Mathematics 5248 (Cryptology and number theory)

Description for prospective students

This is a junior-senior level course in cryptology: the art of

• making codes for secrecy (cryptography)
• breaking such codes (cryptanalysis).

The class focusses on using ideas from probability, algebra and number theory to explain modern approaches to the design and breaking of codes for encryption of messages and data. The necessary probability, algebra and number theory is developed along the way, and is not required in advance. Precise statements and theorems do come up in the class, but theorem/proof is usually not the main focus.

Topics include modular arithmetic (used heavily), elementary probability, pre-modern encryption techniques/ciphers and attacks on them, along with modern ciphers such as RSA, Diffie-Hellman key exchange, and ElGamal. Methods for primality testing, and factorization are discussed.

Because the course teaches some algebra of modular arithmetic, groups and rings, it is among those that can be counted toward a math major's algebra requirement (part of "Column X") as described here .

To suggest some flavor of Math 5248, a student might learn ...

• a good bit of algebra related to modular arithmetic, such as Fermat's "Little" Theorem, and the Gauss's celebrated Theorem of Quadratic Reciprocity.
• how to attack standard substitution ciphers, e.g. rendering standard newspaper cryptogram puzzles no fun at all,
• how your credit card number is kept safely encrypted, even though it is sent in data packets on the internet that almost anyone can pick up,
• efficient testing for whether a huge number is (likely) prime versus composite,
• factoring of large-but-not-huge numbers in intelligent ways, along with why these methods become inefficient when the number gets huge.

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