ElGamal example over GF(11) field

 This very simple example of ElGamal with small field. It's highly recommended that use large fields.

• Our group is GF(11) = {1,2,3,4,5,6,7,8,9,10}
• Lets take n = 10 and  α = 2. Bob randomly select b = 5 then α^b =2⁵ = 10.Have public key (n,α,α^b) = (10,2,10) and private key b = 5.
• Alice chooses k = 7 and calculates α^k = 2⁷ = 7.
• Alice looks up α^b = 10 and encodes message as m = 3 then calculates           m(α^b)^k = 3 * 10⁷ = 8.
• Alice sends (α^k,mα^bk) = (7,8).
• Bob Calculates mα^bk((α^k)^b)^-1 = 8(7⁵)^-1 = 3.
• Thus Bob receives the message, 3, as sent by Alice

You need some idea about finite fields and modular arithmetic