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zc
2020-02-24 13:59:00
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我们要做的是强行将式子转换为bsgs能求解的形式(求y^x\equiv z \pmod p)

X_{i+1} \equiv aX_i+b \pmod p \\ X_{i+1} + \frac ba \equiv aX_i+b+\frac ba \pmod p \\ aX_{i+1} + b \equiv a^2X_i+ab+b \pmod p \\ X_{i+2} \equiv a^2X_i+ab + b \pmod p

继续往下推,并根据等比数列的公式可以得到:

X_i \equiv a^{i-1}X_1+\frac{b(1-a^{i-1})}{1-a} \pmod p

带入t\equiv X_i \pmod p,得

t \equiv a^{i-1}X_1+\frac{b(1-a^{i-1})}{1-a} \pmod p \\ (1-a)t \equiv (1-a)a^{i-1}X_1+b(1-a^{i-1}) \pmod p \\ t-at \equiv a^{i-1}(X_1-aX_1)+b-b\cdot a^{i-1} \pmod p \\ a^{i-1}(X_1-aX_1-b)\equiv t-at-b \pmod p \\ a^{i-1}\equiv \frac{t-at-b}{X_1-aX_1-b} \pmod p

成功强行转换

可以用bsgs求解了!

还没完,要考虑一下几种特殊情况:

  1. X_1=t

    第一天就可以读到

  2. a=1

    题目变为X_1+kb \equiv t \pmod p

    因为p是质数,所以可以直接用费马小定理求逆元

    b=0时无解

  3. a=0

    判断b=t

#include<bits/stdc++.h>
namespace ZDY{
    #pragma GCC optimize(3)
    #define il __inline__ __attribute__ ((always_inline))
    #define register
    #define ll long long
    #define ull unsigned long long
    #define db double
    #define sht short
    #define MB template <class T>il
    #define Fur(i,x,y) for(int i(x);i<=y;++i)
    #define Fdr(i,x,y) for(int i(x);i>=y;--i)
    #define fl(i,x) for(int i(head[x]),to;to=e[i].to,i;i=e[i].nxt)
    #define clr(x,y) memset(x,y,sizeof(x))
    #define cpy(x,y) memcpy(x,y,sizeof(x))
    #define fin(s) freopen(s".in","r",stdin)
    #define fout(s) freopen(s".out","w",stdout)
    #define fcin ios::sync_with_stdio(false)
    #define l2(n) ((int)(log2(n)))
    #define inf 2122219134
    MB T ABS(T x){return x>0?x:-x;}
    MB T MAX(T x,T y){return x>y?x:y;}
    MB T MIN(T x,T y){return x<y?x:y;}
    MB T GCD(T x,T y){return y?GCD(y,x%y):x;}
    MB void SWAP(T &x,T &y){T t=x;x=y;y=t;}
}using namespace ZDY;using namespace std;
namespace IO{const int str=1<<20;static char in_buf[str],*in_s,*in_t;bool __=0;il char gc(){return (in_s==in_t)&&(in_t=(in_s=in_buf)+fread(in_buf,1,str,stdin)),in_s==in_t?EOF:*in_s++;}il void in(string &ch){ch.clear();if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}ch+=c;while((c=gc())!=EOF&&!isspace(c))ch+=c;if(c==EOF)__=1;}il void in(char &ch){if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)__=1;else ch=c;}il void in(char *ch){*ch='\0';if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}*ch=c;ch++;while((c=gc())!=EOF&&!isspace(c))*ch=c,ch++;if(c==EOF)__=1;*ch='\0';}template<typename T>il void in(T &x){if(__)return;char c=gc();bool f=0;while(c!=EOF&&(c<'0'||c>'9'))f^=(c=='-'),c=gc();if(c==EOF){__=1;return;}x=0;while(c!=EOF&&'0'<=c&&c<='9')x=x*10+c-48,c=gc();if(c==EOF)__=1;if(f)x=-x;}template<typename T,typename ... arr>il void in(T &x,arr & ... y){in(x),in(y...);}const char ln='\n';static char out_buf[str],*out_s=out_buf,*out_t=out_buf+str;il void flush(){fwrite(out_buf,1,out_s-out_buf,stdout);out_s=out_buf;}il void pt(char c){(out_s==out_t)?(fwrite(out_s=out_buf,1,str,stdout),*out_s++=c):(*out_s++=c);}il void out(const char* s){while(*s)pt(*s++);}il void out(char* s){while(*s)pt(*s++);}il void out(char c){pt(c);}il void out(string s){for(int i=0;s[i];i++)pt(s[i]);}template<typename T>il void out(T x){if(!x){pt('0');return;}if(x<0)pt('-'),x=-x;char a[50],t=0;while(x)a[t++]=x%10,x/= 10;while(t--)pt(a[t]+'0');}template<typename T,typename ... arr>il void out(T x,arr & ... y){out(x),out(y...);}}using namespace IO;
const int N=1000007;
int n,p,a,b,X1,t,T;
il int pw(int x,int b){
    int ans=1;
    while(b){
        if(b&1)ans=1ll*ans*x%p;
        b>>=1;x=1ll*x*x%p;
    }
    return ans;
}
il int inv(int x){return pw(x,p-2);}
struct hash{
    int head[N],cnt=0;
    struct edge{int to,nxt,w;}e[N];
    il void add(int x,int y,int w){e[++cnt].to=y;e[cnt].nxt=head[x];head[x]=cnt;e[cnt].w=w;}
    il void clear(){cnt=0;clr(head,0);}
    il void ins(int x,int v){add(x%N,x,v);}
    il int ask(int x){
        for(int i=head[x%N];i;i=e[i].nxt)
            if(e[i].to==x)return e[i].w;
        return -1;
    }
}S;

il int bsgs(int y,int z){
    S.clear();
    int m=sqrt(p)+1,x=z%p,f=pw(y,m);
    Fur(i,0,m)S.ins(x,i),x=1ll*x*y%p;
    x=1;
    Fur(i,1,m){
        x=1ll*x*f%p;
        int k=S.ask(x);
        if(~k)return ((1ll*i*m-k)%p+p)%p;
    }
    return -2;
}
int main(){
    in(T);
    while(T--){
        in(p,a,b,X1,t);
        if(X1==t){out("1\n");continue;}
        if(!a){out((b==t)?2:-1,ln);continue;}
        if(a==1){
            if(!b)out("-1\n");
            else out(1ll*inv(b)*(((t-X1)%p+p)%p)%p+1,ln);
            continue;
        }
        int x=1ll*(t-(1ll*a*t%p)+p-b+p)%p*(inv(X1-1ll*a*X1%p+p-b+p)%p)%p;
        out(bsgs(a,x)+1,ln);
    }
    flush();
}
LG 3306 [SDOI2013]随机数生成器
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