The Tate pairing has plenty of attractive applications, e.g., ID-based cryptosystems, short signatures, etc. Recently several fast implementations of the Tate pairing has been reported, which make it appear that the Tate pairing is capable to be used in practical applications. The computation time of the Tate pairing strongly depends on underlying elliptic curves and definition fields. However ...

It is proved that the supersingular parameters α of the elliptic curve E3(α) : Y 2 + αXY + Y = X3 in Deuring normal form satisfy α = 3 + γ3, where γ lies in the finite field Fp2 . This is accomplished by finding explicit generators for the normal closure N of the finite extension k(α)/k(j(α)), where α is an indeterminate over k = Fp2 and j(α) is the j-invariant of E3(α). The function field N is...

A Self-pairing es(P,P ) is a special subclass of bilinear pairing where both input points in a group are equal. Self-pairings have some interesting applications in cryptographic scheme and protocols. Recently some novel methods for constructing self-pairings on supersingular elliptic curves have been proposed. In this paper we first give the construction of self-pairings on some supersingular e...

Even first semester calculus students are aware of how calculus, hence analysis, is used to solve problems in engineering. In recent decades the engineering world is gaining more exposure to algebra through the powerful problem solutions it provides. One area that algebra has made significant contributions to is cryptography and, more specifically, public key cryptography. In this paper we aim ...

Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g > 1 and we give several examples in genus 2.

There are two types of elliptic curves, ordinary curves and supersingular curves. In 2012, Sutherland proposed an efficient almost deterministic algorithm for determining whether a given curve is or supersingular. Sutherland's based on sequences isogenies started from the input curve, computation each isogeny requires square root computations, which dominant cost algorithm. this paper, we reduc...

A Trace Zero Variety is a specific subgroup of the group of the divisor classes on a hyperelliptic curve C/Fq, which are rational over a small degree extension Fqr of the definition field. Trace Zero Varieties (TZV) are interesting for cryptographic applications since they enjoy properties that can be exploited to achieve fast arithmetic and group construction. Furthermore, supersingular TZV al...

We focus on the implementation and security aspects of cryptographic protocols that use Type 1 and Type 4 pairings. On the implementation front, we report improved timings for Type 1 pairings derived from supersingular elliptic curves in characteristic 2 and 3 and the first timings for supersingular genus-2 curves in characteristic 2 at the 128-bit security level. In the case of Type 4 pairings...