Modular Inverse Finder
Calculate modular multiplicative inverses using the Extended Euclidean Algorithm
Modular Inverse Finder
Find the modular multiplicative inverse using the Extended Euclidean Algorithm. Calculate a⁻¹ mod m where a × a⁻¹ ≡ 1 (mod m).
The number to find the inverse of
The modulus (must be coprime with a)
Quick Examples
Understanding Modular Inverse
The modular multiplicative inverse is a fundamental concept in number theory and cryptography. Given integers a and m, the modular inverse of a modulo m is an integer x such that:
a × x ≡ 1 (mod m)Existence Conditions
A modular inverse exists if and only if gcd(a, m) = 1. When this condition is met, we say that a and m are coprime or relatively prime.
Applications
Cryptography
- RSA encryption key generation
- Digital signature algorithms
- Elliptic curve cryptography
- Key exchange protocols
Mathematics
- Solving linear congruences
- Number theory research
- Abstract algebra
- Computational mathematics
Extended Euclidean Algorithm
The Extended Euclidean Algorithm is the most efficient method for finding modular inverses. It computes the GCD and finds Bézout coefficients that express the GCD as a linear combination.