766 research outputs found

    Nonlinarity of Boolean functions and hyperelliptic curves

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    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

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    Let pp and qq be distinct prime numbers, with q≡1(mod12)q\equiv 1\pmod{12}. Let NN be a positive integer that is coprime to pqpq. We prove a formula relating the Hasse--Weil zeta function of the modular curve X0(qN)FqX_0(qN)_{\mathbb{F}_q} to the Ihara zeta function of the pp-isogeny graphs of supersingular elliptic curves defined over Fq‾\overline{\mathbb{F}_q} equipped with a Γ0(N)\Gamma_0(N)-level structure. When N=1N=1, 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

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    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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