200. Cracking RSA
time limit per test: 0.25
sec.
memory limit per test: 65536
KB
input: standard
output: standard
The following problem is somehow related to the final stage of many famous integer factorization algorithms involved in some cryptoanalytical problems, for example cracking wellknown RSA public key system.
The most powerful of such algorithms, so called quadratic sieve descendant algorithms, utilize the fact that if n = pq where p and q are large unknown primes needed to be found out, then if v^{2}=w^{2} (mod n), u ≠ v (mod n) and u ≠ v (mod n), then gcd(v + w, n) is a factor of n (either p or q).
Not getting further in the details of these algorithms, let us consider our problem. Given m integer numbers b_{1}, b_{2}, ..., b_{m} such that all their prime factors are from the set of first t primes, the task is to find such a subset S of {1, 2, ..., m} that product of b_{i} for i from S is a perfect square i.e. equal to u^{2} for some integer u. Given such S we get one pair for testing (product of S elements stands for v when w is known from other steps of algorithms which are of no interest to us, testing performed is checking whether pair is nontrivial, i.e. u ≠ v (mod n) and u ≠ v (mod n)). Since we want to factor n with maximum possible probability, we would like to get as many such sets as possible. So the interesting problem could be to calculate the number of all such sets. This is exactly your task.
Input
The first line of the input file contains two integers t and m (1 ≤ t ≤ 100, 1 ≤ m ≤ 100). The second line of the input file contains m integer numbers b_{i} such that all their prime factors are from t first primes (for example, if t = 3 all their prime factors are from the set {2, 3, 5}). 1 ≤ b_{i} ≤ 10^{9} for all i.
Output
Output the number of nonempty subsets of the given set {b_{i}}, the product of numbers from which is a perfect square
Sample test(s)
Input
3 4
9 20 500 3
Output
3
Author:  Andrew Stankevich

Resource:  Petrozavodsk Winter Trainings 2003

Date:  20030206

Server time: 20160526 19:23:18  Online Contester Team © 2002  2016. All rights reserved. 

