Highly resilient correctors for polynomials

Abstract

We consider the problem of correcting programs that compute multivariate polynomials over large finite fields and give an efficient procedure to transform any program that computes a multivariate polynomial ⨍ correctly on a $\text{1}\text{2+δ}$ fraction of its inputs (δ>0) into a randomized program that computes ⨍ correctly on every input with high probability. This shows that programs computing polynomials are “resilient” to a high fraction of errors. The resilience shown in this paper is better than that of the previously known correction procedures and is close to the information theoretic optimum. The running time of the correction procedure is polynomial in the degree of ⨍, the number of variables, and $\text{1}\text{δ}$, where calls to the incorrect program are assessed a unit cost per call. An important consequence of this result is that the nxn permanent is resilient to errors of up to $\text{1}\text{2}\text{−p(n)}$ for any polynomial p(n).