1. What is Modular Multiplicative Inverse?

The modular multiplicative inverse of an integer a modulo m is an integer x such that the product a × x is congruent to 1 modulo m. It exists if and only if a and m are coprime (their greatest common divisor is 1).

2. How Does the Calculator Work?

The calculator uses the Extended Euclidean Algorithm to find integers x and y such that:

When gcd(a, m) = 1, the coefficient x is the modular inverse of a modulo m.

Explanation: The algorithm recursively finds the gcd while keeping track of the coefficients that express the gcd as a linear combination of the original numbers.

3. Importance of Modular Inverse

Details: Modular inverses are crucial in cryptography (especially RSA algorithm), computer algebra systems, and solving linear congruences. They're fundamental in many areas of number theory and discrete mathematics.

4. Using the Calculator

Tips: Enter positive integer values for a and m (m must be greater than 1). The calculator will find x such that a × x ≡ 1 mod m, or indicate if no inverse exists.

5. Frequently Asked Questions (FAQ)

Q1: When does a modular inverse exist?
A: A modular inverse exists if and only if a and m are coprime (gcd(a, m) = 1).

Q2: Can there be more than one inverse?
A: The inverse is unique modulo m. If x is an inverse, then x + km for any integer k is also an inverse.

Q3: What's the time complexity of finding modular inverse?
A: The Extended Euclidean Algorithm runs in O(log min(a, m)) time, making it very efficient.

Q4: How is this used in cryptography?
A: In RSA, modular inverses are used to compute the private key from the public key.

Q5: What if m is prime?
A: When m is prime, every integer a not divisible by m has an inverse, and Fermat's Little Theorem provides an alternative method to find it.