Pseudorandom generators for low degree polynomials

We investigate constructions of pseudorandom generators that fool polynomial tests of degree d in m variables over finite fields F. Our main construction gives a generator with seed length O(d4 log m(1 + log(d/ )/ log log m) + log |F|) bits that achieves arbitrarily small bias and works whenever |F| is at least polynomial in d, log m, and 1/ . We also present an alternate construction that uses a seed that can be described by O(c2 d8 m6/(c-2) log(d/ ) + log |F|) bits (more precisely, O(c2 d8 m6/(c-2) ) field elements, each chosen from a set of size poly(cd/ ), plus two field elements ranging over all of F), works whenever |F| is at least polynomial in c, d, and 1/ , and has the property that every element of the output is a function of at most c field elements in the input. Both generators are computable by small arithmetic circuits. The main tool used in the construction is a reduction that allows us to transform any "dense" hitting set generator for polynomials into a pseu...

• Type: Conference
• Year: 2005
• Where: STOC
• Authors: Andrej Bogdanov
• Keywords: Algorithms, Field Elements, Length O(d4 Log, Small Arithmetic Circuits, STOC 2005