An Algorithm for Generating Irreducible Cubic Trinomials over Prime Field

This paper proposes an algorithm for generating irreducible cubic trinomials in the form x(3) + ax + b, b ∈ F(p), where a is a certain fixed non-zero element in the prime field F(p). The proposed algorithm needs a certain irreducible cubic trinomial over F(p) to be previously given as a generator; however, the proposed algorithm can generate irreducible cubic polynomials one after another by changing a certain parameter in F(p). In this paper, we compare the calculation cost and the average computation time for generating an irreducible cubic polynomial, especially trinomial, among Hiramoto et al. irreducibility testing algorithm, Berlekamp-Massey minimal polynomial determining algorithm, and the proposed algorithm. From the experimental results, it is shown that the proposed algorithm is the fastest among the three algorithms for generating irreducible cubic trinomials

Elliptic curve cryptography: Generation and validation of domain parameters in binary Galois Fields

Elliptic curve cryptography (ECC) is an increasingly popular method for securing many forms of data and communication via public key encryption. The algorithm utilizes key parameters, referred to as the domain parameters. These parameters must adhere to specific characteristics in order to be valid for use in the algorithm. The American National Standards Institute (ANSI), in ANSI X9.62, provides the process for generating and validating these parameters. The National Institute of Standards and Technology (NIST) has identified fifteen sets of parameters; five for prime fields, five for binary fields, and five for Koblitz curves. The parameter generation and validation processes have several key issues. The first is the fast reduction within the proper modulus. The modulus chosen is an irreducible polynomial having degree greater than 160. Choosing irreducible polynomials of a particular order is less critical since they have isomorphic properties, mathematically. However, since there are differences in performance, there are standards that determine the specific polynomials chosen. The NIST standards are also based on word lengths of 32 bits. Processor architecture, primality, and validation of irreducibility are other important characteristics. The area of ECC that is researched is the generation and validation processes, as they are specified for binary Galois Fields F (2m). The rationale for the parameters, as computed for 32 bit and 64 bit computer architectures, and the algorithms used for implementation, as specified by ANSI, NIST and others, are examined. The methods for fast reduction are also examined as a baseline for understanding these parameters. Another aspect of the research is to determine a set of parameters beyond the 571-bit length that meet the necessary criteria as determined by the standards

Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic

FPGA IMPLEMENTATION FOR ELLIPTIC CURVE CRYPTOGRAPHY OVER BINARY EXTENSION FIELD

Elliptic curve cryptography plays a crucial role in network and communication security. However, implementation of elliptic curve cryptography, especially the implementation of scalar multiplication on an elliptic curve, faces multiple challenges. One of the main challenges is side channel attacks (SCAs). SCAs pose a real threat to the conventional implementations of scalar multiplication such as binary methods (also called doubling-and-add methods). Several scalar multiplication algorithms with countermeasures against side channel attacks have been proposed. Among them, Montgomery Powering Ladder (MPL) has been shown an effective countermeasure against simple power analysis. However, MPL is still vulnerable to certain more sophisticated side channel attacks. A recently proposed modified MPL utilizes a combination of sequence masking (SM), exponent splitting (ES) and point randomization (PR). And it has shown to be one of the best countermeasure algorithms that are immune to many sophisticated side channel attacks [11]. In this thesis, an efficient hardware architecture for this algorithm is proposed and its FPGA implementation is also presented. To our best knowledge, this is the first time that this modified MPL with SM, ES, and PR has been implemented in hardware

Pseudo-boundaries in discontinuous 2-dimensional maps

It is known that Kolmogorov-Arnold-Moser boundaries appear in sufficiently smooth 2-dimensional area-preserving maps. When such boundaries are destroyed, they become pseudo-boundaries. We show that pseudo-boundaries can also be found in discontinuous maps. The origin of these pseudo-boundaries are groups of chains of islands which separate parts of the phase space and need to be crossed in order to move between the different sub-spaces. Trajectories, however, do not easily cross these chains, but tend to propagate along them. This type of behavior is demonstrated using a ``generalized'' Fermi map.Comment: 4 pages, 4 figures, Revtex, epsf, submitted to Physical Review E (as a brief report

Extracteur aléatoires multi-sources sur les corps finis et les courbes elliptiques

International audienceWe propose two-sources randomness extractors over finite fields and on elliptic curves that can extract from two sources of information without consideration of other assumptions that the starting algorithmic assumptions with a competitive level of security. These functions have several applications. We propose here a description of a version of a Diffie-Hellman key exchange protocol and key extraction.Nous proposons des extracteurs d'aléas 2-sources sur les corps finis et sur les courbes elliptiques capables d'extraire à partir de plusieurs sources d'informations sans considération d'autres hypothèses que les hypothèses algorithmiques de départ avec un niveau de sécurité compétitif. Ces fonctions possèdent plusieurs applications. Nous proposons ici une version du protocole d'échange de clé Diffie-Hellman incluant la phase d'extraction