bzoj1420/1319 Discrete Root

Posted 2020-09-22 逢山开路 遇水架桥

传送门：http://www.lydsy.com/JudgeOnline/problem.php?id=1420

http://www.lydsy.com/JudgeOnline/problem.php?id=1319

【题解】

求x^A=B(mod P)，其中P是质数。

考虑对两边取log，设g为P的原根。

Alog(x) = log(B) (mod P-1)

log(x)表示以g为底的log

那么log(B) = y，其中g^y = B (mod P)，用BSGS求出即可。

我们要求的是x，不妨先求log(x)，设ans=log(x)

那么A*ans + (P-1)*k = y。这是一个exgcd的形式，所以我们可以求出ans的所有解（由于相当于指数，所以必须小于P-1）

然后快速幂即可。

# include <map>
# include <math.h>
# include <stdio.h>
# include <assert.h>
# include <string.h>
# include <iostream>
# include <algorithm>
// # include <bits/stdc++.h>

using namespace std;

typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
const int M = 5e5 + 10;
const int mod = 1e9+7;

# define RG register
# define ST static

ll A, B, P, B0;
ll g, ans[M]; int ansn=0;

inline ll pwr(ll a, ll b, ll P) {
    ll ret = 1; a %= P;
    while(b) {
        if(b&1) ret = ret * a % P;
        a = a * a % P;
        b >>= 1;
    }
    return ret;
}

ll y[M];
inline ll G(ll x) {
    ll t = x; int nn = 0;
    for (int i=2; i*i<=x; ++i) {
        if(x%i) continue;
        y[++nn] = i;
        while(x%i == 0) x/=i;
    }
    if(x != 1) y[++nn] = x;
    for (ll g=2; ; ++g) {
        bool flag = 1;
        for (int i=1; i<=nn; ++i)
            if(pwr(g, t/y[i], P) == 1) {
                flag = 0;
                break;
            }
        if(flag) return g;
    }
}

map<ll, int> mp;
inline ll BSGS(ll A, ll B, ll P) {
    mp.clear();
    int m = ceil(sqrt(1.0 * P));
    ll t = B, g;
    for (int i=0; i<m; ++i) {
        if(!mp[t]) mp[t] = i;
        t = t * A % P;
    }
    g = pwr(A, m, P); t = g;
    for (int i=1, ps; i<=m+1; ++i) {
        if(mp.count(t)) return (ll)i*m - mp[t];
        t = t * g % P;
    }
    return -1;
}

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

int main() {
    cin >> P >> A >> B;
    g = G(P-1);
//    cout << g << endl;
    // x^A = B (mod P)
    // A log_g(x) = log_g(B) (mod P-1)
    B0 = BSGS(g, B, P);
    assert(B0 != -1);
//    cout << B0 << endl;
    ll tx, ty, GCD;
    GCD = exgcd(A, P-1, tx, ty);
    if(B0 % GCD) {
        puts("0");
        return 0;
    }
    
    ty = (P-1)/GCD;
    tx = (tx % ty + ty) % ty;
    tx = (tx * B0/GCD) % ty;
    while(tx < P-1) {
        ans[++ansn] = pwr(g, tx, P);
        tx += ty;
    }
    sort(ans+1, ans+ansn+1);
    
    cout << ansn << endl;
    for (int i=1; i<=ansn; ++i) 
        printf("%lld\n", ans[i]);
    return 0;
}

View Code