9 ElGamal Method

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ElGam

alCryptos

ystem

BySafar M. Asaad

Lecture 7 4thYear


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Outlines

! An introduction to ElGamal Cryptosystem.! ElGamal Key generation! ElGamal encryption! ElGamal decryption

!ElGamal encryption and decryption examples.

! Home Work.! Summary! References.


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ElGamal Method! In 1984, Taher ElGamal introduced the concept of ElGamal

Encryption.

! Asymmetric key encryption algorithm.! We will also take a look at the ElGamal public key cipher

system for a number of reasons:o To show that RSA is not the only one public key system.o To exhibit a public key system based on a different one way

function.


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ElGamal Key generation

! Choose a large prime p." pmust be large enough to hide the message.

! Generate a special numberg.! The number gmust be between 1 and p 1, of the Group Z*p.

"The number gmust be a primitive element modulo p.

! Choose a private key x." This can be any number bigger than 1 and smaller than p-2

! Compute public key yfrom x,pand g"

The public key yis graised to the power of the private keyxmodulop. In other words:

y = gxmod p


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ElGamal Key generation: Example! Letp= 23! Select a primitive element g= 11! Choose a private key x= 6! Compute y= 116(mod 23)! = 9

! Public key is 9! Private key is 6


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ElGamal encryption! After receiving the Public key by the message sender as triplet

(p, g, y).

! The first job is to represent the plaintext as a series of numbers(m) modulo p.

! Then for each message block mshe takes the following steps:1. She generates a random number kbetween 1 and p-1.2. Compute two values C1and C2, where

C1= gkmod p, and C2= My

kmod p

3. Send the ciphertext C, which consists of the two separatevalues C1 and C2. The cipher text is represented as a pair

(C1,C2).


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ElGamal encryption: Example! To encrypt M = 10 using Public key 9

1. Generate a random number k = 3.2. Compute:

" C1= 113mod 23 = 20" C2= 10 x 93mod 23

= 10 x 16 = 160 mod 23 = 22

3. Ciphertext C = (20 , 22 )! Do the same steps for the remaining blocks.


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ElGamal decryption! C1= gkmod p C2= Mykmod p

1. The receiver begins by using their private key x totransform C1into something more useful:

C1x= (gk)xmod p

NOTE: C1x= (gk)x= (gx)k= (y)k= ykmod p

2. This is a very useful quantity because if you divide C2by ityou get M. In other words:

C2/ yk

= (Myk

) / yk

= M mod p


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ElGamal decryption: Example

! To decrypt C = (20 , 22 )1. Compute 206= 16 mod 232. Compute 22 / 16 = 10 mod 233. Plaintext = 10

! Note:! C1x= ykmod p.! C2/ yk= M mod p.

!


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ElGamals Security

! The ElGamel public key cryptosystem is based upon the difficultyof solving the discrete logarithm problem (DLP) which is as

follows:

! Given a prime p and values g and y, find x such thaty = gxmod p

! For a small value of p, it is easy to solve a DLP by trial and erroror exhaustive search.

! For example, given p = 11, g = 2 and y = 9, we can try differentvalues of x until we reach the correct solution for 2x mod 11 = 9.

! However, for a large value of p, i.e., if p has around 100 decimaldigits, then it is not possible to solve a DLP using current

technology.


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Comparing RSA and ElGamel

RSA

! Secur i ty based on thedifficulty of the factorisation

problem.

! The ciphertext is just onevalue c which is roughly the

same size as the message m.

! T h e e n c r y p t i o n a n ddecryption algorithms are the

s a m e ( m o d u l a r

exponentiation).

El Gamel

! Security based on thedifficulty of the discrete log

problem.

! The ciphertext is two valuesc and r and so is twice the

size of the message m.

! T h e e n c r y p t i o n a n ddecryption algorithms are

different (although both takeabout the same time to

perform).


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Home Work

! What is Steganography?! Uses of Steganography?! Its Relation to Cryptography?! What are the types of Steganography?


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Summary

We have covered:! An introduction to ElGamal Cryptosystem.

! ElGamal Key generation with example.! ElGamal encryption and decryption.

! ElGamal encryption and decryption examples.


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References

" William Stallings, Cryptography and Network Security, 3rdEdition, Prentice Hall, 2003.

" Behrouz Forouzan, Cryptography and Network Security.McGraw Hill.

"Kaufman, Perlman and Speciner, Network Security: PrivateCommunication in a Public World. Pearson - Prentice Hall.

" Lecture notes, Dr Keith Martin, Introduction to Cryptographyand Security Mechanisms

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9 ElGamal Method