On the Correlation Between Parity and Modular Polynomials

Abstract

We consider the problem of bounding the correlation between parity and modular polynomials over Zq, for arbitrary odd integer q ≥ 3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bounds on the size necessary to compute parity by depth-3 circuits of certain form. Our technique is based on a new representation of the correlation using exponential sums. Our results include Goldmann's result [Go] on the correlation between parity and degree one polynomials as a special case. Our general expression for representing correlation can be used to derive the bounds of Cai, Green, and Thierauf [CGT] for symmetric polynomials, using ideas of the [CGT] proof. The classes of polynomials for which we obtain exponentially small upper bounds include polynomials of large degree and with a large number of terms, that previous techniques did not apply to. 2 n x∈{0,1} n (−1) f1(x) (−1) f2(x). Our interest in this question is motivated by its relevance to circuit complexity lower bounds. In addition, we believe that the question is also interesting on its own right.

Uniqueness of optimal mod 3 polynomials for parity

“…In fact, it is believed that one can still obtain exponentially small upper bounds on the exponential sum even for polynomials of degree O (log k n) for any k. Some evidence supporting this comes from the fact that such a bound exists for O (log k n) degree symmetric polynomials [3]. Furthermore, the bounds obtained by Gál and Trifonov [6] apply to polynomials of very high degree (although they again do not hold for general polynomials).…”

“…These bounds have been improved in subsequent work by Viola and Wigderson [12] and Chattopadhyay [4]. In [6], Gál and Trifonov prove exponentially decreasing upper bounds for special classes of polynomials modulo m.…”

On the Power of Small-Depth Computation

“…Other special classes of polynomials, for example symmetric polynomials, are studied in [CGT96,GT06,BEHL08]. We finally mention that many of the works we discussed, such as Theorems 2, 4, and 7 can be sometimes extended to polynomials modulo m = 2.…”