Mathematical Foundations of Crytography

Instructor: Sam Buss

**Date/Time:** Winter Quarter, Monday-Wednesday-Friday. 1:00-2:00.

**Place:** HSS 2152. University of California, San Diego.

**Course announcement **

This one-quarter course will cover the mathematical foundations of cryptography. The tentative course outline includes: introduction to one-way functions and applications to secure communications, descriptions of the usual candidates for secure encryption, pseudo-random number generators, the conversion of weak one-way functions into strong one-way functions, obtaining one-way functions from psuedo-random number generators and obtaining pseudo-random number generators from one-way functions, and other topics as time permits, such as stream and block cryptosystems, DES, trapdoor functions, cryptographic protocols.

**Day
1:** Jeremy Martin. Introduction. One-time pad and
pseudorandom number generators. P and NP.

**Day
2:** Rob Ellis. Feasibility, Randomization, BPP, RP, PP.

**Day
3:** Jason Ribando. P/poly. Function Ensembles.
Definition of pseudorandom number generators.

**Day
4:** Chris Pollett. One-way functions. Examples. Public
input.

**Day
5:** Dell Kronewitter. Pseudo-random number generators are
one-way functions. Definition of weak one-qay functions.
input.

**Day
6:** Mike Mastropietro. From weak one-way functions to one
way functions.

**Day
7:** Tyler McIntosh. Reverse Expansion. Conclusion of
proof of one-way functions from weak one-way functions.

**Day
8:** Jennifer Wagner. Weak one-way permutations. One-way
permutations from weak one-way permutations. Square-roots and
finding non-trivial factors.

**Day
9:** Preeti Mehta. Finding square roots versus finding
non-trivial factors. Next-bit unpredictability.

**Day
10:** Imre Tuba.
Stretching the output of pseudorandom number generators.

**Day
11:** Bill Wood.
Private key stream cryptosystems. Passive attacks. Plaintext
attacks.

**Day
12:** David Meyer.
More on plaintext attacks.
Block-cryptosystems. Definition of pseudorandom function
generators.

**Day
13:** Roland Meyer.
Block cryptosystems based on pseudorandom function generators. Construcing a pseudorandom function generator from a
pseudorandom number generator.

**Day
14:** Christian Gromoll. Trapdoor functions and RSA.

**Day
15:** Tin Yen Lee. Square root extraction. Existence of
pseudorandom number generators.

**Day
16:** Anand Desai. Simple probability. Markov inequality,
Chebychev inequality, Chernoff bounds. Pairwise independent sampling
theorem. Hidden inner product bits.

**Day
17:** David Little. Hidden Bit Theorem. Hidden Bit
Technical Lemma.

**Day
18:** Howard Skogman. Many Hidden Bits.

**Day
19:** Jeremy Martin. Statistical distinguishability.
Computation indistinguishability. Hidden Bit Theorems revisited.
Entropy and Information.

**Day
20:** Anand Desai. Information and Entropy.
Kullback-Liebler inequality.

**Day
21:** Jennifer Wagner. Prefix-free codes. Kraft inequality.
Huffman codes.

**Day
22:** Robert Ellis. Pseudorandom number generators from
one-way functions. Hash functions and one-way hash functions.
The Birthday Attack.

**Day
23:** Tin Yen Lee. Applications of hash functions. The
Birthday Attack again.

**Day
24:** Sam Buss. Thwarting the birthday attack. Blinded
signatures.