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The connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, Lisoněk exploited it to reformulate the Charpin-Gong characterization of a large class of hyper-bent functions in terms of numbers of points on hyperelliptic curves. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin-Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by Mesnager. Doing so, a polynomial time and space test for the hyper-bentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong. [ABSTRACT FROM AUTHOR]

In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, the non-adjacent form (NAF) and its generalizations (e.g., the generalized non-adjacent form (GNAF) and the radix- r non-adjacent form ( rNAF)) have been proposed for minimizing the non-zero densities in the representations of the scalars. On the other hand, for subfield elliptic curves, the Frobenius expansions of the scalars can be used for improving efficiency. Unfortunately, there are only a few methods apply the techniques of NAF or its analogue to the Frobenius expansion, namely τ-adic NAF techniques on Koblitz curves and hyperelliptic Koblitz curves. In this paper, we try to combine these techniques, namely recoding methods for reducing non-zero density and the Frobenius expansion, and propose two new efficient recoding methods of scalars on more general family of subfield elliptic curves in odd characteristic. We also prove that the non-zero densities for the new methods are same as those for the original GNAF and rNAF. We estimate scalar multiplication costs on the above subfield elliptic curves in terms of elliptic curve operations and finite field operations for several previous methods and the proposed methods. In addition, we implement scalar multiplication on an subfield elliptic curve belonging to the above family, for the previous methods and a proposed method. As a result, our estimation and implementation show that the speed of the proposed methods improve between 8% and 50% over that for the Frobenius expansion method. [ABSTRACT FROM AUTHOR]

A new type of public key cryptosystem, called MST3, has been recently introduced on the basis of covers and logarithmic signatures for non-abelian finite groups. The class of Suzuki 2-groups has been proposed for a possible realization of the generic scheme. Due to their simple structure, the groups enable us to study the security of the system and also provide an efficient implementation. An earlier relevant result of the cryptanalysis has shown that the transversal logarithmic signatures are unfit for use in this realization. In this paper we present a revised version of MST3 for the Suzuki 2-groups and show a thorough study of its security. Using heuristic and algebraic methods we establish strong lower bounds for the workload of conceivable direct attacks on the private key of the scheme. We then develop a powerful chosen plaintext attack which allows us to rule out the usage of a certain class of logarithmic signatures. In addition, we show a class of logarithmic signatures withstanding this attack and thus to our knowledge they could be used in the realization of the scheme. Finally, we describe and discuss the implementation issues of the scheme in detail and include data of its performance obtained from an experimental result. [ABSTRACT FROM AUTHOR]

We propose a new simple and efficient family of hash functions based on matrix-vector multiplications with a competitive software implementation. The hash design combines a hard mathematical problem based on solving a system of linear equations with special-random requirements and the fast computation of the convolution product algorithm. Such a mixing was often unrealizable. For security, the one-way and collision resistant criteria are based on the fact that inverting the compression function for random values is infeasible in reasonable time. In a subsequent result, we conjecture a general framework for producing secure matrix multiplication hash functions. [ABSTRACT FROM AUTHOR]

We apply combinatorics on words to develop an approach to multicollisions in generalized iterated hash functions. Our work is based on the discoveries of A. Joux and on generalizations provided by M. Nandi and D. Stinson as well as J. Hoch and A. Shamir. We wish to unify the existing diverse notation in the field, bring basic facts together, reprove some previously published results and produce some new ones. A multicollision attack method informally described by Hoch and Shamir is laid on a sound statistical basis and studied in detail. [ABSTRACT FROM AUTHOR]

In this work we re-examine a modular multiplication and a modular exponentiation method. The multiplication method, proposed by Hayashi in 1998, uses knowledge of the factorization of both N ++ 1 and N ++ 2 to compute a multiplication modulo N. If both N ++ 1 and N ++ 2 can be factored into k equally sized relatively prime factors then the computations are done modulo each of the factors and then combined using the Chinese Remainder Theorem. It was suggested that the (asymptotic) computational costs of the method is 1/ k of simply multiplying and reducing modulo N. We show, however, that the computational costs of the method is (asymptotically) at least as costly as simply multiplying and reducing modulo N for both squarings and general multiplications when efficient arithmetic is used. The exponentiation method, proposed by Hwang, Su, Yeh and Chen in 2005, is based on Hayashi's method and uses knowledge of the factorization of P ++ 1 and P -- 1 to compute an exponentiation modulo an odd prime P. We begin by showing that the method cannot be used as a general purpose exponentiation method and then modify the method so that it can work as a general purpose modular multiplication method. Like Hayashi's method, however, this method is at best (asymptotically) only as efficient as simply multiplying and reducing modulo P. [ABSTRACT FROM AUTHOR]