Codeforces 1106F（数论）

Posted 2021-02-16 alphawa

要点

• 998244353的原根g = 3，意味着对于任意[1 <= x,y<p][x eq y][g^x\%p eq g^y\%p]因此可以有构造序列(q(a)与a一一对应，g^{q(a)}\%p=a)。那么对应到这道题上，因为(f_i)是%p的，所以构造(h_i)序列，使得[g^{h_i}\%p=f_i=prod_{j=1}^{k}(f_{i-j})^{b_j}\%p=g^{sum_{j=1}^k{h_{i-j} imes b_j}}\%p][ecause 原根的唯一对应性质且g^{p-1}\%p=1][ herefore h_iequiv sum_{j=1}^kh_{i-j} imes b_j(mod p-1)]
• 以上就是本题全部关键了，接下来就是数论复习内容了。
• 首先看到这个熟悉的式子想到我们可以(\%(p-1))意义下矩阵快速幂求解，往常是给前面的项求第n项，这次是有(h_n)求(h_k)。
• 其中(h_n)的求法是BSGS算法
• 矩阵快速幂以后，因为题面说初始除了(f_k)以外都是1，所以(h_{1…k-1})都是0，故而有[h_nequiv Matrix[0][0] imes h_k(mod p-1)]
• 这就变成了(ah_xequiv c(\%b))，变形为(ax+by=c)即可用扩展欧几里得求解，若有解，用快速幂求得(f_k)，否则输出-1.

#include <cstdio>
#include <cstring>
#include <cmath>
#include <iostream>
#include <algorithm>
using namespace std;

typedef long long ll;
const int p = 998244353, g = 3;
int K, b[101], n, fn, hn, hk;

struct Matrix {
    int n;
    int v[101][101];

    Matrix(int n) { memset(v, 0, sizeof v); this->n = n; }

    friend Matrix operator * (Matrix A, Matrix B) {
        int n = A.n;
        Matrix ret(n);
        for (int i = 0; i < n; i++)
            for (int j = 0; j < n; j++)
                for (int k = 0; k < n; k++) {
                    ret.v[i][j] = ((ll)ret.v[i][j] + (ll)A.v[i][k] * B.v[k][j] % (p - 1)) % (p - 1);
                }
        return ret;
    }

    friend Matrix operator ^ (Matrix A, int k) {
        int n = A.n;
        Matrix ret(n);
        for (int i = 0; i < n; i++)
            ret.v[i][i] = 1;
        for (; k; k >>= 1) {
            if (k & 1)  ret = ret * A;
            A = A * A;
        }
        return ret;
    }
};

namespace BSGS {
    const int maxm = 1e5 + 1000;
    int hash_table[maxm], val[maxm];

    int ksm(int a, int b, int mod) {
        int res = 1;
        for (; b; b >>= 1) {
            if (b & 1)  res = (ll)res * a % mod;
            a = (ll)a * a % mod;
        }
        return res;
    }

    int find(int n) {
        int id = n % maxm;
        while (hash_table[id] >= 0 && hash_table[id] != n)
            id = (id + 1) % maxm;
        return id;
    }

    int bsgs(int a, int b, int p) {
        a %= p, b %= p;
        if (!a) return b ? -1 : 1;
        memset(hash_table, -1, sizeof hash_table);

        int m = sqrt(p) + 1;
        int now = b;
        hash_table[now % maxm] = now;
        val[now % maxm] = 0;
        for (int i = 1; i <= m; i++) {
            now = (ll)now * a % p;
            int pos = find(now);
            hash_table[pos] = now;
            val[pos] = i;
        }

        int t = ksm(a, m, p);
        now = 1;
        for (int i = 1; i <= m; i++) {
            now = (ll)now * t % p;
            int pos = find(now);
            if (hash_table[pos] >= 0) {
                return i * m - val[pos];
            }
        }

        return -1;
    }
}

namespace EXGCD {
    int gcd(int a, int b) { return b ? gcd(b, a % b) : a; }

    ll exgcd(ll a, ll b, ll &x, ll &y) {
        if (!b) {
            x = 1, y = 0;
            return a;
        }
        ll q = exgcd(b, a % b, y, x);
        y -= a / b * x;
        return q;
    }

    int solve(int a, int b, int c) {//ax = c (% b)求x的解
        if (!c) return 0;
        int q = gcd(a, b);
        if (c % q)  return -1;

        a /= q, b /= q, c /= q;
        ll ans, __;
        exgcd((ll)a, (ll)b, ans, __);
        ans = (ans * c % b + b) % b;
        return ans;
    }
}

int main() {
    scanf("%d", &K);
    for (int i = 0; i < K; i++)
        scanf("%d", &b[i]), b[i] %= p - 1;
    scanf("%d%d", &n, &fn);

    hn = BSGS::bsgs(g, fn, p);
    Matrix A(K);
    for (int i = 0; i < K; i++)
        A.v[0][i] = b[i];
    for (int j = 1; j < K; j++)
        A.v[j][j - 1] = 1;
    A = A ^ (n - K);
    hk = EXGCD::solve(A.v[0][0], p - 1, hn);

    if (hk >= 0) {
        printf("%d
", BSGS::ksm(g, hk, p));
    } else {
        printf("-1
");
    }
    return 0;
}