Suitable Curves for Genus-4 HCC over Prime Fields: Point Counting Formulae for Hyperelliptic Curves of Type y 2=x $^{\rm 2{\it k}+1}$ +ax

Abstract

AbstractComputing the order of the Jacobian group of a hyperelliptic curve over a finite field is very important to construct a hyperelliptic curve cryptosystem (HCC), because to construct secure HCC, we need Jacobian groups of order in the form l c where l is a prime greater than about 2160 and c is a very small integer. But even in the case of genus two, known algorithms to compute the order of a Jacobian group for a general curve need a very long running time over a large prime field. In this article, we give explicit formulae of the order of Jacobian groups for hyperelliptic curves over a finite prime field of type y 2=x \(^{\rm 2{\it k}+1}\)+ax, which allows us to search suitable curves for HCC. By using these formulae, we can find many suitable curves for genus-4 HCC and show some examples.KeywordsCharacteristic PolynomialHyperelliptic CurveDiscrete Logarithm ProblemElliptic Curve CryptosystemCryptology ePrint ArchiveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.