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Modular multiplicative inverse

Modular multiplicative inverse is a fundamental concept in number theory and modular arithmetic. Given two positive integers a and m, the modular multiplicative inverse of a modulo m is an integer x such that:

a * x ≡ 1 (mod m)

In other words, x is the inverse of a modulo m, and the product of a and x (mod m) is equal to 1. This means that x is the number which when multiplied by a and reduced modulo m, gives the remainder of 1.

The modular multiplicative inverse of a modulo m exists only when a and m are relatively prime (i.e., they have no common factors other than 1). In such cases, the modular multiplicative inverse can be found using the Extended Euclidean Algorithm, which is a recursive algorithm for finding the greatest common divisor (GCD) of two numbers and their corresponding Bezout coefficients.

The modular multiplicative inverse of a modulo m can be used to solve various problems in number theory and cryptography, including RSA encryption, Chinese remainder theorem, and primality testing.

Here given code implementation process.

// C Program
// Modular multiplicative inverse
#include <stdio.h>

// This function find the inverse of a (a^-1) under m
void invert_modulo(int a, int m)
{
	int num = a % m;
	//Loop controlling variable
	int i = 1;
	//result variable
	int result = -1;
	//Display given data information
	printf("\n Number  [a] : %d", a);
	printf("\n Modular [m] : %d", m);
	while (i < m && result == -1)
	{
		if ((num * i) % m == 1)
		{
			//When get result
			result = i;
		}
		i++;
	}
	if (result == -1)
	{
		//When result are not possible
		printf("\n Inverse are not found\n");
	}
	else
	{
		printf("\n Result %d\n", result);
	}
}
int main()
{
	printf("\n Test Modular multiplicative inverse\n");
	//Test function
	invert_modulo(7, 34);
	invert_modulo(9, 28);
	invert_modulo(28, 9);
	return 0;
}

Output

 Test Modular multiplicative inverse

 Number  [a] : 7
 Modular [m] : 34
 Result 5

 Number  [a] : 9
 Modular [m] : 28
 Result 25

 Number  [a] : 28
 Modular [m] : 9
 Result 1
// Java program
// Modular multiplicative inverse
class ModularInverse
{
	// This function find the inverse of a (a^-1) under m
	public void invert_modulo(int a, int m)
	{
		int num = a % m;
		//Loop controlling variable
		int i = 1;
		//result variable
		int result = -1;
		//Display given data information
		System.out.print("\n Number [a] : " + a);
		System.out.print("\n Modular [m] : " + m);
		while (i < m && result == -1)
		{
			if ((num * i) % m == 1)
			{
				//When get result
				result = i;
			}
			i++;
		}
		if (result == -1)
		{
			//When result are not possible
			System.out.print("\n Inverse are not found\n");
		}
		else
		{
			System.out.print("\n Result " + result + "\n");
		}
	}
	public static void main(String[] args)
	{
		ModularInverse obj = new ModularInverse();
		System.out.print("\n Test Modular multiplicative inverse\n");
		// Test Case
		obj.invert_modulo(7, 34);
		obj.invert_modulo(9, 28);
		obj.invert_modulo(28, 9);
	}
}

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1
//Include header file
#include <iostream>

using namespace std;
// C++ program
// Modular multiplicative inverse
class ModularInverse
{
	public:
		// This function find the inverse of a (a^-1) under m
		void invert_modulo(int a, int m)
		{
			int num = a % m;
			//Loop controlling variable
			int i = 1;
			//result variable
			int result = -1;
			//Display given data information
			cout << "\n Number [a] : " << a;
			cout << "\n Modular [m] : " << m;
			while (i < m && result == -1)
			{
				if ((num * i) % m == 1)
				{
					//When get result
					result = i;
				}
				i++;
			}
			if (result == -1)
			{
				//When result are not possible
				cout << "\n Inverse are not found\n";
			}
			else
			{
				cout << "\n Result " << result << "\n";
			}
		}
};
int main()
{
	ModularInverse obj = ModularInverse();
	cout << "\n Test Modular multiplicative inverse\n";
	// Test Case
	obj.invert_modulo(7, 34);
	obj.invert_modulo(9, 28);
	obj.invert_modulo(28, 9);
	return 0;
}

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1
//Include namespace system
using System;

// C# program
// Modular multiplicative inverse

class ModularInverse
{
	// This function find the inverse of a (a^-1) under m
	public void invert_modulo(int a, int m)
	{
		int num = a % m;
		//Loop controlling variable
		int i = 1;
		//result variable
		int result = -1;
		//Display given data information
		Console.Write("\n Number [a] : " + a);
		Console.Write("\n Modular [m] : " + m);
		while (i < m && result == -1)
		{
			if ((num * i) % m == 1)
			{
				//When get result
				result = i;
			}
			i++;
		}
		if (result == -1)
		{
			//When result are not possible
			Console.Write("\n Inverse are not found\n");
		}
		else
		{
			Console.Write("\n Result " + result + "\n");
		}
	}
	public static void Main(String[] args)
	{
		ModularInverse obj = new ModularInverse();
		Console.Write("\n Test Modular multiplicative inverse\n");
		// Test Case
		obj.invert_modulo(7, 34);
		obj.invert_modulo(9, 28);
		obj.invert_modulo(28, 9);
	}
}

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1
<?php
// Php program
// Modular multiplicative inverse
class ModularInverse
{
	// This function find the inverse of a (a^-1) under m
	public	function invert_modulo($a, $m)
	{
		$num = $a % $m;
		//Loop controlling variable
		$i = 1;
		//result variable
		$result = -1;
		//Display given data information
		echo "\n Number [a] : ". $a;
		echo "\n Modular [m] : ". $m;
		while ($i < $m && $result == -1)
		{
			if (($num * $i) % $m == 1)
			{
				//When get result
				$result = $i;
			}
			$i++;
		}
		if ($result == -1)
		{
			//When result are not possible
			echo "\n Inverse are not found\n";
		}
		else
		{
			echo "\n Result ". $result ."\n";
		}
	}
}

function main()
{
	$obj = new ModularInverse();
	echo "\n Test Modular multiplicative inverse\n";
	// Test Case
	$obj->invert_modulo(7, 34);
	$obj->invert_modulo(9, 28);
	$obj->invert_modulo(28, 9);
}
main();

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1
// Node Js program
// Modular multiplicative inverse
class ModularInverse
{
	// This function find the inverse of a (a^-1) under m
	invert_modulo(a, m)
	{
		var num = a % m;
		//Loop controlling variable
		var i = 1;
		//result variable
		var result = -1;
		//Display given data information
		process.stdout.write("\n Number [a] : " + a);
		process.stdout.write("\n Modular [m] : " + m);
		while (i < m && result == -1)
		{
			if ((num * i) % m == 1)
			{
				//When get result
				result = i;
			}
			i++;
		}
		if (result == -1)
		{
			//When result are not possible
			process.stdout.write("\n Inverse are not found\n");
		}
		else
		{
			process.stdout.write("\n Result " + result + "\n");
		}
	}
}

function main()
{
	var obj = new ModularInverse();
	process.stdout.write("\n Test Modular multiplicative inverse\n");
	// Test Case
	obj.invert_modulo(7, 34);
	obj.invert_modulo(9, 28);
	obj.invert_modulo(28, 9);
}
main();

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1
#  Python 3 program
#  Modular multiplicative inverse
class ModularInverse :
	#  This function find the inverse of a (a^-1) under m
	def invert_modulo(self, a, m) :
		num = a % m
		# Loop controlling variable
		i = 1
		# result variable
		result = -1
		# Display given data information
		print("\n Number [a] : ", a, end = "")
		print("\n Modular [m] : ", m, end = "")
		while (i < m and result == -1) :
			if ((num * i) % m == 1) :
				# When get result
				result = i
			
			i += 1
		
		if (result == -1) :
			# When result are not possible
			print("\n Inverse are not found\n", end = "")
		else :
			print("\n Result ", result ,"\n", end = "")
		
	

def main() :
	obj = ModularInverse()
	print("\n Test Modular multiplicative inverse\n", end = "")
	#  Test Case
	obj.invert_modulo(7, 34)
	obj.invert_modulo(9, 28)
	obj.invert_modulo(28, 9)

if __name__ == "__main__": main()

Output

 Test Modular multiplicative inverse

 Number [a] :  7
 Modular [m] :  34
 Result  5

 Number [a] :  9
 Modular [m] :  28
 Result  25

 Number [a] :  28
 Modular [m] :  9
 Result  1
#  Ruby program
#  Modular multiplicative inverse
class ModularInverse

	#  This function find the inverse of a (a^-1) under m
	def invert_modulo(a, m)
	
		num = a % m
		# Loop controlling variable
		i = 1
		# result variable
		result = -1
		# Display given data information
		print("\n Number [a] : ", a)
		print("\n Modular [m] : ", m)
		while (i < m && result == -1)
		
			if ((num * i) % m == 1)
			
				# When get result
				result = i
			end
			i += 1
		end
		if (result == -1)
		
			# When result are not possible
			print("\n Inverse are not found\n")
		else
		
			print("\n Result ", result ,"\n")
		end
	end
end
def main()

	obj = ModularInverse.new()
	print("\n Test Modular multiplicative inverse\n")
	#  Test Case
	obj.invert_modulo(7, 34)
	obj.invert_modulo(9, 28)
	obj.invert_modulo(28, 9)
end
main()

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1

// Scala program
// Modular multiplicative inverse
class ModularInverse
{
	// This function find the inverse of a (a^-1) under m
	def invert_modulo(a: Int, m: Int): Unit = {
		var num: Int = a % m;
		//Loop controlling variable
		var i: Int = 1;
		//result variable
		var result: Int = -1;
		//Display given data information
		print("\n Number [a] : " + a);
		print("\n Modular [m] : " + m);
		while (i < m && result == -1)
		{
			if ((num * i) % m == 1)
			{
				//When get result
				result = i;
			}
			i += 1;
		}
		if (result == -1)
		{
			//When result are not possible
			print("\n Inverse are not found\n");
		}
		else
		{
			print("\n Result " + result + "\n");
		}
	}
}
object Main
{
	def main(args: Array[String]): Unit = {
		var obj: ModularInverse = new ModularInverse();
		print("\n Test Modular multiplicative inverse\n");
		// Test Case
		obj.invert_modulo(7, 34);
		obj.invert_modulo(9, 28);
		obj.invert_modulo(28, 9);
	}
}

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1
// Swift program
// Modular multiplicative inverse
class ModularInverse
{
	// This function find the inverse of a (a^-1) under m
	func invert_modulo(_ a: Int, _ m: Int)
	{
		let num: Int = a % m;
		//Loop controlling variable
		var i: Int = 1;
		//result variable
		var result: Int = -1;
		//Display given data information
		print("\n Number [a] : ", a, terminator: "");
		print("\n Modular [m] : ", m, terminator: "");
		while (i < m && result == -1)
		{
			if ((num * i) % m == 1)
			{
				//When get result
				result = i;
			}
			i += 1;
		}
		if (result == -1)
		{
			//When result are not possible
			print("\n Inverse are not found\n", terminator: "");
		}
		else
		{
			print("\n Result ", result ,"\n", terminator: "");
		}
	}
}
func main()
{
	let obj: ModularInverse = ModularInverse();
	print("\n Test Modular multiplicative inverse\n", terminator: "");
	// Test Case
	obj.invert_modulo(7, 34);
	obj.invert_modulo(9, 28);
	obj.invert_modulo(28, 9);
}
main();

Output

 Test Modular multiplicative inverse

 Number [a] :  7
 Modular [m] :  34
 Result  5

 Number [a] :  9
 Modular [m] :  28
 Result  25

 Number [a] :  28
 Modular [m] :  9
 Result  1
fn main() 
{
	print!("\n Test Modular multiplicative inverse\n");
	//Test function
	invert_modulo(7, 34);
	invert_modulo(9, 28);
	invert_modulo(28, 9);
	
}
fn invert_modulo(a: i32, m: i32)
{
	let num: i32 = a % m;
	//Loop controlling variable
	let mut i: i32 = 1;
	//result variable
	let mut result: i32 = -1;
	//Display given data information
	print!("\n Number [a] : {}", a);
	print!("\n Modular [m] : {}", m);
	while i < m && result == -1
	{
		if (num * i) % m == 1
		{
			//When get result
			result = i;
		}
		i += 1;
	}
	if result == -1
	{
		//When result are not possible
		print!("\n Inverse are not found\n");
	}
	else
	{
		print!("\n Result {}\n", result);
	}
}

Output

 Test Modular multiplicative inverse

 Number [a] : 7
 Modular [m] : 34
 Result 5

 Number [a] : 9
 Modular [m] : 28
 Result 25

 Number [a] : 28
 Modular [m] : 9
 Result 1

This C program finds the modular multiplicative inverse of a given number 'a' with respect to a given modulus 'm'. The modular multiplicative inverse of a number 'a' is another number 'b' such that their product is congruent to 1 modulo m, i.e., (a * b) ≡ 1 (mod m). If 'b' exists, then 'a' is said to be invertible modulo 'm'.

The program defines a function called 'invert_modulo' that takes two integer arguments 'a' and 'm'. Inside the function, it first calculates the residue of 'a' modulo 'm' using the modulus operator '%'. Then, it initializes a loop controlling variable 'i' to 1 and a result variable 'result' to -1. The loop runs from 1 to 'm' (exclusive) and checks if the product of 'num' and 'i' modulo 'm' is equal to 1. If it is, then it sets the 'result' variable to 'i' and breaks out of the loop. If the loop completes without finding a suitable 'i', then the 'result' variable remains as -1, indicating that the modular inverse does not exist. Finally, the function prints the given values of 'a' and 'm' and the computed modular inverse value (if it exists) using printf statements.

Last updated on by Kalkicode




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