Daniel Sadornil

On the linear complexity of the Naor-Reingold sequence with elliptic curves

The Naor-Reingold sequences with elliptic curves are used in cryptography due to their nice construction and good theoretical properties. Here we provide a new bound on the linear complexity of these sequences. Our result improves the previous one obtained by I.E. Shparlinski and J.H. Silverman and holds in more cases.

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.

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.

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.

A PRIMALITY TEST FOR Kp n + 1 NUMBERS

In this paper we generalize the classical Proths theorem for inte- gers of the form N = Kpn + 1. For these families, we present a primality test whose computational complexity is e O(log 2 (N)) and, what is more important, that requires only one modular exponentiation similar to that of Fermats test. Consequently, the presented test improves the most often used one, derived from Pocklingtons theorem, which usually requires the computation of several modular exponentiations together with some GCDs.

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.

Volcanoes of ℓ-isogenies of elliptic curves over finite fields :

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.

Volcanoes of

This paper is devoted to the study of the volcanoes of

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