SDOI2013 随机数生成器

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;
}