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Problem 1352 Discrete Logging

Accept: 15    Submit: 23
Time Limit: 1000 mSec    Memory Limit : 32768 KB

Problem 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
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) .

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

Source

Waterloo 020126

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