Multiplicative order

In number theory, given an integer A and a positive integer N with gcd( A , N) = 1, the multiplicative order of a modulo N is the smallest positive integer k with A^k( mod N ) = 1. ( 0 < K < N ) 

Examples : 

Input : A = 4 , N = 7 
Output : 3
explanation : GCD(4, 7) = 1 
  A^k( mod N ) = 1 ( smallest positive integer K )
 4^1 = 4(mod 7) = 4
 4^2 = 16(mod 7) = 2
 4^3 = 64(mod 7) = 1
 4^4 = 256(mod 7) = 4
 4^5 = 1024(mod 7) = 2
 4^6 = 4096(mod 7) = 1

smallest positive integer K = 3 

Input : A = 3 , N = 1000 
Output : 100 (3^100 (mod 1000) == 1) 

Input : A = 4 , N = 11 
Output : 5 

If we take a close look then we observe that we do not need to calculate power every time. we can be obtaining next power by multiplying 'A' with the previous result of a module. 

Explanation : 
A = 4 , N = 11 
initially result = 1 
with normal with modular arithmetic (A * result)
4^1 = 4 (mod 11 ) = 4 || 4 * 1 = 4 (mod 11 ) = 4 [ result = 4]
4^2 = 16(mod 11 ) = 5 || 4 * 4 = 16(mod 11 ) = 5 [ result = 5]
4^3 = 64(mod 11 ) = 9 || 4 * 5 = 20(mod 11 ) = 9 [ result = 9]
4^4 = 256(mod 11 )= 3 || 4 * 9 = 36(mod 11 ) = 3 [ result = 3]
4^5 = 1024(mod 5 ) = 1 || 4 * 3 = 12(mod 11 ) = 1 [ result = 1]

smallest positive integer 5 

Run a loop from 1 to N-1 and Return the smallest +ve power of A under modulo n which is equal to 1. 

Below is the implementation of above idea.  

Output : 

3

Time Complexity: O(N) 

Space Complexity: O(1)

Reference: https://en.wikipedia.org/wiki/Multiplicative_order