Self-correcting polynomial programs

The theory of self-testing/correcting programs by random self-reducibility (RSR) [1, 2, 5] is a novel and powerful tool to approach the problem of program correction. It allows real-time testing and correcting of programs. Moreover, it allows the simultaneous checking of the hardware and software without requiring any knowledge of the implementation of a program. Basically, a function is random self-reducible of order k if its value at a given point can be efficiently reconstructed from its evaluation a t k random points. The research conducted in this domain focuses on two main issues: the enlargement of the set of random self-reducible functions and the reduction of the complexity of self-correction schemes based on RSR. In this paper, we are interested in reducing the complexity of the self-correction scheme for mulfi-variate polynomials. We investigate two methods to achieve the complexity reduction: 1) by reducing the order of RSR in some special cases (e.g., studying polynomials defined over extension fields or introducing some determinism in the choice of the queries) and 2) by substituting a new generic self-correction algorithm to the simple majority vote. In Section 2, we recall some known results on the RSR of polynomials. In Section 3, we give a lower bound of RSR of polynomials. In Section 4, we introduce new random self-reducibility formulas. In Section 5, we introduce a new algorithm for the self-correction of polynomial programs. The proofs of the theorems can be found in [3].