Number theory seminar in memory of F. Momose
Hyperelliptic curve cryptosystems are based on the discrete logarithm problem (DLP) of hyperelliptic curves defined over finite fields.
Presenters
Mahoro Shimura
Abstract
Let $k$ be a finite field, and let $k_d$ be its extension field of degree $d$. If an hyperelliptic curve $C_0$ defined over $k_d$ has a covering curve $C$ defined over $k$, we can transfer the DLP of $J(C_0 )$ into the DLP of $J(C)$ ($J(C)$ is the Jacobian variety of $C$). When the latter is easier than former, this attack (so-called GHS attack) works and we call $C_0$ is a weak curve. In this talk, we present a classification of hyperelliptic curves defined over $k_d$ with even characteristic, which have $(2, 2, ...2)$-coverings over $k$ therefore can be attacked by GHS attack. In particular, we show the density of the weak curves in the case that $C_0$ are elliptic curves. When the degree of $k$ over $\mathbb{F}_2$ is even, the density of weak curves is about $3/4$, and when the degree is odd, the density is about a half. Therefore the even characteristic case has more weak curves than the odd characteristic case.
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