Randomizing polynomials: A new representation with applications to round-efficient secure computation

Motivated by questions about secure multi-party computation, we introduce and study a new natural representation of functions by polynomials, which we term randomizing polynomials. "Standard" low-degree polynomials over a finite field are easy to compute with a small number of communication rounds in virtually any setting for secure computation. However, most Boolean functions cannot be evaluated by a polynomial whose degree is smaller than their input size. We get around this barrier by relaxing the requirement of evaluating f into a weaker requirement of randomizing f: mapping the inputs of f along with independent random inputs into a vector of outputs, whose distribution depends only on the value of f. We show that degree-3 polynomials are sufficient to randomize any function f, relating the efficiency of such a randomization to the branching program size of f. On the other hand, by characterizing the exact class of Boolean functions which can be randomized by degree-2 polynomials, we show that 3 is the minimal randomization degree of most functions. As an application, randomizing polynomials provide a powerful, general, and conceptually simple tool for the design of round-efficient secure protocols. Specifically, the secure evaluation of any function can be reduced to a secure evaluation of degree-3 polynomials. One corollary of this reduction is that two (respectively, three) communication rounds are sufficient for k parties to compute any Boolean function f of their inputs, with perfect information-theoretic [k-1/3]-privacy (resp., [k-1/2]-privacy), and communication complexity which is at most quadratic in the branching program size of f (with a small probability of one-sided error).