# Problem B: Discrete Logging

Problem B: Discrete Logging

## Problem B: Discrete Logging

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

```    BL == N (mod P)
```

Read several lines of input, each containing P,B,N separated by a space, and 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) .
```

``````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
``````

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

``````#include <stdio.h>
#include <string.h>
#include <map>
#include <malloc.h>
using namespace std;
struct species {
char s[81];
} buf;

bool operator==(const species &a, const species &b) {
return !strcmp(a.s, b.s);
}

bool operator<(const species &a, const species &b) {
return strcmp(a.s, b.s) < 0;
}

map<species,int> cnt;

main(){
int tot = 0;
while(gets(buf.s)){
cnt[buf]++;
tot++;
}
for (map<species,int>::iterator i = cnt.begin(); i!=cnt.end(); *i++) {
printf("%s %0.4lf\n",i->first.s, 100.0*i->second/tot);
}
}

/**************************************************************
Problem: 1270
User: zhblue
Language: C++
Result: Compile Error
****************************************************************/

``````

``````#include <stdio.h>
#include <string.h>
#include <map>
#include <malloc.h>
using namespace std;
struct species {
char s[81];
} buf;

bool operator==(const species &a, const species &b) {
return !strcmp(a.s, b.s);
}

bool operator<(const species &a, const species &b) {
return strcmp(a.s, b.s) < 0;
}

map<species,int> cnt;

main(){
int tot = 0;
while(gets(buf.s)){
cnt[buf]++;
tot++;
}
for (map<species,int>::iterator i = cnt.begin(); i!=cnt.end(); *i++) {
printf("%s %0.4lf\n",i->first.s, 100.0*i->second/tot);
}
}

/**************************************************************
Problem: 1270
User: zhblue
Language: C++
Result: Compile Error
****************************************************************/

``````