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In this problem, we explore the Diffie-Hellman (DH) public-key encryption algorithm, which allows two entities to agree on a shared key. The DH algorithm makes use of a large prime number p and another large number g less than p. Both p and g are made public (so that an attacker would know them). In DH, Alice and Bob each independently choose secret keys, SA and SB, respectively. Alice then computes her public key, TA, by raising g to SA and then taking mod p. Bob similarly computes his own public key TB by raising g to SB and then taking mod p. Alice and Bob then exchange their public keys over the Internet. Alice then calculates the shared secret key S by raising TB to SA and then taking mod p. Similarly, Bob calculates the shared key S麓 by raising TA to SB and then taking mod p. a. Prove that, in general, Alice and Bob obtain the same symmetric key, that is, prove S = S麓. b. With p = 11 and g = 2, suppose Alice and Bob choose private keys SA = 5 and SB = 12, respectively. Calculate Alice鈥檚 and Bob鈥檚 public keys, TA and TB . Show all work. c. Following up on part (b), now calculate S as the shared symmetric key. Show all work. d. Provide a timing diagram that shows how Diffie-Hellman can be attacked by a man-in-the-middle. The timing diagram should have three vertical lines, one for Alice, one for Bob, and one for the attacker Trudy

Step by step solution

01

Prove S = S'

The shared secret keys for Alice and Bob, denoted by S and S', respectively, should satisfy:1. Alice computes S by \( S = (TB)^{SA} \, \text{mod} \, p \).2. Bob computes S' by \( S' = (TA)^{SB} \, \text{mod} \, p \).Recall TA = \( g^{SA} \, \text{mod} \, p \) and TB = \( g^{SB} \, \text{mod} \, p \).Start by substituting TB into S:\[S = (g^{SB} \, \text{mod} \, p)^{SA} \, \text{mod} \, p = g^{SA \cdot SB} \, \text{mod} \, p\]Similarly, substituting TA into S':\[S' = (g^{SA} \, \text{mod} \, p)^{SB} \, \text{mod} \, p = g^{SB \cdot SA} \, \text{mod} \, p\]Since \(SA \cdot SB = SB \cdot SA\), it follows that \( S = S' \), proving that they compute the same key.

02

Calculate Public Keys TA and TB

Given \( p = 11 \), \( g = 2 \), \( SA = 5 \), and \( SB = 12 \):- Alice's public key TA: \[ TA = g^{SA} \, \text{mod} \, p = 2^5 \, \text{mod} \, 11 = 32 \, \text{mod} \, 11 = 10 \]- Bob's public key TB: \[ TB = g^{SB} \, \text{mod} \, p = 2^{12} \, \text{mod} \, 11 = 4096 \, \text{mod} \, 11 = 4 \]These calculations give the public keys as \( TA = 10 \) and \( TB = 4 \).

03

Calculate Shared Symmetric Key S

Following the exchange of public keys:- Alice computes the shared key S: \[ S = TB^{SA} \, \text{mod} \, p = 4^5 \, \text{mod} \, 11 \] Calculate \( 4^5 \): \[ 4^5 = 1024 \] Now find \( 1024 \, \text{mod} \, 11 \): \[ 1024 \, \text{mod} \, 11 = 9 \]Thus, the shared key is \( S = 9 \).

04

Timing Diagram for Man-in-the-Middle Attack

Illustrate the man-in-the-middle attack:1. **Alice** initiates communication: - Sends \( TA \) to Bob.2. **Trudy (Attacker)** intercepts \( TA \): - Computes and sends \( T_{TrudyA} \) to Bob.3. **Bob** receives \( T_{TrudyA} \), believing it is \( TA \): - Computes \( S' \) with \( T_{TrudyA} \). - Sends \( TB \) to Alice.4. **Trudy** intercepts \( TB \): - Computes and sends \( T_{TrudyB} \) to Alice.5. **Alice** receives \( T_{TrudyB} \), believing it is \( TB \): - Computes \( S \) with \( T_{TrudyB} \).In the end, Trudy can establish two different shared keys, one with Alice and one with Bob, instead of them sharing a key directly.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Public-Key Encryption

Public-key encryption is a fascinating method in cryptography. It allows secure communication over an insecure channel. The Diffie-Hellman (DH) algorithm is one prominent example of this method. Public-key encryption involves using two separate keys: a public key, available to anyone, and a private key, kept secret by its owner.
A sender uses the recipient's public key to encrypt a message, which can only be decrypted by the corresponding private key. This ensures that even if someone intercepts the encrypted message, they cannot read it without the private key.
One key advantage of public-key encryption is facilitating secure key exchange over an open network, like the Internet. The DH algorithm helps two parties to securely share a cryptographic key, which they can later use for encrypting future communications.

Shared Symmetric Key

The shared symmetric key is a critical component of secure communication in the Diffie-Hellman algorithm. Despite using a public-key method to exchange information, the result is a symmetric key, which both parties use.
To understand this better, let's explore the process further:

• Each participant generates a private secret and computes a corresponding public key.
• They exchange their public keys over the network.
• Using the received public key and their private secret, each participant can compute the same shared symmetric key.

This key is considered symmetric because the same key is used for both encryption and decryption by both parties. The main advantage of this approach is the enhanced security it provides, while also being computationally efficient.

Man-in-the-Middle Attack

In cybersecurity, a man-in-the-middle (MITM) attack is a significant threat when using protocols like Diffie-Hellman. During a MITM attack, an attacker intercepts the communication between two parties, potentially altering messages without either party knowing.
Here's how a MITM attack might happen in the Diffie-Hellman exchange:

• The attacker eavesdrops on the communication channel.
• They block and replace each party's public key with their own.
• This results in both parties unknowingly assigning their shared keys to the attacker instead of each other.

Using this method, the attacker can decrypt, modify, or fake messages between the parties involved, significantly comprising security. Understanding and preventing MITM attacks is vital for ensuring secure communication.

Prime Number

Prime numbers play a crucial role in various cryptographic algorithms, including Diffie-Hellman. A prime number is a natural number greater than 1, which has no positive divisors other than itself and 1.
In the context of the Diffie-Hellman algorithm:

• A large prime number, denoted as \(p\), is used as a modulus for computations.
• This number is typically chosen to be large enough to ensure security.
• The property of prime numbers helps secure the algorithm against specific types of mathematical attacks.

Choosing the right prime number is essential for maintaining the robustness of cryptographic methods, preventing attackers from easily deciphering the key exchange.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, known as the modulus. It's an integral part of the Diffie-Hellman key exchange process.
Let's understand modular arithmetic with some basic principles:

• Operations like addition, subtraction, and multiplication are performed as usual, but the result is taken modulo a fixed number \(p\).
• In the DH algorithm, this concept ensures that even large numbers are managed within a fixed range defined by the prime modulus.
• This helps in creating complex keys that are computationally difficult for an attacker to break.

Modular arithmetic provides the backbone for the Diffie-Hellman algorithm, contributing to the difficulty of reversing the key exchange without knowing the secret keys.