Johan Håstad

A Pseudorandom Generator from any One-way Function

Pseudorandom generators are fundamental to many theoretical and applied aspects of computing. We show how to construct a pseudorandom generator from any one-way function. Since it is easy to construct a one-way function from a pseudorandom generator, this result shows that there is a pseudorandom generator if and only if there is a one-way function.

Some optimal inapproximability results

We prove optimal, up to an arbitrary ε > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.

Almost optimal lower bounds for small depth circuits

We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when the depth is restricted to k 1. Our Main Lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely. Warning: Essentially this paper has been published in Advances for Computing and is hence subject to copyright restrictions. It is for personal use only.

Simple Constructions of Almost k‐wise Independent Random Variables

We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1)) (log log n + k/2 + log k + log 1/ϵ), where ϵ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ϵ < 1/(k log n)). An additional advantage of our constructions is their simplicity.

Clique is hard to approximate within n/sup 1-/spl epsiv//

The author proves that unless NP=coR, Max Clique is hard to approximate in polynomial time within a factor n/sup 1-/spl epsiv// for any /spl epsiv/>0. This is done by, for any /spl delta/>0, constructing a proof system for NP which uses /spl delta/ amortized free bits. A central lemma, which might be of independent interest, gives sufficient conditions (in the form of a certain type of agreement) for creating a global function from local functions certain local consistency conditions.

Does co-NP have short interactive proofs?

Abstract Babai (1985) and Goldwasser,Micali and Rackoff (1985) introduced two probabilistic extensions of the complexity class NP. The two complexity classes, denoted AM[Q] and IP[Q] respectively, are defined using randomized interactive proofs between a prover and a verifier. Goldwasser and Sipser (1986) proved that the two classes are equal. We prove that if the complexity class co -NP is contained in IP[k] for some constant k (i.e., if every language in co -NP has a short interactive proof), then the polynomial-time hierarchy collapses to the second level. As a corollary, we show that if the Graph Isomorphism problem is NP-complete, then the polynomial-time hierarchy collapses.

Tensor rank is NP-complete

Abstract We prove that computing the rank of a three-dimensional tensor over any finite field is NP-complete. Over the rational numbers the problem is NP-hard.

Majority gates vs. general weighted threshold gates

In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following:1.A single threshold gate with weights cannot in general be replaced by a polynomial fan-in unweighted threshold gate of parity gates.2.On the other hand it can be replaced by a depth 2 unweighted threshold circuit of polynomial size. An extension of this construction is used to prove that whatever can be computed by a depthd polynomial size threshold circuit with weights can be computed by a depthd+1 polynomial size unweighted threshold circuit, whered is an arbitrary fixed integer.3.A polynomial fan-in threshold gate (with weights) of parity gates cannot in general be replaced by a depth 2 unweighted threshold circuit of polynomial size.

Solving simultaneous modular equations of low degree

We consider the problem of solving systems of equations

Everything provable is provable in zero-knowledge

P_i (x) \equiv 0(\bmod n_i )i = 1 \cdots k