Counting Points for Hyperelliptic Curves of type y2=x5+axy^2=x^5+ax over Finite Prime Fields

Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running times. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type $y^2=x^5+ax$ over given large prime fields \Fp, e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form $l\cdot c$ where $l$ is a prime number greater than about $2^{160}$ and $c$ is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type $y^2=x^5+a$ where $a$ is not square in \Fp

Constructing suitable ordinary pairing-friendly curves: A case of elliptic curves and genus two hyperelliptic curves

One of the challenges in the designing of pairing-based cryptographic protocols is to construct suitable pairing-friendly curves: Curves which would provide e�cient implementation without compromising the security of the protocols. These curves have small embedding degree and large prime order subgroup. Random curves are likely to have large embedding degree and hence are not practical for implementation of pairing-based protocols. In this thesis we review some mathematical background on elliptic and hyperelliptic curves in relation to the construction of pairing-friendly hyper-elliptic curves. We also present the notion of pairing-friendly curves. Furthermore, we construct new pairing-friendly elliptic curves and Jacobians of genus two hyperelliptic curves which would facilitate an efficient implementation in pairing-based protocols. We aim for curves that have smaller values than ever before reported for di�erent embedding degrees. We also discuss optimisation of computing pairing in Tate pairing and its variants. Here we show how to e�ciently multiply a point in a subgroup de�ned on a twist curve by a large cofactor. Our approach uses the theory of addition chains. We also show a new method for implementation of the computation of the hard part of the �nal exponentiation in the calculation of the Tate pairing and its varian