## Math 75: Mathematical Cryptography

### Spring 2016

Course Info:

Syllabus:

[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.

 1 28 Mar (M) Introduction 2 30 Mar (W) Shift ciphers HPS 1.3; G 1.1 HW 1: [PDF] [TeX] 3 1 Apr (F) Substitution ciphers, permutations, statistics of English HPS 1.1, 5.1; G 2.3, 2.4, 3.1, 3.3 S Chapter 1 4 4 Apr (M) Affine ciphers, Euclidean algorithm G 1.2, 1.4, 7.3, 7.4 5 6 Apr (W) Algorithms, big Oh G 9.1, 9.2, 9.4 HW 2: [PDF] [TeX] 6 7 Apr (R) Sage, ASCII - 8 Apr (F) No class, JV at CUNY 7 11 Apr (M) Vigenere cipher HPS 5.2; G 1.3 S Chapter 2 8 13 Apr (W) Cryptanalysis of Vigenere cipher HPS 4.1, 4.3, 4.4 HW 3: [PDF] [TeX] 9 15 Apr (F) Block (Hill) ciphers, inverting matrices G 7.5, 8.1, 8.2 10 18 Apr (M) Enigma HPS 1.6 S Chapters 3-4Enigma simulators: Mac, WWW (choose Wehrmacht I), Windows 11 20 Apr (W) Enigma HW 4: [PDF + Reading] [TeX] 12 22 Apr (F) Feistel ciphers, SDES, DES HPS 8.12; G 6.1, 6.2 SDES 13 25 Apr (M) Rijndael (AES) G 6.3 Rijndael animation 14 27 Apr (W) Finite fields HPS 2.10; G 26.1-26.5 HW 5: [PDF] [TeX] 15 29 Apr (F) Public key cryptography, RSA HPS 2.1, 3.2 S Chapter 6 16 2 May (M) RSA HPS 3.4 17 4 May (W) Primality testing HPS 3.3 HW 6: [PDF] [TeX] 18 5 May (R) Sage, attacks on RSA S Chapter 7 19 6 May (F) Pollard rho HPS 5.5; G 24.1 20 9 May (M) Quadratic sieve HPS 3.6; G 25.2, 25.4, 25.5 21 11 May (W) Smooth numbers HPS 3.7 HW 7: [PDF] [TeX] 22 13 May (F) Discrete logs HPS 1.5, 2.2, 2.3; G 7.8 23 16 May (M) Diffie-Hellman, Baby step-giant step HPS 2.6, 2.7, 5.4; G 27.1, 27.2 24 18 May (W) Pohlig-Hellman (index calculus), CRT HPS 2.8, 2.9; G 27.3 HW 8: [PDF] [TeX] 25 20 May (F) Elgamal, signatures HPS 4.1-4.3 26 23 May (M) Elliptic curves HPS 6.1, 6.2; G 28.3, 28.4, 28.5 27 25 May (W) Elliptic curve cryptography, factoring with elliptic curves HPS 3.5, 6.3, 6.4, 6.6; G 24.2, 28.1 28 27 May (F) Quantum cryptography, wrap-up HPS 6.5, 8.11 S Chapter 8 - 30 May (M) No class, Memorial Day

Homework:

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.