### Elgamal Public Key Cryptosystem (Encryption)**Introduction:**In this example, we will perform an encryption using the Elgamal public key cryptosystem. Here are the given parameters:- Prime \( p = 17 \)- Primitive root \( g = 3 \)- Alice's public key \( A = 5 \)- Bob’s message to Alice \( m = 13 \)- Random element chosen by Bob \( k = 2 \)**Steps:**1. **Setup:** Bob wants to send the message \( m = 13 \) to Alice. To do this, he will use the Elgamal encryption method.2. **Step 1: Calculate \( c_1 \):** \[ c_1 = g^k \mod p \] Substitute the values: \[ c_1 = 3^2 \mod 17 = 9 \]3. **Step 2: Calculate \( c_2 \):** \[ c_2 = m \cdot A^k \mod p \] Substitute the values, and calculate \( A^k \mod p \): \[ A^k = 5^2 = 25 \mod 17 = 8 \] Then, \[ c_2 = 13 \cdot 8 \mod 17 = 104 \mod 17 = 2 \]4. **Result:** The pair of numbers \( (c_1, c_2) \) that Bob sends to Alice is: \[ (c_1, c_2) = (9, 2) \]**Note:**This problem only requires the encryption part of the Elgamal cryptosystem. Therefore, the decryption step has been omitted.By following these steps, Bob successfully encrypts his message using the Elgamal public key cryptosystem.

Here below i explain to find (c1,c2):…

(encryption): Elgamal public key cryptosystem Start with the prime p 17 and the primitive root g = 3. = Alice sends Bob the public key A = 5. Bob wants to send the message m = = 13 to Alice. Bob chooses the random element k = 2. Using the Elgamal public key cryptosystem, compute the pair of numbers (C₁, C₂) that Bob sends to Alice. Note: Do not work out how Alice computes the plaintext message m from the ciphertext (C₁, C₂).

(encryption): Elgamal public key cryptosystem Start with the prime p 17 and the primitive root g = 3. = Alice sends Bob the public key A = 5. Bob wants to send the message m = = 13 to Alice. Bob chooses the random element k = 2. Using the Elgamal public key cryptosystem, compute the pair of numbers (C₁, C₂) that Bob sends to Alice. Note: Do not work out how Alice computes the plaintext message m from the ciphertext (C₁, C₂).

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

### Elgamal Public Key Cryptosystem (Encryption)

**Introduction:**
In this example, we will perform an encryption using the Elgamal public key cryptosystem. Here are the given parameters:

- Prime \( p = 17 \)
- Primitive root \( g = 3 \)
- Alice's public key \( A = 5 \)
- Bob’s message to Alice \( m = 13 \)
- Random element chosen by Bob \( k = 2 \)

**Steps:**

1. **Setup:**
Bob wants to send the message \( m = 13 \) to Alice. To do this, he will use the Elgamal encryption method.

2. **Step 1: Calculate \( c_1 \):**
\[
c_1 = g^k \mod p
\]
Substitute the values:
\[
c_1 = 3^2 \mod 17 = 9
\]

3. **Step 2: Calculate \( c_2 \):**
\[
c_2 = m \cdot A^k \mod p
\]
Substitute the values, and calculate \( A^k \mod p \):
\[
A^k = 5^2 = 25 \mod 17 = 8
\]
Then,
\[
c_2 = 13 \cdot 8 \mod 17 = 104 \mod 17 = 2
\]

4. **Result:**
The pair of numbers \( (c_1, c_2) \) that Bob sends to Alice is:
\[
(c_1, c_2) = (9, 2)
\]

**Note:**
This problem only requires the encryption part of the Elgamal cryptosystem. Therefore, the decryption step has been omitted.

By following these steps, Bob successfully encrypts his message using the Elgamal public key cryptosystem.

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