Juan Tena

Weight hierarchy of a product code

Formulas to compute the first members of the weight hierarchy of a product code are given and proved. Basic patterns for the supports of subcodes in which the minimum cardinalities can be found are presented. Hence, formulas for every term of the hierarchy can be given, and the proof for each case will be a generalization of the proofs shown here. >

An algorithm to compute volcanoes of 2-isogenies of elliptic curves over finite fields

The goal of this paper is presenting an algorithm to determine the structure of the volcano of 2-isogenies of a given elliptic curve over a finite field. The core of the algorithm relies on the relationship between the 2-torsion structure of the curves and its level in the volcano, as well as on those results that determine the direction of the different outgoing isogenies from each vertex. The algorithm is specially efficient for the so-called regular volcanoes, where the 2-torsion structure is different at every level.

Computing the height of volcanoes of ℓ-isogenies of elliptic curves over finite fields

The structure of the volcano of l-isogenies, l-prime, of elliptic curves over finite fields has been extensively studied over recent years. Previous works present some results and algorithms concerning the height of such volcanoes in the case of isogenies whose kernels are generated by a rational point. The main goal of this paper is to extend such works to the case of l-isogenies whose kernels are defined by a rational subgroup. In particular, the height of such volcanoes is completely characterized and can be computationally obtained.

On Avoiding ZVP-Attacks Using Isogeny Volcanoes

The usage of elliptic curve cryptography in smart cards has been shown to be efficient although, when considering curves, one should take care about their vulnerability against the Zero-Value Point Attacks (ZVP). In this paper, we present a new procedure to find elliptic curves which are resistant against these attacks. This algorithm finds, in an efficient way, a secure curve by means of volcanoes of isogenies. Moreover, we can deal with one more security condition than Akishita-Takagi method with our search.

On Edwards curves and ZVP-attacks

Elliptic curve cryptography on smart cards is vulnerable under a particular Side Channel Attack: the existence of zero-value points (ZVP). One approach to face this drawback relies on changing the curve for an isogenous one, until a resistant curve is found. This paper focuses on an alternative strategy: exploiting the properties of a recently introduced form of elliptic curves, Edwards curves. We show that these curves achieve conditions for being resistant to ZVP-attacks. Hence, using Edwards curves is a good countermeasure to avoid these attacks.

An algorithm to compute the number of points on elliptic curves ofj-invariant 0 or 1728 over a finite field

We present an algorithm to compute the number ofFq-rational points on elliptic curves defined over a finite fieldFq, withj-invariant 0 or 1728. This algorithm takesO(log3p) bit operations, werep is the characteristic ofFq.

Elliptic curves with j = 0,1728 and low embedding degree

Elliptic curves over a finite field with j-invariant 0 or 1728, both supersingular and ordinary, whose embedding degree k is low are studied. In the ordinary case we give conditions characterizing such elliptic curves with fixed embedding degree with respect to a subgroup of prime order ℓ. For , these conditions give parameterizations of q in terms of ℓ and two integers m, n. We show several examples of families with infinitely many curves. Similar parameterizations for need a fixed kth root of the unity in the underlying field. Moreover, when the elliptic curve admits distortion maps, an example is provided.

A Rao-Nam like Cryptosystem with Product Codes

The purpose of this paper is to design a private key cryptosystem that uses error correcting codes in an efficient way.

Pairing-Based Cryptography on Elliptic Curves

We give a brief overview of a recent branch of Public Key Cryptography, the so called Pairing-based Cryptography or Identity-based Cryptography. We describe the Weil pairing and its applications to cryptosystems and cryptographic protocols based on pairings as well as the elliptic curves suitable for the implementation of this kind of cryptography, the so called pairing-friendly curves. Some recent results of the authors are included.

Volcanoes of ℓ-isogenies of elliptic curves over finite fields : the case ℓ = 3

This paper is devoted to the study of the volcanoes of l-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the l-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case l = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results are also provided.