Which values are used for an elgamal cryptosystem public key?

Question

I know that – within the elgamal cryptosystem – the values of $a$ and $b$ are public. But which values are used to create public keys?

Answer 1

Let $p$ be a prime such that the Discrete Logarithm problem in $({\mathbb Z_p}^*,.)$ is infeasible, and let $\alpha \in {\mathbb Z_p}^*$ be a primitive element.

$$\beta =\alpha^a \bmod p$$

The values $p,\alpha , \beta$ are the public key, and $a$ is a private key. For a secret random number $k$ encryption and decryption are as follow:

• $\operatorname{Enc}(x,k)=(\alpha^k \bmod p,x\beta^k \bmod p)$

• $\operatorname{Dec}(y_1,y_2)=y_2\cdot{({y_1}^a)}^{-1}\bmod p$