Example: Alice chooses $p_A=107,{}_A=2,d_A=67,$ and she computes ${}_A=2^{67}-=94(mod107).$ Her public key is $(p_A,{}_A,{}_A)=(2,67,94),$ and her private key is $d_A=67.$

Bob wants to send the message $"$B$"(66$ in ASCII$)$ to Alice. He chooses a random integer $k=45$ and encrypts $M=66$ as $(r,t)=({}_A^k,{}_A^{k}M)-=(2^{45},{94}^{45}66)-=(28,9)(mod107).He$ sends the encrypted message $(28,9)$ to Alice.

Alice receives the message $(r,t)=(28,9),$ and using her private key $d_A=67$ she decrypts to

$t{r}^{-dA}=9*{28}^{-67}-=9*{28}^{106-67}-=9*43-=66(mod107).$

Alice is using the El Gamal Cryptosystem. Her global public elements are:

$q=43($prime number$)$

$=12($generator$)$

Alice's private key $x_A=11$ and her posted public key is $Y_A=26.$

Bob has sent Alice the following ciphertext: $C_1=40,C_2=20.$