Table of Contents

## What does it mean to be invertible modulo?

It means that a member has an inverse in the modulo. In simple terms, if there is an inverse of a number in a modulus, we call an inverse modulo that number exists. For example: ed = 1 (mod n) [Here ‘=’ means equivalent]

### How do you know if a modular inverse exists?

It can be proven that the modular inverse exists if and only if a and m are relatively prime (i.e. gcd(a,m)=1).

**Can a modulo inverse be negative?**

Modular multiplicative inverse function doesn’t work for negative numbers.

**How do you find the modulo of a negative number?**

Adding a thumb rule to all the answers above: negative number modulo k = k minus positive number modulo k. To find (−n)\%k just find k−(n\%k). Ex: (−144)\%5=5−(144\%5)=5−(4)=1.

## What does it mean to have an inverse modulo?

It means that a member has an inverse in the modulo. In simple terms, if there is an inverse of a number in a modulus, we call an inverse modulo that number exists. For example:

### What is a modular multiplicative inverse of an integer?

In mathematics, particularly in the area of number theory, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as ≡ (),

**How do you know if a division is modular?**

When is modular division defined? Modular division is defined when modular inverse of the divisor exists. The inverse of an integer ‘x’ is a another integer ‘y’ such that (x*y) \% m = 1 where m is the modulus. When does inverse exist? As discussed here, inverse a number ‘a’ exists under modulo ‘m’ if ‘a’ and ‘m’ are co-prime, i.e., GCD of them is 1.

**How do you find the modular multiplicative inverse of k mod 2W?**

One solution is as follows: Use the extended Euclidean algorithm to compute k−1, the modular multiplicative inverse of k mod 2w, where w is the number of bits in a word. This inverse will exist since the numbers are odd and the modulus has no odd factors.