Pagini recente » Cod sursa (job #2294091) | Clasament minune9 | Cod sursa (job #427582) | Cod sursa (job #1706277) | Cod sursa (job #2654829)
#include <bits/stdc++.h>
using namespace std;
ifstream fin("inversmodular.in");
ofstream fout("inversmodular.out");
/**
a ^ n % P = 43210
n = 25 = 2 ^ 4 + 2 ^ 3 + 2 ^ 0 = (11001)
a ^ 25 = a ^ (2 ^ 4) * a ^ (2 ^ 3) * a ^ (2 ^ 0)
p = 1 * a * a ^ 8 * a ^ 16
a = a * a
a = a * a
a = a * a
a = a * a
Invers Modular
a ^ phi(P) % P = 1, P - prim
phi(n) - nr de numere < n si prime cu acesta
phi(12) = 4 (1,5,7,11)
Problema: Se dau a > 1 si P. Sa se determine un nr nat b a.i. a * b % P = 1
ex: a = 5, P = 17 => b = 7
a * b % P = 1 => b = 1/a % P
a ^ phi(P) % P = 1 => a * a ^ (phi(P) - 1) % P
=> b = a ^ (phi(P) - 1)
x / y % P = x * y ^ (phi(P) - 1) % P
Daca P - prim => phi(P) = P - 1
=> inversul modular al lui y este y ^ (P - 2) % P
*/
int Fact(int n, int P)
{
int rez = 1;
for(int i = 2;i <= n;i++)
rez = rez * i % P;
return rez;
}
int LogP(int a, int n, int P)
{
int rez = 1;
while(n > 0)
{
if(n % 2 == 1)
rez = 1LL * rez * a % P;
n /= 2;
a = 1LL * a * a % P;
}
return rez;
}
int Phi(int n)
{
int f,exp,phi = n;
f = 2;
while(n > 1)
{
exp = 0;
while(n % f == 0)
{
n /= f;
exp++;
}
if(exp > 0) phi = phi / f * (f - 1);
if(f * f < n) f++;
else f = n;
}
return phi;
}
int main()
{
int a, N;
fin >> a >> N;
fout << LogP(a, Phi(N) - 1, N);
return 0;
}