(4) Elgamal public key cryptosystem (encryption): Start with the prime p = 29 and the primitive root g = 2. Alice sends Bob the public key A = 3. Bob wants to send the message m₁ = 15 to Alice. Bob chooses the random element k = 5. Using the Elgamal public key cryptosystem, compute the pair of numbers (C₁, C₂) that Bob sends to Alice.
(4) Elgamal public key cryptosystem (encryption): Start with the prime p = 29 and the primitive root g = 2. Alice sends Bob the public key A = 3. Bob wants to send the message m₁ = 15 to Alice. Bob chooses the random element k = 5. Using the Elgamal public key cryptosystem, compute the pair of numbers (C₁, C₂) that Bob sends to Alice.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Elgamal Public Key Cryptosystem (Encryption):
Consider the following example to understand the Elgamal public key cryptosystem for encryption:
1. **Initialization**:
- Start with the prime \( p = 29 \) and the primitive root \( g = 2 \).
2. **Public Key Exchange**:
- Alice sends Bob the public key \( A = 3 \).
3. **Message Preparation**:
- Bob wants to send the message \( m_1 = 15 \) to Alice.
4. **Random Element Selection**:
- Bob chooses the random element \( k = 5 \).
5. **Encryption**:
- Using the Elgamal public key cryptosystem, compute the pair of numbers \( (c_1, c_2) \) that Bob sends to Alice.
#### Computation:
To compute the pair \( (c_1, c_2) \):
1. Calculate \( c_1 \):
\[
c_1 = g^k \mod p
\]
Substituting the values:
\[
c_1 = 2^5 \mod 29
\]
\[
c_1 = 32 \mod 29
\]
\[
c_1 = 3
\]
2. Calculate \( c_2 \):
\[
c_2 = m_1 \cdot A^k \mod p
\]
Substituting the values:
\[
c_2 = 15 \cdot 3^5 \mod 29
\]
\[
3^5 = 243
\]
\[
243 \mod 29 = 11
\]
\[
c_2 = 15 \cdot 11 \mod 29
\]
\[
c_2 = 165 \mod 29
\]
\[
c_2 = 20
\]
So, the pair of numbers \( (c_1, c_2) \) that Bob sends to Alice are:
\[
(c_1, c_2) = (3, 20)
\]
### Summary:
- Prime \( p \): \( 29 \)
- Primitive Root \( g \): \( 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa332eac-d846-4704-9340-0a50b86bfcea%2Fd89a8063-3f3b-4107-aae3-8a03352fb9d8%2Fmeha2op_processed.png&w=3840&q=75)
Transcribed Image Text:### Elgamal Public Key Cryptosystem (Encryption):
Consider the following example to understand the Elgamal public key cryptosystem for encryption:
1. **Initialization**:
- Start with the prime \( p = 29 \) and the primitive root \( g = 2 \).
2. **Public Key Exchange**:
- Alice sends Bob the public key \( A = 3 \).
3. **Message Preparation**:
- Bob wants to send the message \( m_1 = 15 \) to Alice.
4. **Random Element Selection**:
- Bob chooses the random element \( k = 5 \).
5. **Encryption**:
- Using the Elgamal public key cryptosystem, compute the pair of numbers \( (c_1, c_2) \) that Bob sends to Alice.
#### Computation:
To compute the pair \( (c_1, c_2) \):
1. Calculate \( c_1 \):
\[
c_1 = g^k \mod p
\]
Substituting the values:
\[
c_1 = 2^5 \mod 29
\]
\[
c_1 = 32 \mod 29
\]
\[
c_1 = 3
\]
2. Calculate \( c_2 \):
\[
c_2 = m_1 \cdot A^k \mod p
\]
Substituting the values:
\[
c_2 = 15 \cdot 3^5 \mod 29
\]
\[
3^5 = 243
\]
\[
243 \mod 29 = 11
\]
\[
c_2 = 15 \cdot 11 \mod 29
\]
\[
c_2 = 165 \mod 29
\]
\[
c_2 = 20
\]
So, the pair of numbers \( (c_1, c_2) \) that Bob sends to Alice are:
\[
(c_1, c_2) = (3, 20)
\]
### Summary:
- Prime \( p \): \( 29 \)
- Primitive Root \( g \): \( 2
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