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(4) Elgamal public key cryptosystem (encryption): Start with the prime p = 13 and the primitive root g = 2. Alice sends Bob the public key A = 3. Bob wants to send the message m = 10 to Alice. Bob chooses the random element k = 3. 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).

(4) Elgamal public key cryptosystem (encryption): Start with the prime p = 13 and the primitive root g = 2. Alice sends Bob the public key A = 3. Bob wants to send the message m = 10 to Alice. Bob chooses the random element k = 3. 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
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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### Elgamal Public Key Cryptosystem (Encryption) #### Example Problem: Given: - Prime \( p = 13 \) - Primitive root \( g = 2 \) **Steps:** 1. **Public Key Exchange:** - Alice sends Bob the public key \( A = 3 \). 2. **Message Encryption:** - Bob wants to send the message \( m = 10 \) to Alice. - Bob chooses the random element \( k = 3 \). 3. **Compute Ciphertext Pair \( (c_1, c_2) \):** Using the Elgamal cryptosystem, Bob computes the pair \((c_1, c_2)\) to send to Alice. **Note:** The problem requires computing the ciphertext \((c_1, c_2)\) but does not require demonstrating how Alice retrieves the plaintext message \( m \) from \((c_1, c_2)\). To compute the ciphertext: - Calculate \( c_1 \): \[ c_1 = g^k \mod p \] Substituting the values: \[ c_1 = 2^3 \mod 13 = 8 \] - Calculate \( c_2 \): \[ c_2 = m \cdot A^k \mod p \] Substituting the values: \[ c_2 = 10 \cdot 3^3 \mod 13 \] First, compute \( 3^3 \mod 13 \): \[ 3^3 = 27 \quad \text{and} \quad 27 \mod 13 = 1 \] Then: \[ c_2 = 10 \cdot 1 \mod 13 = 10 \] Thus, the pair \((c_1, c_2)\) that Bob sends to Alice is \( (8, 10) \).
Transcribed Image Text:### Elgamal Public Key Cryptosystem (Encryption) #### Example Problem: Given: - Prime \( p = 13 \) - Primitive root \( g = 2 \) **Steps:** 1. **Public Key Exchange:** - Alice sends Bob the public key \( A = 3 \). 2. **Message Encryption:** - Bob wants to send the message \( m = 10 \) to Alice. - Bob chooses the random element \( k = 3 \). 3. **Compute Ciphertext Pair \( (c_1, c_2) \):** Using the Elgamal cryptosystem, Bob computes the pair \((c_1, c_2)\) to send to Alice. **Note:** The problem requires computing the ciphertext \((c_1, c_2)\) but does not require demonstrating how Alice retrieves the plaintext message \( m \) from \((c_1, c_2)\). To compute the ciphertext: - Calculate \( c_1 \): \[ c_1 = g^k \mod p \] Substituting the values: \[ c_1 = 2^3 \mod 13 = 8 \] - Calculate \( c_2 \): \[ c_2 = m \cdot A^k \mod p \] Substituting the values: \[ c_2 = 10 \cdot 3^3 \mod 13 \] First, compute \( 3^3 \mod 13 \): \[ 3^3 = 27 \quad \text{and} \quad 27 \mod 13 = 1 \] Then: \[ c_2 = 10 \cdot 1 \mod 13 = 10 \] Thus, the pair \((c_1, c_2)\) that Bob sends to Alice is \( (8, 10) \).
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