# The Mathematics of ElGamal

Let’s learn the basic mathematics behind ElGamal.

## We'll cover the following

The version of the ElGamal cryptosystem also relies on modular arithmetic.

## ElGamal public keys

Recall that each ElGamal public key involves three numbers. The first is a prime **$p$**, while the second number **$g$** has the special property of a *primitive* element. We’ll now explain the significance of this.

### Primitive elements

Let **$p$** be a prime. A number **$g$** between 1 and **$p - 1$** is said to be *primitive* (or is a *primitive element*) modulo **$p$** if the numbers:

$g, g^2, g^3, g^4, g^5, g^6, ... , g^{p-1} \; mod \; p$

are all different. If this is the case, since there are **$p - 1$** of them, they must consist of **$1, 2, 3, ... , p - 1$** in some order.

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