Josep M. Miret

A secure elliptic curve-based RFID protocol

Nowadays, the use of Radio Frequency Identification (RFID) systems in industry and stores has increased. Nevertheless, some of these systems present privacy problems that may discourage potential users. Hence, high confidence and effient privacy protocols are urgently needed. Previous studies in the literature proposed schemes that are proven to be secure, but they have scalability problems. A feasible and scalable protocol to guarantee privacy is presented in this paper. The proposed protocol uses elliptic curve cryptography combined with a zero knowledge-based authentication scheme. An analysis to prove the system secure, and even forward secure is also provided.

Determining the 2-Sylow subgroup of an elliptic curve over a finite field

In this paper we describe an algorithm that outputs the order and the structure, including generators, of the 2-Sylow subgroup of an elliptic curve over a finite field. To do this, we do not assume any knowledge of the group order. The results that lead to the design of this algorithm are of inductive type. Then a right choice of points allows us to reach the end within a linear number of successive halvings. The algorithm works with abscissas, so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field. Efficient methods for this computation determine the efficiency of our algorithm.

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.

Simple and efficient hash-based verifiable mixing for remote electronic voting

Remote voting permits an election to be carried out through telecommunication networks. In this way, its participants are not required to physically move to the polling place. Votes are automatically collected and counted so that once the election ends, the results can be published after a very short delay. Security is a key aspect of any remote application. A remote voting system must be secure in the sense that the result of the election cannot be manipulated and the privacy of participants is preserved. This paper presents a novel mix-type remote voting system that permits to verify the correctness of a voting process without requiring complex and costly zero-knowledge proofs. It is based on a very efficient and lightweight hash-based construction that makes use of the homomorphic properties of ElGamal cryptosystem.

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.

Non existence of some mixed Moore graphs of diameter 2 using SAT

Mixed graphs with maximum number of vertices regarding to a given maximum degree and given diameter are known as mixed Moore graphs. In this paper we model the problem of the existence of mixed Moore graphs of diameter 2 through the Boolean satisfiability problem. As a consequence, we prove the non existence of mixed Moore graphs of order 40, 54 and 84.

Computing the ℓ-power torsion of an elliptic curve over a finite field

The algorithm we develop outputs the order and the structure, including generators, of the l-Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive l-divisions leads to generators of the subgroup under consideration. After the computation of a couple of polynomials, each division step relies on finding rational roots of polynomials of degree l. We specify in complete detail the case l = 3, when the complexity of each trisection is given by the computation of cubic roots in finite fields.

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.

A hybrid approach to vector-based homomorphic tallying remote voting

Vector-based homomorphic tallying remote voting schemes provide an efficient protocol for vote tallying, but they require voters to prove in zero-knowledge that the ballots they cast have been properly generated. This is usually achieved by means of the so-called zero-knowledge range proofs, which should be verified by the polling station before tallying. In this paper, we present an end-to-end verifiable hybrid proposal in which ballots are proven to be correct by making use of a zero-knowledge proof of mixing but still using a homomorphic tallying for gathering the election results. Our proposal offers all the advantages of the homomorphic tallying paradigm, while it avoids the elevated computational cost of range proofs. As a result, ballot verification performance is improved in comparison with the equivalent homomorphic systems. The proposed voting scheme is suitable for multi-candidate elections as well as for elections in which the votes have different weights.

Explicit 2-power torsion of genus 2 curves over finite fields

We give an efficient explicit algorithm to find the structure and generators of the maximal 2-subgroup of the Jacobian of a genus 2 curve over a finite field of odd characteristic. We use the 2-torsion points as seeds to successively perform a chain of halvings to find divisors of increasing 2-power order. The halving loop requires a solution to certain degree 16 polynomials over the base field, and the termination of the algorithm is based on the description of the graph structure of the maximal 2-subgroup. The structure of our algorithm is the natural extension of the even characteristic case.