洛咕 P3306 [SDOI2013]随机数生成器

洛咕 P3306 [SDOI2013]随机数生成器

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大力推式子？？？

\(X_{i}=\underbrace{a(a(\cdots(a(a}_{i-1个a}X_1+b)))\cdots)\)

\(=b+ba+ba^2+\cdots+ba^{i-3}+ba^{i-2}+X_1a^{i-1}\equiv t(\text{mod }p)\)

\(b\frac{a^{i-1}-1}{a-1}+a^{i-1}x_1\equiv t(\text{mod }p)\)

拆分一波，提出\(a^{i-1}\)

\((X_1+\frac{b}{a-1})a^{i-1}\equiv \frac{b}{a-1}+t(\text{mod }p)\)

\(a^{i-1}\equiv \frac{\frac{b}{a-1}+t}{\frac{b}{a-1}+X_1} (\text{mod }p)\)

然后bsgs即可。

这题还要加一堆特判。。。懒得写了。。。丧心病狂

#include<bits/stdc++.h>
#define il inline
#define vd void
typedef long long ll;
il int gi(){
    int x=0,f=1;
    char ch=getchar();
    while(!isdigit(ch)){
        if(ch=='-')f=-1;
        ch=getchar();
    }
    while(isdigit(ch))x=x*10+ch-'0',ch=getchar();
    return x*f;
}
ll T,mod,a,b,X1,t;
il ll pow(ll a,ll b){
    ll ret=1;
    while(b){
        if(b&1)ret=ret*a%mod;
        a=a*a%mod;b>>=1;
    }
    return ret;
}
std::map<ll,ll>M;
int main(){
#ifndef ONLINE_JUDGE
    freopen("3306.in","r",stdin);
    freopen("3306.out","w",stdout);
#endif
    T=gi();
    while(T--){
        mod=gi(),a=gi(),b=gi(),X1=gi(),t=gi();
        ll B=(t+b*pow(a-1,mod-2)%mod)%mod,A=(B-t+X1+mod)%mod,C=B*pow(A,mod-2)%mod;
        ll s=sqrt(mod)+1,ans=1e18;
        if(t==X1){puts("1");continue;}
        if(a==1){
            A=b,B=(t-X1+mod)%mod;
            if(b==0||A%std::__gcd(B,mod))puts("-1");
            else printf("%lld\n",((t-X1+mod)*pow(b,mod-2)%mod)%mod+1);
            continue;
        }
        if(a==0){printf("%lld\n",b==t?2ll:(-1ll));continue;}
        M.clear();
        for(ll i=0,j=C%mod;i<s;++i,j=j*a%mod)M[j]=i;
        ll P=pow(a,s);
        for(ll i=1,j=P;i<=s+1;++i,j=j*P%mod)if(M.find(j)!=M.end())ans=std::min(ans,i*s-M[j]);
        if(ans==1e18)puts("-1");
        else printf("%lld\n",ans+1);
    }
    return 0;
}