# Pre-Class Assignments, Math 202 Cryptography Calculus I, Fall 2020

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

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

All chapter references are to the text Understanding Cryptography by Paar and Pelzl.

### For Sunday August 23

#### Modular arithmetic and affine ciphers

• Chapter 1, pp 1 - 23, although you can demphasize Section 1.4.2 Integer Rings
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse. ~13 minutes total
• Week 1: Overview
• Week 1: Modular Arithmetic

#### Pre-Class Questions

• No questions to turn in for the first week.
If you didn't see this before Monday's class, be sure to complete before Wednesday.

### Due Sunday August 30 @ midnight

• Chapter 2: through Section 2.2.2 The One-Time Pad, pp 29 - 38
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse. ~14 minutes total
• Week 2: Encoding messages as numeric values
• Week 2: A few notes on stream ciphers
• Week 2: Note on the size of the key space
• Download the notebook aug31.nb and open it in Mathematica. Follow the instructions to work through the notebook. You'll use this to answer the last two Pre-Class Questions.
• In case Mathematica is being tempermental for you, here's a pdf of the notebook so that you can at least take a look at the content.

#### 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. Use the Mathematica notebook to encrypt the message "Verbing weirds language" using a Caesar cipher with a shift of k = 17. (You should remove any spaces). What is the ciphertext?
4. Use the Mathematica notebook to decrypt the message "CNAVXVSZXFGECBLXGEDXOBISNAXABUNPAIPAVPYSBEUFIIPVBOLPAISBD" using an affine cipher with a key of k= = (23, 13). What is the plaintext?

### Due Sunday September 6 @ midnight

#### AES and finite fields

• Chapter 3: Section 3.1, pp 55 - 58, and Section 3.8 & 3.9, pp 81 - 82
• Chapter 4: Sections 4.1 - 4.3, pp 87 - 99
Don't get boggged down in the notation in Section 4.3. We'll hit the high points we need in the videos and in class.
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse
• For general background, History/Intro to AES (~2:00 - ~18:00) by Christof Paar
• AES explained as a Flash animation (~5 minutes)
• Week 3: Clarification on Problem 3.12 from Problem Set #2
• Week 3: GCDs and multiplicative inverses mod m
• Week 3: Introduction to AES and finite fields

#### Pre-Class Questions

1. Does an affine cipher perfom confusion? Why or why not?
2. What is the purpose of diffusion in encryption algorithm?
3. Let f(x)=x 2 + 1 and g(x)=x+1 be polynomials in ℤ2[x]. What is f(x) ∙ g(x) ?

### Due Sunday September 13 @ midnight

#### AES Data path and key path

##### To Watch
Links in "Echo360 and other course videos" topic in onCourse

#### Pre-Class Questions

1. Perform the following matrix multiplication and addition: $\left( \begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{matrix} \right) \cdot \left( \begin{matrix} 0 \\ 1 \\ 1 \end{matrix} \right) + \left( \begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \right) \mod \ 2$
2. In which layer(s) of AES does confusion occur? Why?
3. In which layer(s) of AES does confusion NOT occur? Why?

### Due Sunday September 20 @ midnight

#### Public key cryptography and RSA

• Chapter 6: Sections 6.1-6.3, pp 149 - 157
• Chapter 7: Sections 7.1-7.3, pp 173 - 179
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse

#### 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?
4. For this question, we'll use the notation from the text for the RSA keys.
Bob publishes the public key kpub=(n,e)=(25021,3).
1. If Alice uses Bob's public key to encrypt the message x=1315, what is the ciphertext y?
2. Bob's private key is kpr=d=16467. Decrypt the ciphertext y from part (a) and verify that you obtain the correct plaintext.

### Due Sunday September 27 @ midnight

#### The Euclidean Algorithm and Fast Exponentiation

• Chapter 7: Section 7.4, pp 179 - 182
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse
No questions to submit today since you shoud be prepping for Exam 1.

### Due Sunday October 4 @ midnight

#### The Prime Number Theorem and RSA Digital Signatures

##### To Watch
Links in "Echo360 and other course videos" topic in onCourse
• Introduction to Digital Signatures by Jeff Suzuki (~5 minutes)
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.

#### Pre-Class Questions

1. Based on the Prime Number Theorem, approximately how many primes are less than 21024?
2. What is the purpose of a digital signature?
3. 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?

### Due Sunday October 11 @ midnight

#### Diffie-Hellman Key Exchange and Hash Functions

• Chapter 8: Section 8.1, pp 205 - 208
• Chapter 11: Section 11.1, pp 293 - 296, and the list of Properties of Hash Functions at the end of Section 11.2, pg 303
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse

#### 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?
3. What is one purpose of a hash function?

### Due Sunday October 18 @ midnight

#### The Discrete Log Problem

• Re-read the Tutorial Problems from last week to remind yourself of the terminology introduced there.
• Chapter 8: Skim Section 8.2. 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 $$\mathbb{Z}_{31}^*$$?
3. Explain the connection between questions 1 & 2.

### Due Sunday October 25 @ midnight

#### Elgamal Digital Signatures

• Section 10.3: The Elgamal Digitial Signature Scheme. Focus on 10.3.1 and 10.3.2, pp 270 - 273
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse
• Week 10: Elgamal digital signatures

#### Pre-Class Questions

1. What is the hard math problem underlying Elgamal digital signatures? Where does this problem appear?
2. What is the purpose of the ephemeral key kE used in an Elgamal digital signature?
3. If Bob receives a message signed with Elgamal, (x,(r,s)) from Alice, what values does Bob need to verify the signature?

### Due Sunday November 1 @ midnight

#### The Digital Signature Algorithm

• Chapter 10: Section 10.4, pp. 277 - 284
##### To Watch
Links in "Echo360 and other course videos" topic in onCourse
• Week 11: The Digital Signature Algorithm

#### 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? How does this compare to Elgamal signatures?
3. The text suggests using SHA-1 for the hash in DSA. Do a little research. Is this hash function still recommended?

### Due Sunday November 8 @ midnight

#### Counter Modes, M-I-T-M Attacks, and TLS

##### To Watch
Links in "Echo360 and other course videos" topic in onCourse

#### Pre-Class Questions

None for today since I was so deliquent in getting this posted.