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.  

// C++ program to implement multiplicative order
#include<bits/stdc++.h>
using namespace std;

// function for GCD
int GCD ( int a , int b )
{
    if (b == 0 )
        return a;
    return GCD( b , a%b ) ;
}

// Function return smallest +ve integer that
// holds condition A^k(mod N ) = 1
int multiplicativeOrder(int A, int  N)
{
    if (GCD(A, N ) != 1)
        return -1;

    // result store power of A that raised to
    // the power N-1
    unsigned int result = 1;

    int K = 1 ;
    while (K < N)
    {
        // modular arithmetic
        result = (result * A) % N ;

        // return smallest +ve integer
        if (result  == 1)
            return K;

        // increment power
        K++;
    }

    return -1 ;
}

//driver program to test above function
int main()
{
    int A = 4 , N = 7;
    cout << multiplicativeOrder(A, N);
    return 0;
}

// Java program to implement multiplicative order
import java.io.*;

class GFG {

    // function for GCD
    static int GCD(int a, int b) {
        
        if (b == 0)
            return a;
            
        return GCD(b, a % b);
    }
    
    // Function return smallest +ve integer that
    // holds condition A^k(mod N ) = 1
    static int multiplicativeOrder(int A, int N) {
        
        if (GCD(A, N) != 1)
            return -1;
    
        // result store power of A that raised to
        // the power N-1
        int result = 1;
    
        int K = 1;
        
        while (K < N) {
            
            // modular arithmetic
            result = (result * A) % N;
        
            // return smallest +ve integer
            if (result == 1)
                return K;
        
            // increment power
            K++;
        }
    
        return -1;
    }
    
    // driver program to test above function
    public static void main(String args[]) {
        
        int A = 4, N = 7;
        
        System.out.println(multiplicativeOrder(A, N));
    }
}

/* This code is contributed by Nikita Tiwari.*/

# Python 3 program to implement
# multiplicative order

# function for GCD
def GCD (a, b ) :
    if (b == 0 ) :
        return a
    return GCD( b, a % b ) 

# Function return smallest + ve
# integer that holds condition 
# A ^ k(mod N ) = 1
def multiplicativeOrder(A, N) :
    if (GCD(A, N ) != 1) :
        return -1

    # result store power of A that raised 
    # to the power N-1
    result = 1

    K = 1
    while (K < N) :
    
        # modular arithmetic
        result = (result * A) % N 

        # return smallest + ve integer
        if (result == 1) :
            return K

        # increment power
        K = K + 1
    
    return -1
    
# Driver program
A = 4
N = 7
print(multiplicativeOrder(A, N))

# This code is contributed by Nikita Tiwari.

// C# program to implement multiplicative order
using System;

class GFG {

    // function for GCD
    static int GCD(int a, int b)
    {
        
        if (b == 0)
            return a;
            
        return GCD(b, a % b);
    }
    
    // Function return smallest +ve integer 
    // that holds condition A^k(mod N ) = 1
    static int multiplicativeOrder(int A, int N) 
    {
        
        if (GCD(A, N) != 1)
            return -1;
    
        // result store power of A that 
        // raised to the power N-1
        int result = 1;
    
        int K = 1;
        
        while (K < N) 
        {
            
            // modular arithmetic
            result = (result * A) % N;
        
            // return smallest +ve integer
            if (result == 1)
                return K;
        
            // increment power
            K++;
        }
    
        return -1;
    }
    
    // Driver Code
    public static void Main() 
    {
        
        int A = 4, N = 7;
        
        Console.Write(multiplicativeOrder(A, N));
    }
}

// This code is contributed by Nitin Mittal.

<?php
// PHP program to implement 
// multiplicative order

// function for GCD
function GCD ( $a , $b )
{
    if ($b == 0 )
        return $a;
    return GCD( $b , $a % $b ) ;
}

// Function return smallest
// +ve integer that holds 
// condition A^k(mod N ) = 1
function multiplicativeOrder($A, $N)
{
    if (GCD($A, $N ) != 1)
        return -1;

    // result store power of A 
    // that raised to the power N-1
    $result = 1;

    $K = 1 ;
    while ($K < $N)
    {
        // modular arithmetic
        $result = ($result * $A) % $N ;

        // return smallest +ve integer
        if ($result == 1)
            return $K;

        // increment power
        $K++;
    }

    return -1 ;
}

// Driver Code
$A = 4; $N = 7;
echo(multiplicativeOrder($A, $N));

// This code is contributed by Ajit.
?>

<script>

// JavaScript program to implement 
// multiplicative order

    // function for GCD 
    function GCD(a, b) { 
          
        if (b == 0) 
            return a; 
              
        return GCD(b, a % b); 
    } 
      
    // Function return smallest +ve integer that 
    // holds condition A^k(mod N ) = 1 
    function multiplicativeOrder(A, N) { 
          
        if (GCD(A, N) != 1) 
            return -1; 
      
        // result store power of A that raised to 
        // the power N-1 
        let result = 1; 
      
        let K = 1; 
          
        while (K < N) { 
              
            // modular arithmetic 
            result = (result * A) % N; 
          
            // return smallest +ve integer 
            if (result == 1) 
                return K; 
          
            // increment power 
            K++; 
        } 
      
        return -1; 
    }

// Driver Code
    let A = 4, N = 7; 
          
    document.write(multiplicativeOrder(A, N)); 
 
 // This code is contributed by chinmoy1997pal.
</script>

Output : 

3

Time Complexity: O(N) 

Space Complexity: O(1)

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