Pre-Class Assignments
Math 202 Cryptography, Fall 2024

This page uses MathJax to display mathematical notation, so please let me know if any part isn't clear.

All section numbers refer to Understanding Cryptography: From Established Symmetric and Asymmetric Ciphers to Post-Quantum Algorithms, Second Edition by Paar, Pelzl, and Güneysu

There are a few other resources that you might find useful for reference, including

• The textbook website
• Christof Paar's YouTube channel of lectures for the first edition of the text

Be sure to check back, because this page will be updated often during the semester.

---

Week 1: Before first class meeting on Wednesday August 28

Welcome to Cryptography!
Modular arithmetic

To Read

• Forward: pp vii-viii
• Chapter 1: Section 1.1, pp 1 - 4

Optional videos from Fall 2020

• Week 1: Overview - Link in Canvas
• Week 1: Modular Arithmetic - Link in Canvas

No questions to submit for the first week.

Remember to fill out the Background Questionnaire linked from Canvas.

---

Week 2: Due Tuesday September 3 @ 11:59 pm

Encoding messages as numeric variables

To Read

• Chapter 1: Finish the chapter, pp 4 - 27, although you can de-emphasize Section 1.4.2 Integer Rings

Optional videos from Fall 2020

• Week 2: Encoding messages as numeric values - Link in Canvas

Pre-Class Questions

1. Look up a Unicode UTF-8 table. What is the UTF-8 encoding for the leter 'b'? Give the decimal, hex, and UTF-8 Hex values.
2. Repeat the previous question for '\( \alpha \)', the Greek letter alpha.
3. What is the total number of possible keys for the affine cipher described in Section 1.4.4? Explain.
4. Why is Kerckhoff's Principle a desirable property for a cryptosystem?

Submit answers through Canvas

---

Week 3: Due Tuesday September 10 @ 11:59 pm

One time pads & stream ciphers
Introduction to AES

To Read

• Chapter 2: through Section 2.2.2 The One-Time Pad, pp 37 - 46
• Chapter 3: Section 3.1, pp 73 - 76, and Section 3.8 & 3.9, pp 103 - 105
• Chapter 4: Sections 4.1 - 4.2, pp 111 - 114 (Be sure to look at Figure 4.2 on pg 115, too)

Optional videos from Fall 2020

• Week 3: A few notes on stream ciphers - Link in Canvas
• Week 3: Note on the size of the key space - Link in Canvas
• AES explained as an interactive HTML5 animation

Pre-Class Questions

1. Encrypt the plaintext x = 1101 using a stream cipher with key stream s = 0111.
2. Why are one-time pads unconditionally secure?
3. What is the purpose of confusion in an encryption algorithm?
4. What is the purpose of diffusion in an encryption algorithm?

Submit answers through Canvas

---

Week 4: Due Sunday September 15 @ 11:59 pm

AES and finite fields

To Read

• Chapter 4: skim Sections 4.3 - 4.5, pp 114 - 140. Don't get bogged down in too many of the details in these sections. We'll hit the high points we need in class.
• Chapter 4: Sections 4.6 - 4. 8, pp 140 - 142

Optional videos from Fall 2020

• Week 4: Introduction to AES and finite fields - Link in Canvas

Pre-Class Questions

1. Let f(x)=x ² + x + 1 and g(x)=x+1 be polynomials in ℤ₂[x]. What is f(x) ∙ g(x) ?
2. In which layer(s) of AES does confusion occur? Why?
3. Why does diffusion occur in the ShiftRows sublayer of AES?

Submit answers through Canvas

---

Week 5: Due Sunday September 22 @ 11:59 pm

Principles of Public Key Cryptography
RSA

To Read

• Chapter 6: Sections 6.1-6.2, pp 177 - 186
• Chapter 7: Sections 7.1-7.3, pp 205 - 211

Optional videos from Fall 2020

• Asymmetric encryption - Simply explained gives a nice introductory motivation
• How RSA works by F5 DevCentral
  Note that what they call the "totient of n" is what we've called the Euler phi function \( \phi(n) \).

Pre-Class Questions

1. Is nonrepudiation possible with a symmetric cipher? Explain.
2. Give a real-world example where nonrepudiation is desirable.
3. What is the hard math problem underlying the security of RSA?

Submit answers through Canvas

---

Week 6: Before class on Wednesday October 2

RSA & the Euclidean Algorithm

To Read

• Chapter 7: Section 7.4, pp 211 - 215

Optional videos from Fall 2020

• Extended Euclidean Algorithm Example by John Bowers (I like this better than the text's explanation)
• The Euclidean Algorithm: How and Why, Visually by Proof of Concept gives a visual justification for why the Euclidean Algorithm works

No questions to submit this week with Exam 1.

---

Week 7: Due Sunday October 6 @ 11:59 pm

Digital signatures
Hash functions

To Read

• Chapter 10: Sections 10.1-10.2, pp 299 - 312
• Chapter 11: Section 11.1, pp 335 - 338, and the list of Properties of Hash Functions in the box at the end of Section 11.2, pg 345
• Mathematicians Will Never Stop Proving the Prime Number Theorem from Quanta Magazine

Optional videos from Fall 2020

• Introduction to Digital Signatures by Jeff Suzuki
  The process of RSA signatures is very clear, but one slightly weird thing in the video is that it has Bob specifying the message that Alice signs, rather than Alice signing a message that they want to send to Bob.
• Hashing Algorithms and Security by Computerphile

Pre-Class Questions

1. What is the purpose of a digital signature?
2. Suppose Alice and Bob want to exchange an AES key to use for a secure conversation.
   1. If Alice generates the AES key, whose RSA credentials will they use to encrypt the key to send to Bob?
   2. If Alice wants to sign the message containing the encrypted key, whose RSA credentials will they use to sign the message?
3. Based on the Prime Number Theorem, approximately how many primes are less than 2¹⁰²⁴?

Submit answers through Canvas

---

Week 8: Due Tuesday October 15 @ 11:59 pm

Diffie-Hellman Key Exchange

To Read

• Chapter 8: Section 8.1, pp 241 - 244

Optional videos from Fall 2020

• Public key cryptography - Diffie-Hellman Key Exchange by Art of the Problem

Pre-Class Questions

1. What's an advantage of Diffie-Hellman key exchange over using RSA to exchange a key for AES?
2. Suppose Alice and Bob agree on values \(p=37\) and \( \alpha=5 \) for Diffie-Hellman.
   1. If Alice picks a private key of a=9, what will be their public key A?
   2. If Bob picks a private key of b=14, what will be their public key B?
   3. What will be the shared key?

Submit answers through Canvas

---

Week 9: Due Sunday October 20 @ 11:59 pm

The Discrete Log Problem

To Read

• Chapter 8: Skim Section 8.2, pp 244 - 252. There's a lot here, so don't get too bogged down. We'll talk about the major results we need during class.

Pre-Class Questions

1. Solve the discrete log problem \( 3^x \equiv 26 \mod 31 \)
2. What is ord(26) in \( \mathbb{Z}_{31}^* \)?

Submit answers through Canvas

---

Week 10: Due Sunday October 27 @ 11:59 pm

The Digital Signature Algorithm

To Read

• Chapter 10: Section 10.4, pp. 318 - 324

Optional videos from Fall 2020

• Week 10: The Digital Signature Algorithm - Link in Canvas

Pre-Class Questions

1. What is the hard math problem underlying the Digital Signature Algorithm? Where does this problem appear?
2. What are the possible lengths of the message x that can be securely signed by DSA?
3. What is the purpose of the ephemeral key k_E used in DSA?

Submit answers through Canvas

---

Week 11: Due Sunday November 3 @ 11:59 pm

Transport Layer Security
Passkeys

To Read

• Chapter 5: Section 5.1 through 5.1.2, pp 147 - 155 (the videos below will help, too)
• Chapter 14: Sections 14.4 - 14.8, pp 497 - 515
• AES GCM (Advanced Encryption Standard in Galois Counter Mode) (video from Computerphile) is a really nice explanation
• For more information on Transport Layer Security specifically:
  • What is Transport Layer Security (TLS)? (Cloudflare)
  • What Happens in a TLS Handshake? | SSL Handshake (Cloudflare)
• Passkeys: an Overview. Pay extra attention to the section WebAuthn: How does it work under the hood?

Optional videos from Fall 2020

• Cipher Block Chaining Mode (Udacity)
• Counter Mode (Udacity)
  I think this is a super clever, simple idea that uses a block cipher like in AES to create the key used in a stream cipher

No questions to submit this week.

---

Week 12: Due Sunday November 10 @ 11:59 pm

Implications of Quantum Computing

To Read

• Chapter 12: Section 12.1, pp 379 - 386
• What Makes Quantum Computing So Hard to Explain? from Quanta Magazine
• Post-quantum encryption contender is taken out by single-core PC and 1 hour from Arstechnica

No questions to submit this week.

---

Week 13: Before class on Wednesday November 20

Introduction to blockchains

To Read

• Types of Blockchains: PoW, PoS, and Private by Cryptopedia
• This Is What Happens When Bitcoin Miners Take Over Your Town from ProPublica
• Central Bank Digital Currency Development Enters the Next Phase from the IMF

To Watch

• But how does bitcoin actually work? by 3Blue1Brown

---

Week 14: Before class on Monday November 25

Blockchain "mining"

To Read

• Review last week's material

No questions to submit this week.