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Code Mathematics

# Modular multiplicative inverse

Modular multiplicative inverse is a crucial concept in number theory and cryptography. Given two integers 'a' and 'm', where 'a' is coprime to 'm' (i.e., gcd(a, m) = 1), the modular multiplicative inverse of 'a' modulo 'm' is another integer 'x' such that (a * x) % m = 1. In simpler terms, it's finding the number that, when multiplied by 'a' and then taken modulo 'm', results in 1. This article discusses the process of finding modular multiplicative inverses and presents a C program to compute them.

## Problem Statement

Given two integers 'a' and 'm', the problem is to find an integer 'x' such that (a * x) % m = 1.

## Example

For example, let's take 'a' = 7 and 'm' = 34. The modular multiplicative inverse of 7 modulo 34 is 5, because (7 * 5) % 34 = 35 % 34 = 1. Similarly, for 'a' = 9 and 'm' = 28, the modular multiplicative inverse is 25, as (9 * 25) % 28 = 225 % 28 = 1. For 'a' = 28 and 'm' = 9, the modular multiplicative inverse is 1, as (28 * 1) % 9 = 28 % 9 = 1.

## Idea to Solve

To solve this problem, we need to iterate through all possible values of 'x' until we find the one that satisfies the condition (a * x) % m = 1. Since 'm' can be relatively large, it's not efficient to check all possible values of 'x' linearly. Instead, we can use a loop to iterate through integers and use the property that (a * x) % m = 1 is equivalent to (a * x) ≡ 1 (mod m). This can be solved using the Extended Euclidean Algorithm, which finds 'x' and 'y' such that (a * x) + (m * y) = gcd(a, m) = 1.

## Pseudocode

``````function invert_modulo(a, m):
num = a % m
i = 1
result = -1
while i < m and result == -1:
if (num * i) % m == 1:
result = i
i++
if result == -1:
Print "Inverse not found"
else:
Print "Result:", result

function main():
invert_modulo(7, 34)
invert_modulo(9, 28)
invert_modulo(28, 9)``````

## Algorithm Explanation

1. The `invert_modulo` function takes two parameters, 'a' and 'm'.
2. It calculates the remainder of 'a' when divided by 'm' (num = a % m).
3. It initializes a loop variable 'i' to 1 and a result variable to -1.
4. The loop iterates from 1 to 'm-1':
• If (num * i) % m equals 1, it means the modular multiplicative inverse is found (result = i).
5. If no result is found, it prints "Inverse not found", otherwise, it prints the value of the result.
6. The `main` function tests the `invert_modulo` function with different values.

## Code Solution

``````// 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``````

## Time Complexity

The time complexity of this code is linear, O(m), where 'm' is the modulus value. The loop iterates up to 'm-1' times, and each iteration involves a constant number of operations. The execution time grows linearly with the modulus value.

## Comment

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