Inverse of Number Mod N Calculator

What is a Modular Inverse?

In modular arithmetic, the inverse of a number a modulo n is a number x such that:

This means that when a is multiplied by x, the result leaves a remainder of 1 when divided by n. Not all numbers have inverses modulo n. A number a has an inverse modulo n if and only if a and n are coprime (their greatest common divisor is 1).

Modular inverses are used in various fields including:

• Cryptography (RSA algorithm)
• Error detection and correction codes
• Solving linear congruences
• Finite field arithmetic in computer science

How to Find the Modular Inverse

There are several methods to find the modular inverse of a number:

Brute Force Method

For small values of n, you can test all possible values of x from 1 to n-1 until you find one that satisfies the equation a × x ≡ 1 mod n.

Extended Euclidean Algorithm

This is the most efficient method for finding modular inverses. It finds integers x and y such that:

If gcd(a, n) = 1, then x is the modular inverse of a modulo n. The algorithm works by repeatedly applying the Euclidean algorithm to find the greatest common divisor and then backtracking to find the coefficients.

Fermat's Little Theorem

If n is prime and a is not divisible by n, then the modular inverse of a modulo n is given by:

This method is efficient when n is prime and a is not a multiple of n.

Using the Calculator

Our calculator uses the Extended Euclidean Algorithm to find the modular inverse. Here's how to use it:

1. Enter the number for which you want to find the inverse in the "Number" field.
2. Enter the modulus n in the "Modulo" field.
3. Click the "Calculate" button.
4. The calculator will display the modular inverse if it exists, or indicate that no inverse exists.

Examples

Example 1: Finding the Inverse of 3 Modulo 11

We need to find x such that 3 × x ≡ 1 mod 11.

Using the Extended Euclidean Algorithm:

1. 11 = 3 × 3 + 2
2. 3 = 2 × 1 + 1
3. 2 = 1 × 2 + 0

Working backwards:

1. 1 = 3 - 2 × 1
2. 2 = 11 - 3 × 3
3. 1 = 3 - (11 - 3 × 3) × 1 = 4 × 3 - 11 × 1

Thus, x = 4 is the modular inverse of 3 modulo 11.

Example 2: Finding the Inverse of 7 Modulo 26

We need to find x such that 7 × x ≡ 1 mod 26.

Using the Extended Euclidean Algorithm:

1. 26 = 7 × 3 + 5
2. 7 = 5 × 1 + 2
3. 5 = 2 × 2 + 1
4. 2 = 1 × 2 + 0

Working backwards:

1. 1 = 5 - 2 × 2
2. 2 = 7 - 5 × 1
3. 1 = 5 - (7 - 5 × 1) × 2 = 3 × 5 - 7 × 2
4. 1 = 3 × (26 - 7 × 3) - 7 × 2 = 3 × 26 - 11 × 7

Thus, x = 19 is the modular inverse of 7 modulo 26.