Number Theoretic Transform (NTT)

:::note INCLUDE

• ModularArithmetic::ModInt ➝ /cp/math/modint

:::

namespace Math {
    template<typename Mint> class NTT {
        static constexpr auto MOD  = Mint::MOD;
        static constexpr auto ROOT = Mint::primitiveRoot();

    public:
        /** @brief: Assumes a.size() is a power of two.
         */
        static void ntt(std::vector<Mint>& a, bool inverse) {
            if (a.empty()) return;
            auto n = static_cast<std::int32_t>(a.size());
            assert((n ^ (n & -n)) == 0);

            std::int32_t i = 0;
            for (std::int32_t j = 1; j < n - 1; ++j) {
                for (std::int32_t k = n >> 1; k > (i ^= k); k >>= 1);
                if (j < i) swap(a[i], a[j]);
            }

            auto h = ROOT.pow((MOD - 1) / n);
            if (inverse) h = h.inv();

            for (std::int32_t m = 1; m < n; m <<= 1) {
                const auto m2 = m << 1;
                const auto base = h.pow(n / m2);
                Mint w = 1;
                for (std::int32_t x = 0; x < m; ++x) {
                    for (std::int32_t s = x; s < n; s += m2) {
                        auto u = a[s], d = a[s + m] * w;
                        a[s] = u + d; a[s + m] = u - d;
                    }
                    w *= base;
                }
            }

            if (inverse) {
                const auto n_inv = Mint(n).inv();
                for (auto &x : a) x *= n_inv;
            }
        }

        static void conv1D(std::vector<Mint>& a, std::vector<Mint> b) {
            const auto conv_size = static_cast<std::int32_t>(a.size() + b.size() - 1);
            auto n = conv_size;
            while (n ^ (n & -n)) n ^= n & -n;
            if (n < conv_size) n <<= 1;
            a.resize(n); ntt(a, false);
            b.resize(n); ntt(b, false);
            for (std::int32_t i = 0; i < n; ++i) a[i] *= b[i];
            ntt(a, true); a.resize(conv_size);
        }
    };
}

int main() {
    vector<Mint> a = {1, 4, 2, 3, 4};
    vector<Mint> b = {3, 2, 0, 2, 1, 2};
    Math::NTT<Mint>::conv1D(a, b);
    for (auto &x : a) cout << x << ' ';
    cout << '\n';
}