Optimization of Tree Modes for Parallel Hash Functions: A Case Study

This paper focuses on parallel hash functions based on tree modes of operation for an inner Variable-Input-Length function. This inner function can be either a single-block-length (SBL) and prefix-free MD hash function, or a sponge-based hash function. We discuss the various forms of optimality that can be obtained when designing parallel hash functions based on trees where all leaves have the same depth. The first result is a scheme which optimizes the tree topology in order to decrease the running time. Then, without affecting the optimal running time we show that we can slightly change the corresponding tree topology so as to minimize the number of required processors as well. Consequently, the resulting scheme decreases in the first place the running time and in the second place the number of required processors.Comment: Preprint version. Added citations, IEEE Transactions on Computers, 201

Lower bounds on the class number of algebraic function fields defined over any finite field

We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and asymptotics for the class number, depending mainly on the number of places of a certain degree. We give examples of towers of algebraic function fields having a large class number.Comment: 24 page

A Digital Signature Scheme for Long-Term Security

In this paper we propose a signature scheme based on two intractable problems, namely the integer factorization problem and the discrete logarithm problem for elliptic curves. It is suitable for applications requiring long-term security and provides a more efficient solution than the existing ones

On the existence of dimension zero divisors in algebraic function fields defined over F_q

Let F/F_q be an algebraic function field of genus g defined over a finite field F_q. We obtain new results on the existence, the number and the density of dimension zero divisors of degree g-k in F where k is a positive integer. In particular, for q=2,3 we prove that there always exists a dimension zero divisor of degree \gamma-1 where \gamma is the q-rank of F. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree g-k for a hyperelliptic field F in terms of its Zeta function.Comment: 18 page

The second weight of generalized Reed-Muller codes in most cases

The second weight of the Generalized Reed-Muller code of order $d$ over the finite field with $q$ elements is now known for $d (n-1)(q-1)$. In this paper, we determine the second weight for the other values of $d$ which are not multiple of $q-1$ plus 1. For the special case $d=a(q-1)+1$ we give an estimate.Comment: This version corrects minor misprints and gives a more detailed proof of a combinatorial lemm

Remarks on low weight codewords of generalized affine and projective Reed-Muller codes

We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they cannot be second, thirds or fourth weight depending on the hypothesis. In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.Comment: New version taking into account recent results from Elodie Leducq on the characterization of the next-to-minimal codewords (cf. arXiv:1203.5244