Simple ElGamal operation 

When using ElGamal designers have two choices. Prime p and primitive element of modp (generator) g may be parts of the system, thus common for all the users, or they may be different for every user. In this simple example the first approach is taken for clarity:

p= and g=

Example: p= 11, g= 2

Some other good values for p with g equal to 2:
-1073741827
-2147483658
-9223372036854775907

If p and g are written common in a given system, then public keys consist of y values calculated just like the y values in Diffie-Hellman protocol. Let assume that Bob wants to send a message to Alice. Then he will need her public key yB. This kay is created by Alice in the following way:

xA is choosen randomly from the range (0, p - 1)

yA is calculated yA = (g^xA)modp

Example:
      xA= 5
      yA = 2^5mod11 = 32mod11 = 10

Press the generate button to generate this values.

xA becomes Alice's secret key, and yA becomes her public key:

Now Bob can use Alice's public key to send her a message m (), where 0 ≤ m ≤ p - 1. The encryption is berformed in the following way:

k is choosen randomly from the range (0, p - 1), new key should be choosen for every block of message

Example: m = 9, k = 3

K is calculated K =  (yA^k)modp 

Example: K = 10^3mod11 = 10

The ciphertext is then the pair (c1; c2), where:

c1 = (g^k)modp

c2 = (K * m)modp

[ElGamal suggest other invertible operation may be userd e.g addition modulo p]

Example: 
      c1 = 2^3mod11 = 8mod11 = 8

      c2 = 10*9mod11 = 90mod11 = 2

Press the button below to calculate the ciphertext

In order to decrypt the ciphertext (c1;c2) alice performs the following steps:

-Calulates (c1^p-1-xA)modp=(c1^(-xA))modp

-calculates m
m=(c1^(-xA)*c2)modp

Example: 
      c1^(p-1-xA)mod11 = 8^(11-1-5)mod11 = 8^5mod11 = 32768mod11=10

      m = 10*2mod11 = 20mod11 = 9