Math 75: Mathematical Cryptography

Spring 2016


Course Info:



[PDF] Syllabus

We live an information age, with technology increasingly integrated into our daily lives. As a result, the security of our information is of the utmost concern, even as the interconnectedness of the Internet makes our data more vulnerable to attack. The ability to encrypt secrets and to conduct a trusted exchange of digital information, once a subject of interest primarily to governments and the military, is now a matter of necessity for us all.

At the end of the day, the foundation of modern cryptography relies upon the difficulty of solving certain mathematical problems; this course is intended to address them from both a mathematical and algorithmic point of view. We will cover some subset of the following topics: conventional encryption techniques, the Hill cipher, DES and SDES, RSA, the Rijndael cipher, discrete logarithms and the Diffie-Hellman key exchange, and elliptic curve cryptography.

All mathematical objects will be defined, so the essential prerequisite is familiarity with abstract algebra and a healthy mathematical and computational appetite. Some experience with number theory would also be helpful. We will also be writing some simple computer programs in Python.

128 Mar(M)Introduction
230 Mar(W)Shift ciphersHPS 1.3; G 1.1HW 1: [PDF] [TeX]
31 Apr(F)Substitution ciphers, permutations,
statistics of English
HPS 1.1, 5.1; G 2.3, 2.4, 3.1, 3.3S Chapter 1
44 Apr(M)Affine ciphers, Euclidean algorithmG 1.2, 1.4, 7.3, 7.4
56 Apr(W)Algorithms, big OhG 9.1, 9.2, 9.4HW 2: [PDF] [TeX]
67 Apr(R)Sage, ASCII
-8 Apr(F)No class, JV at CUNY
711 Apr(M)Vigenere cipherHPS 5.2; G 1.3S Chapter 2
813 Apr(W)Cryptanalysis of Vigenere cipherHPS 4.1, 4.3, 4.4HW 3: [PDF] [TeX]
915 Apr(F)Block (Hill) ciphers, inverting matricesG 7.5, 8.1, 8.2
1018 Apr(M)EnigmaHPS 1.6S Chapters 3-4
Enigma simulators:
Mac, WWW (choose Wehrmacht I), Windows
1120 Apr(W)EnigmaHW 4: [PDF + Reading] [TeX]
1222 Apr(F)Feistel ciphers, SDES, DESHPS 8.12; G 6.1, 6.2SDES
1325 Apr(M)Rijndael (AES)G 6.3Rijndael animation
1427 Apr(W)Finite fieldsHPS 2.10; G 26.1-26.5HW 5: [PDF] [TeX]
1529 Apr(F)Public key cryptography, RSAHPS 2.1, 3.2S Chapter 6
162 May(M)RSAHPS 3.4
174 May(W)Primality testingHPS 3.3HW 6: [PDF] [TeX]
185 May(R)Sage, attacks on RSAS Chapter 7
196 May(F)Pollard rhoHPS 5.5; G 24.1
209 May(M)Quadratic sieveHPS 3.6; G 25.2, 25.4, 25.5
2111 May(W)Smooth numbersHPS 3.7HW 7: [PDF] [TeX]
2213 May(F)Discrete logsHPS 1.5, 2.2, 2.3; G 7.8
2316 May(M)Diffie-Hellman, Baby step-giant stepHPS 2.6, 2.7, 5.4; G 27.1, 27.2
2418 May(W)Pohlig-Hellman (index calculus), CRTHPS 2.8, 2.9; G 27.3HW 8: [PDF] [TeX]
2520 May(F)Elgamal, signaturesHPS 4.1-4.3
2623 May(M)Elliptic curvesHPS 6.1, 6.2; G 28.3, 28.4, 28.5
2725 May(W)Elliptic curve cryptography,
factoring with elliptic curves
HPS 3.5, 6.3, 6.4, 6.6; G 24.2, 28.1
2827 May(F)Quantum cryptography, wrap-upHPS 6.5, 8.11S Chapter 8
-30 May(M)No class, Memorial Day



The homework assignments will be assigned on a weekly basis and will be posted above. Homework is due in one week; no late homework will be accepted.

Reading out of Singh "The Code Book" is due the day listed (e.g., read Chapter 1 for Friday, 1 April).

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.

[PDF] Homework Submission Guidelines


Final cipher challenge:

A final cipher challenge was assigned in place of a final exam.