Math/CS 295: Mathematical Cryptography

Fall 2012


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 prerequisites are minimal. Really, all one needs is a healthy mathematical and computational appetite. The class will be driven by applications and examples.



Late homework will not be accepted. Standard weekly homework assignments, counting for 60% of the grade, will be due on Fridays.

Be sure to show your work and explain how you got your answer. Correct but incomplete answers will only receive partial credit. Part of the beauty of mathematics is in the elegance of its proofs, and one goal of this course is for you to learn to write mathematics excellently.

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.

Plagiarism, collusion, or other violations of the Code of Academic Integrity will be referred to the The Center for Student Ethics and Standards.

[PDF] Homework Submission Guidlines

Homework is due on the same day as the row in which it appears. For an exercise with *, use of a computer is recommended.

Chapter 1: An Introduction to Cryptography
127 Aug(M)Introduction
229 Aug(W)1.1
331 Aug(F)1.2"Code Book", Chapter 1
3 Sep(M)No class, Labor Day
45 Sep(W)Algorithms1.3, 1.4(a), 1.5
57 Sep(F)Sage
610 Sep(M)1.3
712 Sep(W)1.41.2*, 1.9(a), 1.10(a), 1.11, 1.12*
814 Sep(F)1.5
917 Sep(M)1.6"Code Book", Chapter 3
1019 Sep(W)Enigma1.22, 1.24, 1.25*, 1.33, 1.A, 1.B, 1.C
1121 Sep(F)Enigma
1224 Sep(M)Enigma
1326 Sep(W)1.7-1.7.4 video"Code Book", Chapter 4
1428 Sep(F)1.7.5, 1.7.61.D, 1.E
Chapter 2: Discrete Logarithms and Diffie-Hellman
151 Oct(M)2.1, 2.2
163 Oct(W)2.31.40, 1.41, 1.45
175 Oct(F)2.4, 2.5"Code Book", Chapter 6
188 Oct(M)2.6
1910 Oct(W)2.72.3(b)(c), 2.4, 2.6, 2.8, 2.A
2012 Oct(F)2.9
Chapter 3: Integer Factorization and RSA
2115 Oct(M)3.1"Code Book", Chapter 7
2217 Oct(W)3.22.12, 2.16, 2.17(a), 2.17(b)(c)*, 2.28(a), 2.B
2319 Oct(F)3.3
2422 Oct(M)3.4
2524 Oct(W)3.53.1(a), 3.1(e)*, 3.6*, 3.8(b), 3.9(a), 3.10, 3.A, 3.B
2626 Oct(F)3.6
2729 Oct(M)3.7
2831 Oct(W)Enigma
292 Nov(F)3.73.13(a), 3.14(a)(b)*, 3.21(b)*, 3.23(a), 3.25(b)*
Chapter 4: Combinatorics, Probability, and Information Theory
305 Nov(M)4.1
317 Nov(W)4.23.26(e), 3.C
329 Nov(F)4.2.2"Code Book", Chapter 2
3312 Nov(M)4.4
3414 Nov(W)4.54.11(b), 4.15(a), 4.A*, 4.19
3516 Nov(F)AES
19-23 Nov(M-F)No class, Thanksgiving Recess
Chapter 5: Elliptic Curves and Cryptography
3626 Nov(M)5.1
3728 Nov(W)5.1
3830 Nov(F)5.2
393 Dec(M)5.3, 5.4
405 Dec(W)5.5, 5.65.2, 5.6(a), 5.7, 5.8, 5.A*


Computational Resources:

Certain problems will be computational in nature and the use of computer algebra packages is encouraged. Please print out and attach your work.

Sage is free software for algebra and number theory. See the download instructions to install it on your machine.

Alternatively, you can open a Sage worksheet by connecting to (link will only work on campus or on a machine with a VPN client installed); unfortunately, her sister is very sick right now, but as a backup you may use Or just go to the Sage notebook, which is hosted by William Stein at the University of Washington.


Final Cipher Challenge:

There will be no exams in the course. In place of a final exam, there will be a final cipher challenge.

[TeX] [PDF] Final Cipher Challenge (due Thursday, 13 Dec, 10:30 a.m.)



There are additional resources on the 295 Fall 2010 website.

[PDF] Typos in An Introduction to Mathematical Cryptography

The Enigma simulator for Windows.

The Enigma sim manual.

The Enigma simulator for Mac.