Meier Andreas The ElGamal Cryptosystem - Exercises.pdf

The ElGamal Cryptosystem
Exercises
Andreas V. Meier
March 30, 2005
This exercises will help you to understand the Elgamal Cryptosystem from two sides.
At the beginning, you will create a keypair (yours). Next, you take the sending side and
send yourself a message. This part can also be done with a colleague, where you publish
your public keys to eachother and send an encrypted message back.
In any case, you have received an encrypted message that you can try to decipher. In
the second exercise, you are given a keypair and a message that has been sent to you.
You are now asked to decipher the message.
In the last exercise you will take the role of the eavesdropper, who has intercepted a
message and wants to get to the plaintext.
Please note that, for the sake of simplicity, the messages here consists of only digits
from 1-10, where the 10 is used as 0. (Including the 0 in the message space would make
no sense obviously.)
1 Key Generation
In this exercise you will generate your public/private Elgamal keypair so you can be sent
encrypted messages.
Prime p: The target alphabet are the numbers from 0 to 9. Please find the smallest
prime that can be used to encrypt messages from this alphabet (representing the
digit 0 as 10).
Generator k: Although there usually exist multiple possible generators, in this exercise
we want to be able to compute the algorithm manually. Please find the lowest
usable generator for the group defined by your prime p.
Private key a: Please select now a private key a that fullfills 1 < a ≤ p − 2. (You might
want to choose it such that Figure 1 is helpful.)
Public Key (p, g, g a ): You have now your public key triplet.
1

n 1 2 3 4 5 6 7 8 9 11
2 n 2 4 8 16 32 64 128 256 512 2048
2 n mod 11 2 4 8 5 10 9 7 3 6 2
2 Encryption
Figure 1: Exponent table for p = 11
Take the public key you generated above and encrypt your date of birth specified as
MMDDYYYY to yourself. You may also encrypt the message to your neighbour, using
his public key. If he congratulates you next time you celebrate your brithday, everything
went fine.
Random k: Select a random k with 1 ≤ k ≤ p − 2.
Decryption factor γ: Compute γ = g k mod p
Encryption factor φ: Compute φ = (g a ) k mod p
Replace each occurence of 0 in your message with 10. Now for every digit m i in your
message (with i = |m|), compute δ i = m i ∗ φ.
3 Decryption
You receive the following encrypted message, γ = 7 and δ = {3, 6, 6, 2, 3, 5, 7, 5}, which
contains information about a historical date. The sender used your public key (p, g, g a ) =
(11, 2, 5) for the encryption. You are curious and decrypt the message with the help of
your private key a = 4.
4 Interception
Now we look at the Elgamal Cryptosystem from the eavesdroppers side. You intercepted
an encrypted message, γ = 6 and c = {8, 2, 3, 2, 3, 5, 4, 4}, and you assume it contains
an important date. You acquire the public key (11, 2, 4) of the receiver through public
means. How can you decipher the Message?
5 Solution
4. - Taher Elgamal was born August 18th, 1955. (hint: compute logg g a , it’s easy here.
3. - Linus Torvalds was born December 28th, 1969.
2. - Example: 11301980: m = (1, 1, 3, 0, 1, 9, 8, 0), k = 6: γ = 5, δ = (3, 3, 9, 8, 3, 5, 2, 8)
1. - e.g. a = 3, public key: (11, 2, 8)
2