Incomplete Quadratic Exponential Sums in Several Variables

We consider incomplete exponential sums in several variables of the form S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}} x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree d with coefficients in Z/mZ. We investigate the conjecture, originating in a problem in computational complexity, that for each fixed d and m the maximum norm of S(f,n,m) converges exponentially fast to 0 as n grows to infinity. The conjecture is known to hold in the case when m=3 and d=2, but existing methods for studying incomplete exponential sums appear to be insufficient to resolve the question for an arbitrary odd modulus m, even when d=2. In the present paper we develop three separate techniques for studying the problem in the case of quadratic f, each of which establishes a different special case of the conjecture. We show that a bound of the required sort holds for almost all quadratic polynomials, a stronger form of the conjecture holds for all quadratic polynomials with no more than 10 variables, and for arbitrarily many variables the conjecture is true for a class of quadratic polynomials having a special form.Comment: 31 pages (minor corrections from original draft, references to new results in the subject, publication information

Roots of the derivative of the Riemann zeta function and of characteristic polynomials

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.Comment: 24 pages, 6 figure

A PUBLIC-KEY THRESHOLD CRYPTOSYSTEM BASED ON RESIDUE RINGS

Abstract. We present a generalization of Pedersen’s public-key threshold cryptosystem. Pedersen’s protocol relies on the field properties of Zp. We generalize the protocol so that the calculations can be performed in residue rings that are not necessarily fields. The protocol presented here is polynomialtime equivalent to Pedersen’s