Bzoj 3122 Random number generator

http://www.lydsy.com/JudgeOnline/problem.php?id=3122

Test instructions: Give the p,a,b,x1,t

Known xn=a*xn-1+b%p, to find the smallest n order xn=t

First, if x1=t, returns 1

If a=0, then if B=t returns 2, no solution

If a=1, then t=t-x1+p%p, you can list the equation

B*x+p*y==t% P

If a>=2, then according to geometric series and available

xn=t=x1*a^ (n-1) +b* (a^ (n-1)-1)/(A-1)%p

Because P is prime, so c=inv[a-1]= (A-1) ^ (p-2)

x1*a^ (n-1) +b*c* (a^ (n-1)) ==b*c+t%p

(x1+b*c) * (a^ (n-1)) ==b*c+t% p

Make A=x1+b*c,b=p,c=b*c+t

Is the solution A*x+b*y==c%p

Solution out x=a^ (n-1), then this with bsgs to ask for it.

1#include <algorithm>2#include <cstdio>3#include <cmath>4#include <cstring>5#include <iostream>6#include <map>7 #definell Long Long8 ll P;9 ll Read () {Ten     CharCh=getchar (); ll t=0, f=1; One      while(ch<'0'|| Ch>'9'){if(ch=='-') f=-1; ch=GetChar ();} A      while('0'<=ch&&ch<='9') {t=t*Ten+ch-'0'; ch=GetChar ();} -     returnt*F; - } the ll Pow (ll X,ll y) { -ll res=1; -     if(x<0) x= (x+p)%p; -      while(y) { +         if(y%2) res= (res*x)%p; -x= (x*x)%p; +Y/=2; A     } at     returnRes; - } - ll gcd (ll A,ll b) { -     if(b==0)returnA; -     Else returnGCD (b,a%b); - } in voidEXGCD (ll a,ll b,ll &x,ll &y) { -     if(b==0){ tox=1; +y=0; -         return; the     } *EXGCD (b,a%b,x,y); $ll t=x;Panax Notoginsengx=y; -Y=t-(A/b) *y; the } + ll Reverse (ll X) { All a=x,b=p; the ll x, y; + EXGCD (a,b,x,y); -     return(x%p+p)%p; $ } $ ll work (ll A,ll b) { -a%=p; -     if(a==0){ the         if(b==0)return 1; -         Else return-1;Wuyi     } theStd::map<ll,int>MP; -ll M=sqrt (p) +1, i=1, Im=pow (a,p-1-m), t=1; WuMp.clear (); mp[1]=m+1; -      for(intI=1; i<m;i++){ Aboutt=t*a%p; $         if(!mp[t]) mp[t]=i;  -     } -      for(intk=0; k<m;k++){ -         inti=mp[i*b%p]; A         if(i) { +             if(i==m+1) i=0; the             returni+k*m; -         } $i=i*im%p; the     } the     return-1; the } the ll solve (ll a,ll b,ll x1,ll t) { -     if(t==x1) { in             return 1; the     } the     if(a==0){ About             if(b==t) { the                 return 2; the             } the             return-1; +     } -     if(a==1){ thell a=b,b=p,t= (t-x1+p)%p;Bayill d=gcd (A, b); the             if(t%D) { the                 return-1; -             } -T/=D; the ll x, y; the EXGCD (a,b,x,y); thex= (x*t)%p; the              while(x<0) x+=p; -             returnx+1; the     } thell C=pow (A-1, P-2); thell a= (b*c+x1)%p,b=p,t= (b*c+t)%p;94     if(a<0) a= (a+p)%p; the     if(b<0) b= (b+p)%p; thell d=gcd (A, b); the     if(t%D) {98         return-1; About     } -T/=D;101 ll x, y;102 EXGCD (a,b,x,y);103      while(x<0) x= (x+p)%p;104x= (x*t)%p; thell ans=Work (a,x);106     if(ans!=-1)returnans+1;107     Else returnans;108 }109 intMain () { the     intTcase;111scanf"%d",&Tcase); the      while(tcase--){113 ll A,b,x1,t; theP=read (); A=read (); B=read (); X1=read (); t=read (); theprintf"%lld\n", Solve (a,b,x1,t)); the     }117}

Bzoj 3122 Random number generator