Richard Moloney

Binary Kloosterman Sums Modulo 256 and Coefficients of the Characteristic Polynomial

Kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory. We use Stickelbergers theorem and the Gross-Koblitz formula to determine the value of the binary Kloosterman sum at a modulo 64, modulo 128, and modulo 256 in terms of coefficients of the characteristic polynomial of a.

Ternary Kloosterman sums modulo 18 using Stickelberger's theorem

A result due to Helleseth and Zinoviev characterises binary Kloosterman sums modulo 8. We give a similar result for ternary Kloosterman sums modulo 9. This leads to a complete characterisation of values that ternary Kloosterman sums assume modulo 18. The proof uses Stickelbergers theorem and Fourier analysis.

Two kinds of division polynomials for twisted Edwards curves

This paper presents two kinds of division polynomials for twisted Edwards curves. Their chief property is that they characterise the n-torsion points of a given twisted Edwards curve. We present recursions for the division polynomials, which differ in their flavour. We prove a uniqueness form for elements of the function field of an Edwards curve. We also present results concerning the coefficients of these polynomials, which may aid computation.