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++
// C++ program to implement multiplicative order#include<bits/stdc++.h>using namespace std;// function for GCDint 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 ) = 1int 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 functionint main(){ int A = 4 , N = 7; cout << multiplicativeOrder(A, N); return 0;} |
Java
// Java program to implement multiplicative orderimport 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.*/ |
Python3
# Python 3 program to implement# multiplicative order# function for GCDdef 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 ) = 1def 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 programA = 4N = 7print(multiplicativeOrder(A, N))# This code is contributed by Nikita Tiwari. |
C#
// C# program to implement multiplicative orderusing 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// PHP program to implement // multiplicative order// function for GCDfunction GCD ( $a , $b ){ if ($b == 0 ) return $a; return GCD( $b , $a % $b ) ;}// Function return smallest// +ve integer that holds // condition A^k(mod N ) = 1function multiplicativeOrder($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.?> |
Javascript
<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
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