| Week | Date | Topic | Comments | Quiz Dates |
| One | 19 and 21 January 2021 | History | Codes and Ciphers: From Queen Mary of Scots to the Navajo Code Talkers of WWII. See lecture notes on Canvas-- this material is not in our textbook. | |
| Two | 26 and 28 January 2021 | Background Number Theory | Hierarchy of numbers; formal definition of division; divisors and proper divisors; prime numbers; GCD (You should review the Euclidean and Extended Euclidean Algorithm on your own in Ch 1); relatively prime numbers; Euler's Phi function; modular arithmetic; equivalence relation | |
| Three | 2 and 4 Feb | Group Theory, Fermat's Little Theorem | Equivlance relations revisited; examples of Groups both finite and infinite; order of an element in a group; order of a group; Theorem 2.9.2; order of a subgroup of a group (Theorem 2.10.9); Grand Finale: proof of Fermat's Little Theorem (needed for RSA later on) | |
| Four | 9 and 11 Feb | Beginning of Encyrption Tools | Encryption Schemes, Alphabets and Words, Permutations, Block Ciphers, Simple Example of Stream Cipher, Affine Ciphers. | Quiz 0 due no later than the end of class on Canvas on Thursday. With your partner, open book, open notes, open friends, open internet. Write up your own solutions! |
| Five | 16 and 18 Feb | Secret Sharing Schemes | Affine and stream ciphers. Secret Sharing Schemes: you should read and review basic properties of square matrices (See review of determinants ). The main scheme we will cover is the Threshold Scheme (two different approaches). There is a Mathematica nobebook on Canvas with a secret sharing tutorial in this week's module. | |
| Six | 23 and 25 Feb | Public Key Cryptography and RSA (Chapter 6) | Global Procedure (a high level view of RSA); tranlsation of plaintext to a numerical message; Encryption process [public key=(n,e)]; Decryption process [private key=(p,q,d)]; False proof of correctness; True Proof of correctness; Digital Signature; Discussion of the security of RSA and its history. Peter Shor's 1994 quantum algorithm that led to the breakage of RSA on any classical computer: that is, integer factorization can be done in polynomial time. | Quiz 1 due no later than the end of class on Canvas on Thursday. With your partner, open book, open notes, open friends, open internet. Write up your own solutions together with your partner. |
| Seven | 2 and 4 March | DES: Selections from Chapter 4 in our textbook. | History of DES; Encryption process; Decryption; Expander function; S-boxes and their output; Key; the function f that takes the modified key and part of the text as input; mulitple Rounds of DES; Present-day lack of Security in DES, which led to the new Encryption Standard, namely AES. Warmup for AES: the mathematics of Fields: Galois Fields, particularly the one of order 256 and its relation to the irreducible polynomial x^8 + x^4 + x^3 + x + 1 with coefficients from the field Z_2. | |
| Eight | 9 and 11 March: Project proposal due on March 9th. | AES | Guest speakers: Dalton, Michael, and Julian. Multiple Key Lengths; 10 Rounds; Input/Output sizes; ByteSubTransformation (non-linear); ShiftRow Transformation (diffuses bits over multiple rounds); MixColumn Transformation (same purpose as ShiftRow); AddRoundKey (XOR this key with the output of the previous layer) | Quiz 2 due no later than the end of class on Canvas on Thursday. With your partner, open book, open notes, open friends, open internet. Write up your own solutions together with your partner. |
| Nine | 16 and 18 March | Elliptic Curve Cryptosystems-- Chapter 16 (and a few warmps) | Finite Fields (again); Discrete Log Problem; ElGammal Cryptosystem; Elliptic Curve Cryptosystems; Elliptic Curves mod N; Representing Plaintext on an elliptic curve; Factoring integers using elliptic curves; An Elliptic Curve ElGammal Cryptosystem; | |
| Ten | 23 and 25 March | Guest Lecture | Topic to be announced | Quiz 3 due no later than the end of class on Canvas on Thursday. With your partner, open book, open notes, open friends, open internet. Write up your own solutions together with your partner. |
| Eleven | 30 March and 1 April | Flipping Coins and Playing Poker over the phone | Tools: Fermat's Little Theorem and Quadratic Residues | |
| Twelve | 6 and 8 April | Project Presentations | ||
| Thirteen | 13 and 15 April | Project Presentations | ||
| null | 19-23 April | Spring Break | ||
| Fourteen | 27 and 29 April | Project Presentations | Fifteen | 4 and 6 May | Project Presentations | Sixteen | 10 May | Office hours on Monday to be determined |