Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness.

We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field 𝔽, in the blackbox setting. Our algorithm works for every field 𝔽 with | 𝔽 | ≥ d+1, and uses only d log n + log (1/ e) + O(d log log n) random bits to achieve a success probability 1 - e for some e > 0. In the low degree regime that is d ≪ n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC'14 and Bshouty, ITCS'14) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound. We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions.