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Multiplicative Inverse

The multiplicative inverse of a number, say x, is written as x^-1 or 1/x.

It's a number that when multiplied by x equals 1.

Modular Multiplicative Inverse

In the modular world, a multiplicative inverse of a is x as defined by this formula:

a * x ≡ 1 (mod m)

For example, an inverse of 3 in "mod 11" is 4 because:

3 * 4 ≡ 1 (mod 11)

As another example, an inverse of 5 in "mod 11" is 9 because:

5 * 9 ≡ 1 (mod 11)

Some numbers don't have a multiplicative inverse in a given mod. For example, 2 has no multiplicative inverse in "mod 4" because, in the following equation, there is no x that will result in a congruence:

2 * x ≡ 1 (mod 4)

Who Cares?

This will be important later when we talk about decryption, but it's important to understand that this formula:

y^ax (mod m)

can be reduced to:

y^1 (mod m)

which reduces to:

y (mod m)

Of course, this all assumes that a is the modular multiplicative inverse of x in "mod m".