We discuss abelian equivariant Iwasawa theory for elliptic curves over $${\mathbb {Q}}$$ at good supersingular primes and non-anomalous ordinary primes. Using Kobayashi’s method, we construct Coleman maps, which send the Beilinson–Kato element to p-adic L-functions. Then propose main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application...

It is known that the security of Public Key Cryptosystems can be based on Vector Decomposition Problem (VDP). In this paper, we analyze this problem. In practice, it was shown that the Computational DiffieHellmann Problem (CDHP) is equivalent to VDP for supersingular elliptic curves. Moreover, VDP on a higher genus curve is hard if CDHP is hard on its one dimensional subspace. We propose an enc...

Self-pairings are a special subclass of pairings and have interesting applications in cryptographic schemes and protocols. In this paper, we explore the computation of the self-pairings on supersingular elliptic curves with embedding degree k = 3. We construct a novel self-pairing which has the same Miller loop as the Eta/Ate pairing. However, the proposed self-pairing has a simple final expone...

In this note we review a simple criterion, due to Ekedahl, for superspecial curves defined over finite fields.Using this we generalize and give some simple proofs for some well-known superspecial curves.

The hyperelliptic curve Ate pairing provides an efficient way to compute a bilinear pairing on the Jacobian variety of a hyperelliptic curve. We prove that, for supersingular elliptic curves with embedding degree two, square of the Ate pairing is nothing but the Weil pairing. Using the formula, we develop an X -coordinate only pairing inversion method. However, the algorithm is still infeasible...

In this paper, we describe a quantum algorithm for computing an isogeny between any two supersingular elliptic curves defined over a given finite field. The complexity of our method is in Õ(p) where p is the characteristic of the base field. Our method is an asymptotic improvement over the previous fastest known method which had complexity Õ(p) (on both classical and quantum computers). We also...

The purpose of this paper is to study the Hodge-Arakelov theory of elliptic curves (cf. [Mzk1-4]) in positive characteristic. The first two §’s (§1,2) are devoted to studying the relationship of the Frobenius and Verschiebung morphisms of an elliptic curve in positive characteristic to the Hodge-Arakelov theory of elliptic curves. We begin by deriving a “Verschiebung-Theoretic Analogue of the H...