We show that the Davenport-Stothers inequality fails in any characteristic p > 3. The proof uses elliptic surfaces over P and inseparable base change. We also comment on corrected inequalities. Our ideas are related to supersingular surfaces (in Shioda’s sense). For characteristic 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over P.

Let E be an elliptic curve de ned over a nite eld. Balasubramanian and Koblitz have proved that if the ` roots of unity μ` is not contained in the ground eld, then a eld extension of the ground eld contains μ` if and only if the `-torsion points of E are rational over the same eld extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weiland the ...

Elliptic curve cryptosystem protocols use two main operations, the scalar multiplication and the pairing computation. Both of them are done through a chain of basic operation on the curve. In this paper we present new formulas for supersingular elliptic curve in characteristic 2 and 3. We improve best known formulas by at least one multiplication in the field.

We present Séta (To be pronounced [?e:t?] meaning “walk” in Hungarian.), a new family of public-key encryption schemes with post-quantum security based on isogenies supersingular elliptic curves. It is constructed from trapdoor one-way functions, where the inversion algorithm uses Petit’s so called torsion attacks SIDH to compute an isogeny between curves given endomorphism starting curve and i...

Consider an elliptic curve E over a number field K. Suppose that has supersingular reduction at some prime $$\mathfrak {p}$$ of K lying above the rational p. We completely classify valuations $$p^n$$ -torsion points by valuation coefficient $$p{\text {th}}$$ division polynomial. This classification corrects error in earlier work Lozano-Robledo. As application, we find minimum necessary ramifica...

In a secret sharing scheme, is distributed among several participants in such way that only any authorized subset of able to recover the secret. So far, security many schemes has been based on hardness some mathematical problems, as discrete logarithm and factorization. These problems can be solved polynomial time using Shor?s algorithm for quantum computer. this paper, we propose an efficient ...

This paper grew out of the following question. Given an ordinary elliptic curve E//Fq over a finite field Fq, consider the sequence of integers A(n), n ≥ 1 defined by #E(Fqn) = q + 1− A(n). Is it true that as n grows we have |A(n)| → ∞? Without the hypothesis “ordinary”, the answer can be no, because for a supersingular elliptic curve, one can have A(n) = 0 on entire arithmetic progressions of ...

In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identity-based encryption. At the same time, the security standards for public key cryptosystems are expected to increase, so that in the future they will be cap...