### Part 1: Deriving a Key Using Diffie-Hellman ParametersGiven the following Diffie-Hellman parameters, derive a key for Alice (A) and Bob (B). Show all of your steps.#### Parameters:- \( q = 11 \) (a prime number)- \( \alpha = 2 \) (a primitive root of q)- \( x_A = 5 \) (A's private number)- \( x_B = 8 \) (B's private number)#### Steps to Derive the Key:1. **Compute A's Public Key (\( y_A \))**: \[ y_A = \alpha^{x_A} \mod q \] Substitute the given values: \[ y_A = 2^5 \mod 11 \] Calculating \( 2^5 \): \[ 2^5 = 32 \] Then, compute the modulus: \[ 32 \mod 11 = 10 \] Thus, \( y_A = 10 \).2. **Compute B's Public Key (\( y_B \))**: \[ y_B = \alpha^{x_B} \mod q \] Substitute the given values: \[ y_B = 2^8 \mod 11 \] Calculating \( 2^8 \): \[ 2^8 = 256 \] Then, compute the modulus: \[ 256 \mod 11 = 3 \] Thus, \( y_B = 3 \).3. **Compute the Shared Secret Key (\( K \))**: Both Alice and Bob can compute the shared secret key separately using each other's public keys. - **Alice computes \( K \) using Bob's public key (\( y_B \)):** \[ K_A = y_B^{x_A} \mod q \] Substitute the values: \[ K_A = 3^5 \mod 11 \] Calculating \( 3^5 \): \[ 3^5 = 243 \] Then, compute the modulus: \[ 243 \mod 11 =

In this question we have been given with the diffle-hellman parameters, we need to derive key for…

Part 1: Given the following Diffie-Hellman parameters, derive a key for Alice (A) and Bob (B). Show all of your steps. q=11 a = 2 X₁ = 5 X₂ = 8 (a prime number) (a primitive root of q) (A's private number) (B's private number)

Part 1: Given the following Diffie-Hellman parameters, derive a key for Alice (A) and Bob (B). Show all of your steps. q=11 a = 2 X₁ = 5 X₂ = 8 (a prime number) (a primitive root of q) (A's private number) (B's private number)

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

### Part 1: Deriving a Key Using Diffie-Hellman Parameters

Given the following Diffie-Hellman parameters, derive a key for Alice (A) and Bob (B). Show all of your steps.

#### Parameters:
- \( q = 11 \) (a prime number)
- \( \alpha = 2 \) (a primitive root of q)
- \( x_A = 5 \) (A's private number)
- \( x_B = 8 \) (B's private number)

#### Steps to Derive the Key:

1. **Compute A's Public Key (\( y_A \))**:
\[
y_A = \alpha^{x_A} \mod q
\]
Substitute the given values:
\[
y_A = 2^5 \mod 11
\]
Calculating \( 2^5 \):
\[
2^5 = 32
\]
Then, compute the modulus:
\[
32 \mod 11 = 10
\]
Thus, \( y_A = 10 \).

2. **Compute B's Public Key (\( y_B \))**:
\[
y_B = \alpha^{x_B} \mod q
\]
Substitute the given values:
\[
y_B = 2^8 \mod 11
\]
Calculating \( 2^8 \):
\[
2^8 = 256
\]
Then, compute the modulus:
\[
256 \mod 11 = 3
\]
Thus, \( y_B = 3 \).

3. **Compute the Shared Secret Key (\( K \))**:
Both Alice and Bob can compute the shared secret key separately using each other's public keys.

- **Alice computes \( K \) using Bob's public key (\( y_B \)):**
\[
K_A = y_B^{x_A} \mod q
\]
Substitute the values:
\[
K_A = 3^5 \mod 11
\]
Calculating \( 3^5 \):
\[
3^5 = 243
\]
Then, compute the modulus:
\[
243 \mod 11 =

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