Self-pairings have found interesting applications in cryptographic schemes, such as ZSS short signatures and so on. In this paper, we present a novel method for constructing a self-pairing on supersingular elliptic curves with even embedding degrees, which we call the Ateil pairing. This pairing improves the efficiency of the self-pairing computation on supersingular curves over finite fields w...

Pairings on the Jacobians of (hyper-)elliptic curves have received considerable attention not only as a tool to attack curve based cryptosystems but also as a building block for constructing cryptographic schemes with new and novel properties. Motivated by the work of Scott, we investigate how to use efficiently computable automorphisms to speed up pairing computations on two families of non-su...

At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is ...

We give examples of Drinfeld modules φ of rank 2 and higher over Fq(T ) that have no primes of supersingular reduction. The idea is to construct φ so that the associated mod ` representations are incompatible with the existence of supersingular primes. We also answer a question of Elkies by proving that such obstructions cannot exist for elliptic curves over number fields. Elkies [El1] proved t...

The factorization of the Legendre polynomial of degree (p− e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism √ −2p, by proving an analogue of Deuring’s theorem on supersingular curves with multiplier √ −p. This is used to count th...

We present a general technique for the efficient computation of pairings on supersingular Abelian varieties. This formulation, which we call the eta pairing, generalises results of Duursma and Lee for computing the Tate pairing on supersingular elliptic curves in characteristic three. We then show how our general technique leads to a new algorithm which is about twice as fast as the Duursma-Lee...