We study generalisations to totally real fields of the methods originating with Wiles and Taylor-Wiles ([32], [31]). In view of the results of Skinner-Wiles [26] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some app...

In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over nite elds. Our main results provide a framework in which we give a conceptually simple new proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms i...

In their seminal paper, Miyaji, Nakabayashi and Takano [12] describe a simple method for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6. Such curves are important for the realisation of pairing-based cryptosystems on ordinary (non-supersingular) elliptic curves. We provide an alternative derivation of their results, and extend them to allow for the generation of...

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 ...

Since 2000 pairings became a very useful tool to design new protocols in cryptography. Short signaturesand identity-based encryption became also practical thanks to these pairings.This thesis contains two parts. One part is about optimized pairing implementation on different ellip-tic curves according to the targeted protocol. Pairings are implemented on supersingular elliptic curve...

In implementation of elliptic curve cryptography, three kinds of finite fields have been widely studied, i.e. prime field, binary field and optimal extension field. In pairing-based cryptography, however, pairingfriendly curves are usually chosen among ordinary curves over prime fields and supersingular curves over extension fields with small characteristics. In this paper, we study pairings on...