766 research outputs found
Nonlinarity of Boolean functions and hyperelliptic curves
We study the nonlinearity of functions defined on a finite field with 2^m
elements which are the trace of a polynomial of degree 7 or more general
polynomials. We show that for m odd such functions have rather good
nonlinearity properties. We use for that recent results of Maisner and Nart
about zeta functions of supersingular curves of genus 2. We give some criterion
for a vectorial function not to be almost perfect nonlinear
On the Zeta functions of supersingular isogeny graphs and modular curves
Let and be distinct prime numbers, with . Let
be a positive integer that is coprime to . We prove a formula relating the
Hasse--Weil zeta function of the modular curve to the
Ihara zeta function of the -isogeny graphs of supersingular elliptic curves
defined over equipped with a -level
structure. When , this recovers a result of Sugiyama.Comment: minor changes, accepted for publication in Archiv der Mathemati
Quasi-quadratic elliptic curve point counting using rigid cohomology
We present a deterministic algorithm that computes the zeta function of a
nonsupersingular elliptic curve E over a finite field with p^n elements in time
quasi-quadratic in n. An older algorithm having the same time complexity uses
the canonical lift of E, whereas our algorithm uses rigid cohomology combined
with a deformation approach. An implementation in small odd characteristic
turns out to give very good results.Comment: 14 page
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