1. El Gamal Random R Public
2. El Gamal Public Key Generator Free
3. El Gamal Public Key Generator Download
4. Symmetric Key

• Use the Galois field array function, gf, to implement an ElGamal public key cryptosystem. Key Generation. Find a primitive polynomial and group generator.
• Apr 22, 2013  'ElGamal public-key algorithm (randomized encryption and signature). Signature algorithm The security of the ElGamal signature scheme is based (like DSA) on the discrete.

In the previous lecture, we looked at a public key encryption system that is built from the RSA, or more generally from trapdoors functions. In this lecture, we are going to look at public key encryption schemes that are build from the Diffie-Hellman protocol. So, first recall that a public key encryption system is made up of three algorithms. Use the Galois field array function, gf, to implement an ElGamal public key cryptosystem. Key Generation. Define the polynomial degree, m. M = 15; q = 2^m; Find a primitive polynomial and group generator. Set the random number generator seed to produce a repeatable result.

Prerequisites:

1. Cyclic Groups, Generators
2. Basic Number Theory (Euler's Theorem, Fermat's Little Theorem)

ElGamal Encryption System is an Asymmetric Key Encryption System based on Discrete Logarithm Problem (DLP) and Diffie-Hellman Key Exchange.

It is designed for logging into and executing commands on a remote machine, as well as moving files back and forth between the two machines. SSH has various authentication mechanisms and the most secure is based on keys rather than passwords.

For illustrative purposes, we will consider Alice as the receiver and Bob as the sender.

There are different steps involved while encrypting or decrypting data with ElGamal, let's list them first and then study each of them in detail:

1. Key Generation: Alice generates a public and private key and shares the public key.
2. Encryption: Bob uses Alice's public to encrypt message that he wants to send to Alice, and hence generates a pair of ciphertext (c1, c2). Shares the ciphertext pair.
3. Decryption: Alice then uses her private key to decrypt the ciphertext (c1, c2).

Key Generation

This is the first step in the process of transferring a messages securely, between Alice and Bob. In this step, Alice does the following:

1. Selects a Cyclic Group G of order q and generator g. Note that either the cyclic group should be generated over a safe primep, or the generator g should generate prime-order-subgroups; otherwise it is vulnerable to Small Subgroup Attacks.
2. Generates a random number x such that 1 <= x <= q-1 and assigns it as the private key (Note that q is the group order).
3. Calculates
4. Shares h, g, q as Public Key
5. x is the Private Key which only Alice should know, and that's where the security of the encryption system lies.

Here is a python-2.7 implementation of the above step:

Encryption

Bob receives Alice's Public Key and encrypts the message that he wants to send to Alice as follows:

1. Selects a random number y such that 1 <= y <= q-1. Note that a new value of y is chosen for sending every message. Hence, y is known as ephemeral key.
2. Chooses a message m such that 1 < m < q-1
3. Diffie Hellman Step: Calculates
4. Diffie Hellman Step, also s is known as the shared secret: Calculates
5. Calculates
6. Shares (c₁, c₂) as the ciphertext

El Gamal Random R Public

Here is a python-2.7 implementation of the above step:

Decryption

Alice receives (c₁, c₂), we can write them as:

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1. Alice then calculates the following to get the shared secret s:
2. To get back the message m from c₂, Alice does the following:
   

El Gamal Public Key Generator Download

Here is a python-2.7 implementation of the above step:

Check out the full implementation/example of ElGamal encryption/decryption here

Symmetric Key

References