;
        if (mpow(err, 1 << k, p) != 1)
            x = x * b % p;
        b = mpow(b, 2, p);
        k--;
    }
    return x;
}

ll mod_root(ll k, ll a, ll m) {
    if (a == 0)
        return k ? 0 : -1;
    if (m == 2)
        return a & 1;
    k %= m - 1;
    ll g = gcd(k, m - 1);
    if (mpow(a, (m - 1) / g, m) != 1)
        return -1;
    a = mpow(a, minv(k / g, (m - 1) / g), m);
    FastDiv im(m);

    auto _subroot = [&](ll p, int e, ll a) -> ll { // x^(p^e)==a(mod m)

        ll q = m - 1;
        int s = 0;
        while (q % p == 0) {
            q /= p;
            s++;
        }
        int d = s - e;
        ll pe = mpow(p, e, m),
           res = mpow(a, ((pe - 1) * minv(q, pe) % pe * q + 1) / pe, m), c = 1;
        while (mpow(c, (m - 1) / p, m) == 1)
            c++;
        c = mpow(c, q, m);
        map<ll, ll> mp;
        ll v = 1, block = sqrt(d * p) + 1,
           bs = mpow(c, mpow(p, s - 1, m - 1) * block % (m - 1), m);
        rep(i, 0, block + 1) mp[v] = i, v = v * bs % im;
        ll gs = minv(mpow(c, mpow(p, s - 1, m - 1), m), m);
        rep(i, 0, d) {
            ll err = a * minv(mpow(res, pe, m), m) % im;
            ll pos = mpow(err, mpow(p, d - 1 - i, m - 1), m);
            rep(j, 0, block + 1) {
                if (mp.count(pos)) {
                    res = res *
                          mpow(c,
                               (block * mp[pos] + j) * mpow(p, i, m - 1) %
                                   (m - 1),
                               m) %
                          im;
                    break;
                }
                pos = pos * gs % im;
            }
        }
        return res;
    };

    for (ll d = 2; d * d <= g; d++)
        if (g % d == 0) {
            int sz = 0;
            while (g % d == 0) {
                g /= d;
                sz++;
            }
            a = _subroot(d, sz, a);
        }
    if (g > 1)
        a = _subroot(g, 1, a);
    return a;
}

ull floor_root(ull a, ull k) {
    if (a <= 1 or k == 1)
        return a;
    if (k >= 64)
        return 1;
    if (k == 2)
        return sqrtl(a);
    constexpr ull LIM = -1;
    if (a == LIM)
        a--;
    auto mul = [&](ull &x, const ull &y) {
        if (x <= LIM / y)
            x *= y;
        else
            x = LIM;
    };
    auto pw = [&](ull x, ull t) -> ull {
        ull y = 1;
        while (t) {
            if (t & 1)
                mul(y, x);
            mul(x, x);
            t >>= 1;
        }
        return y;
    };
    ull ret = (k == 3 ? cbrt(a) - 1 : pow(a, nextafter(1 / double(k), 0)));
    while (pw(ret + 1, k) <= a)
        ret++;
    return ret;
}

/**
 * @brief Primitive Function
 */
#line 2 "FPS/multievalgeom.hpp"

template<typename T>vector<T> MultievalGeomSeq(vector<T>& f,T a,T w,int m){
    int n=f.size();
    vector<T> ret(m);
    if(w==0){
        T base=1;
        rep(i,0,n)ret[0]+=base*f[i],base*=a;
        rep(i,1,m)ret[i]=f[0];
        return ret;
    }
    vector<T> tri(n+m-1),itri(n+m-1);
    tri[0]=itri[0]=1;
    T iw=w.inv(),pww=1,pwiw=1;
    for(int i=1;i<n+m-1;i++,pww*=w,pwiw*=iw){
        tri[i]=tri[i-1]*pww;
        itri[i]=itri[i-1]*pwiw;
    }

    Poly<T> y(n),v(n+m-1);
    T pwa=1;
    for(int i=0;i<n;i++,pwa*=a){
        y[i]=f[i]*itri[i]*pwa;
    }
    rep(i,0,n+m-1)v[i]=tri[i];
    reverse(ALL(y));
    y*=v;
    rep(i,0,m)ret[i]=y[n-1+i]*itri[i];
    return ret;
}

/**
 * @brief Multipoint Evaluation on Geometric Sequence
*/
#line 4 "Convolution/multivariatecyclic.hpp"

template<typename T>vector<T> MultivariateCyclic
    (vector<T> f,vector<T> g,vector<int>& a){
    int MO=T::get_mod();
    int k=a.size(),n=1;
    for(auto& x:a)n*=x;
    T pr=getPrimitiveRoot(MO),ipr=T(pr).inv();

    int offset=1;
    rep(x,0,k){
        assert((MO-1)%a[x]==0);
        T w=pr.pow((MO-1)/a[x]);
        rep(i,0,n)if(i%(offset*a[x])<offset){
            vector<T> na(a[x]),nb(a[x]);
            rep(j,0,a[x]){
                na[j]=f[i+offset*j];
                nb[j]=g[i+offset*j];
            }
            na=MultievalGeomSeq(na,T(1),w,a[x]);
            nb=MultievalGeomSeq(nb,T(1),w,a[x]);
            rep(j,0,a[x]){
                f[i+offset*j]=na[j];
                g[i+offset*j]=nb[j];
            }
        }
        offset*=a[x];
    }

    rep(i,0,n)f[i]*=g[i];
    
    offset=1;
    rep(x,0,k){
        T iw=ipr.pow((MO-1)/a[x]);
        rep(i,0,n)if(i%(offset*a[x])<offset){
            vector<T> na(a[x]);
            rep(j,0,a[x])na[j]=f[i+offset*j];
            na=MultievalGeomSeq(na,T(1),iw,a[x]);
            rep(j,0,a[x])f[i+offset*j]=na[j];
        }
        offset*=a[x];
    }
    T ninv=T(n).inv();
    rep(i,0,n)f[i]*=ninv;
    return f;
}

/**
 * @brief Multivarate Convolution Cyclic
*/

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