Number theory seminar in memory of F. Momose

Classification of covering curves of hyperelliptic curves over extension fields: odd characteristics.


Tsutomu Iijima


The GHS attack or cover attack is known as a method to map the discrete logarithm problem (DLP) in the Jacobian of a curve $C_0$ defined over the $d$ degree extension $k_d$ of a finite field $k$, to the DLP in the Jacobian of a covering curve $C$ of $C_0$ over $k$. In this talk, we present a classification of cryptographically used elliptic and hyperelliptic curves $C_0 /k_d$ in odd characteristic case which can be attacked by the GHS attack. Our main approach is to use representation of the extension of $Gal(k_d /k)$ acting on covering group $cov(C/\mathbb{P}^1 )$. We classify genus 1,2,3 hyperelliptic curves $C_0 /k_d$ which possess (2, 2, .., 2)-coverings. Explicit defining equations of such curves $C_0 /k_d$ and existential conditions of a model of $C$ over $k$ are also discussed.


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