## Tuesday, January 20, 2009

### Lecture 3

Lecture 3 covered the following topics: NP vs. coNP and the theme of proof complexity; nondeterministic time hierarchy via lazy diagonalization; Ladner's Theorem statement; Randomized TMs, BPP, RP, ZPP; problems in BPP but not known to be in P (Berlekamp's algorithm vs. factoring cubics mod p) and primality; lack of hierarchy and complete problems for BPP; polynomial identity testing; error amplification for BPP (and statement about randomness reduction); poly-size circuit families; Adleman's Theorem BPP in P/poly.

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Problem set 1 was handed out in today's lecture and is posted on the course webpage. It is due in two weeks on Feb 3.

ReplyDeleteWhen we discussed the problem of finding the roots a_1,a_2,a_3 of a polynomial f(X) = x^3 + b X^2 + c X + d that factors as (X-a_1) (X-a_2) (X-a_3) modulo p, the following question came up: Can we detect (in deterministic polynomial time) if f(X) has this property?

ReplyDeleteThe answer is Yes, and method is very simple: Compute gcd(f(X),X^p-X). If the result equals f(X), f(X) splits into 3 such factors, otherwise not.