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.

CPP

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// 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 ) ;
}
  
// Fucnction 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 rised to
    // the power N-1
    unsigned int result = 1;
  
    int K = 1 ;
    while (K < N)
    {
        // modular arithmetic
        result = (result * A) % N ;
  
        // return samllest +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;
}

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Java














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// 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 rised to 


        // the power N-1 


        int result = 1; 


      


        int K = 1; 


          


        while (K < N) { 


              


            // modular arithmetic 


            result = (result * A) % N; 


          


            // return samllest +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.*/



















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Python3














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# Python 3 program to implement 


# multiplicative order 


  


# funnction for GCD 


def GCD (a, b ) : 


    if (b == 0 ) : 


        return a 


    return GCD( b, a % b )  


  


# Fucnction 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 rised  


    # to the power N-1 


    result = 1


  


    K = 1


    while (K < N) : 


      


        # modular arithmetic 


        result = (result * A) % N  


  


        # return samllest + 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. 



















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C#














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// 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  


        // rised to the power N-1 


        int result = 1; 


      


        int K = 1; 


          


        while (K < N)  


        { 


              


            // modular arithmetic 


            result = (result * A) % N; 


          


            // return samllest +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. 



















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PHP














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<?php 


// PHP program to implement  


// multiplicative order 


  


// function for GCD 


function GCD ( $a , $b ) 


{ 


    if ($b == 0 ) 


        return $a; 


    return GCD( $b , $a % $b ) ; 


} 


  


// Fucnction 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 rised to the power N-1 


    $result = 1; 


  


    $K = 1 ; 


    while ($K < $N) 


    { 


        // modular arithmetic 


        $result = ($result * $A) % $N ; 


  


        // return samllest +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. 


?> 



















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Output :


3



Time Complexity: O(N) 


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

This article is contributed by Nishant Singh . If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.


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