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22346_DiscreteLogging

2022-5-16 18:21| 发布者: Hocassian| 查看: 19| 评论: 0|原作者: 肇庆学院ACM合集

摘要:

C:\Users\Administrator\Downloads\2019-10-12-10-14-5-89506342541400-Problem List-采集的数据-后羿采集器.html

Pro.ID

22346

Title

Discrete Logging

Title链接

http://10.20.2.8/oj/exercise/problem?problem_id=22346

AC

Submit

Ratio

33.33%

时间&空间限制

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)

描述

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)

输入

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

输出

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".

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

Hint

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) .

Source

Waterloo Local 2002.1.26

样例输入

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

样例输出

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

提示

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)