【BSGS】BZOJ3239 Discrete Logging

2023-08-27 12:33:22

3239: Discrete Logging

Time Limit: 1 Sec Memory Limit: 128 MB
Submit: 729 Solved: 485
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Description

Given a prime P, 2 <= P < 2³¹,
an integer B, 2 <= B < P, and an integer N, 2 <= N < P,
compute the discrete logarithm of N, base B, modulo P. That is, find an
integer L such that

    B^L == N (mod P)

Input

Read several lines of input, each containing P,B,N separated by a space,

Output

for each line print the logarithm on a separate line. If there are several, print the smallest; if there is none, print "no solution".

The solution to this problem requires a well known result in number theory that is probably expected of you for Putnam but not ACM competitions. It is Fermat's theorem that states

   B^(P-1) == 1 (mod P)

for any prime P and some other (fairly rare) numbers known as base-B pseudoprimes. A rarer subset of the base-B pseudoprimes, known as Carmichael numbers, are pseudoprimes for every base between 2 and P-1. A corollary to Fermat's theorem is that for any m

   B^(-m) == B^(P-1-m) (mod P) .

Sample Input

5 2 1
5 2 2
5 2 3
5 2 4
5 3 1
5 3 2
5 3 3
5 3 4
5 4 1
5 4 2
5 4 3
5 4 4
12345701 2 1111111
1111111121 65537 1111111111

Sample Output

0
1
3
2
0
3
1
2
0
no solution
no solution
1
9584351
462803587

题解

BSGS模板题

简单说下BSGS（baby step giant step）

用于解决离散对数问题即求解a^x≡b（mod p）中的x

先处理出sqrt（p）范围内的b*a^x的值，丢进map里

然后依次求出k*a^{k*sqrt（p）}（k=1……sqrt（p））看答案是否有出现过

本质上就是分块的思想

代码

//by 减维

#include<set>

#include<map>

#include<queue>

#include<ctime>

#include<cmath>

#include<bitset>

#include<vector>

#include<cstdio>

#include<cstring>

#include<iostream>

#include<algorithm>

#define ll long long

#define il inline

#define rg register

#define db double

#define mpr make_pair

#define maxn

#define inf (1<<30)

#define eps 1e-8

#define pi 3.1415926535897932384626L

using namespace std;

inline int read()

{

    int ret=;bool fla=;char ch=getchar();

    while((ch<''||ch>'')&&ch!='-')ch=getchar();

    if(ch=='-'){fla=;ch=getchar();}

    while(ch>=''&&ch<=''){ret=ret*+ch-'';ch=getchar();}

    return fla?-ret:ret;

}

ll x,y,p;

ll ksm(ll x,ll y,ll p)

{

    ll ret=;x%=p;

    for(;y;y>>=,x=x*x%p)

        if(y&) ret=ret*x%p;

    return ret;

}

ll bsgs(ll x,ll y,ll p)

{

    map<ll,ll> mp;

    x%=p;y%=p;

    if(!x&&!y) return ;

    if(y==) return ;

    if(!x) return -;

    ll m=sqrt(p+0.5);

    ll o=y;

    for(int i=;i<=m;++i,o=o*x%p)

        if(!mp.count(o)) mp[o]=i;

    ll tmp=ksm(x,m,p);o=tmp;

    for(int i=;i<=m;++i,o=o*tmp%p)

        if(mp.count(o)) return ((i*m-mp[o])%p+p)%p;

    return -;

}

int main()

{

    while(scanf("%lld%lld%lld",&p,&x,&y)!=EOF)

    {

        ll ans=bsgs(x,y,p);

        if(ans==-) printf("no solution\n");

        else printf("%lld\n",ans);

    }

    return ;

}

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