Volume 4, Issue 3

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 ( r NAF)) 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 r NAF. 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.

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.

A new type of public key cryptosystem, called MST 3 , 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 MST 3 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.