"Bzoj 3122" [Sdoi2013] random number generator (BSGS)

3122: [Sdoi2013] random number generator time limit:10 Sec Memory limit:256 MB
submit:1442 solved:552 Description

Input

The input contains multiple sets of data, and the first line is a positive integer t, indicating the number of data groups within the test point.

Next T line, each line has five integer p,a,b,x1,t, which represents a set of data. Ensure that both X1 and T are legal page numbers.

Note: P must be prime

Output

A total of T-lines, one integer per line, indicating which day he first read page T. If he would never read page T, output-1.

Sample Input 3
7 1 1) 3 3
7 2 2) 2 0
7 2 2) 2 1


Sample Output
1
3
-1
HINT

0<=a<=p-1,0<=b<=p-1,2<=p<=10^9

Analysis

This problem is almost the pit Father home!! P is a prime number does not say, there are all kinds of disgusting special!!!!

A=0,a=1,b=0 are to be condemned.

Although the introduction of the general formula, but where the division can be reversed where can not know, a beginning-pass various wa!!

Look at other people's puzzle:

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.

Http://www.cnblogs.com/qzqzgfy/p/5581955.html

My Code:

1#include <cstdio>2#include <cstdlib>3#include <cstring>4#include <iostream>5#include <algorithm>6#include <queue>7#include <cmath>8 using namespacestd;9 #defineLL Long LongTen #defineMAXN 35000 One  A structnode - { - LL Id,val; the }T[MAXN]; -  - LL Ax,ay; - ll EXGCD (ll A,ll b) + { -     if(b==0) {ax=1; ay=0;returnA;} +LL G=EXGCD (b,a%b); ALL xx=Ax; atAx=ay;ay=xx-(A/b) *ay; -     returnG; - } -  - BOOLCMP (node X,node y) {return(X.val==y.val)? (x.id<y.id):(x.val<y.val);} -  in LL CNT; -  to ll T_div (ll x) + { -LL l=0, r=CNT; the      while(l<R) *     { $LL mid= (l+r) >>1;Panax Notoginseng         if(t[mid].val==x)returnt[mid].id; -         if(t[mid].val>x) r=mid-1; the         ElseL=mid+1; +     } A     if(t[l].val==x)returnt[l].id; the     return-1; + } -  $ ll Get_ans (ll k,ll a,ll c,ll p) $ { -ll sq= (LL) ceil (sqrt (Double) (p)); -t[0].id=0; t[0].val=k%p; the      for(LL i=1; i<=sq;i++) t[i].val= (t[i-1].val*a)%p,t[i].id=i; -     WuyiSort (t+1, t+1+sq,cmp); theCnt=0; -      for(LL i=1; i<=sq;i++)if(t[i].val!=t[i-1].val) t[++cnt]=T[i]; Wu      -LL bm=1; About      for(LL i=1; i<=sq;i++) bm= (bm*a)%p; $LL g=EXGCD (bm,p); -Ax= (ax% (p/g) + (p/g))% (p/g); -bm=Ax; -      ALL tmp=c%p; +      for(LL i=0; i<=sq;i++) the     { -LL now=T_div (TMP); $         if(now!=-1)returnnow+i*sq; thetmp= (TMP*BM)%p; the     } the     return-1; the } -  in ll Ffind (ll k,ll a,ll c,ll p) the { theLL x=K; About      for(LL i=0; i<= -; i++) the     { the         if(X==C)returni; thex= (x*a)%p; +     } - LL G; thex=0;Bayi      while((G=EXGCD (a,p))! =1) the     { theP/=G; -K= (k* (a/g))%p; -         if(c%g!=0)return-1; theC/=G; theX + +; the     } theLL ans=Get_ans (k,a,c,p); -     if(ans==-1)return-1; the     returnans+x; the } the 94 intMain () the { the     intT,kase=0; thescanf"%d",&T);98      while(t--) About     { -         //printf ("%d:", ++kase);101 LL p,a,b,x,t,c;102scanf"%lld%lld%lld%lld%lld",&p,&a,&b,&x,&t);103         //if (b==0) {printf ("1\n"); continue;}104         if(t==x) {printf ("1\n");Continue;} the         if(a==0&&b==t) {printf ("2\n");Continue;}106         Else if(a==0) {printf ("-1\n");Continue;}107         if(a==1)108         {109LL g=EXGCD (b,p); thec=t-x;c= (c%p+p)%p;111             if(c%g!=0) {printf ("-1\n");Continue;} theax*=c/G;113Ax= (ax% (p/g) + (p/g))% (p/g); theprintf"%d\n", ax+1); the             Continue; the         }117         118 LL Phi;119         if(B% (A-1)==0) phi=b/(A-1); -         Else121         {122             if(EXGCD (A-1, p)! =1) {printf ("-1\n");Continue;}123ax= (ax%p+p)%p;124phi= (b*ax)%p; the         }126LL a=x+phi,b=phi+T;127          -         if(b==0) a=x,b=T;129          theA= (a%p+p)%p; b= (b%p+p)%p;131         //if (a==0) {printf (" -1\n"); continue;} the         133LL ans=Ffind (a,a,b,p);134         135         if(ans==-1) printf ("-1\n");136         Elseprintf"%lld\n", ans+1);137     }138     return 0;139}

Bzoj 3211

2016-09-05 18:24:44

"Bzoj 3122" [Sdoi2013] random number generator (BSGS)