SDOI2013 随机数生成器

Posted 2021-12-02 fengxunling

题目链接：戳我

就是大力推式子，然后上BSGS就行了。

\[x_n\equiv a^{n-1}x_1+b(a^{n-2}+a^{n-3}+...+a)\pmod p\]

\[t\equiv a^{n-1}x_1+b\sum_{i=0}^{n-2}a^i\pmod p\]

\[t\equiv a^{n-1}x_1+b\times \frac{1\times(1-a^{n-1})}{1-a}\pmod p\]

\[t\equiv a^{n-1}x_1+\frac{b}{1-a}-\frac{b}{1-a}\times a^{n-1}\pmod p\]

\[t\equiv a^{n-1}(x_1-\frac{b}{1-a})+\frac{-b}{1-a}\pmod p\]

\[a^{n-1}\equiv \frac{-b+(1-a)\times t}{(1-a)\times x_1-b}\pmod p\]

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#include<map>
#define int long long
#define MAXN 1010
using namespace std;
int T;
map<int,int>ex;
inline int fpow(int x,int y,int mod)
{
    int cur_ans=1; 
    while(y)
    {
        if(y&1) cur_ans=1ll*cur_ans*x%mod;
        x=1ll*x*x%mod;
        y>>=1;
    }
    return cur_ans;
}
inline int inv(int x,int mod){return fpow(x,mod-2,mod);}
inline int bsgs(int x,int y,int mod)
{
    x%=mod,y%=mod;
    ex.clear();
    int m=ceil(sqrt(mod));
    for(int i=0,t=1;i<m;i++,t=1ll*t*x%mod)
        if(!ex.count(t)) ex[t]=i;
    for(int i=0,tt=inv(fpow(x,m,mod),mod),cur=y;i<m;i++,cur=1ll*cur*tt%mod)
        if(ex.count(cur))
            return i*m+ex[cur];
    return -2;
}
signed main()
{
    #ifndef ONLINE_JUDGE
    freopen("ce.in","r",stdin);
    freopen("ce.out","w",stdout);
    #endif
    scanf("%lld",&T);
    while(T--)
    {
        int a,p,x1,t,b;
        // cout<<endl;
        scanf("%lld%lld%lld%lld%lld",&p,&a,&b,&x1,&t);
        if(x1==t){printf("1\n");continue;}
        else if(a==0)
        {
            if(b==t)printf("2\n");
            else printf("-1\n");
            continue;
        }
        else if(a==1)
        {
            if(b==0){printf("-1\n");continue;}
            t=(t-x1+p)%p;
            t=1ll*t*fpow(b,p-2,p)%p;
            printf("%lld\n",t+1);
            continue;
        }
        else
        {
            // cout<<((t-b*inv(1-a,p))%p+p)%p<<endl;
            // cout<<((t-b*inv(1-a,p))%p+p)%p*inv(((x1-b*inv(1-a,p))%p+p)%p,p)<<endl;
            int cur=bsgs(a,((t-b*inv(1-a,p))%p+p)%p*inv(((x1-b*inv(1-a,p))%p+p)%p,p),p)%p;
            printf("%lld\n",cur+1);
        }
    }
    return 0;
}