Alexis Bonnecaze

Cyclic codes and self-dual codes over F/sub 2/+uF/sub 2/

We introduce linear cyclic codes over the ring F/sub 2/+uF/sub 2/={0,1,u,u~=u+1}, where u/sup 2/=0 and study them by analogy with the Z/sub 4/ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z/sub 4/-codes. Unlike the Z/sub 4/ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map.

Quaternary quadratic residue codes and unimodular lattices

We construct new self-dual and isodual codes over the integers module 4. The binary images of these codes under the Gray map are nonlinear, but formally self-dual. The construction involves Hensel lifting of binary cyclic codes. Quaternary quadratic residue codes are obtained by Hensel lifting of the classical binary quadratic residue codes. Repeated Hensel lifting produces a universal code defined over the 2-adic integers. We investigate the connections between this universal code and the codes defined over Z/sub 4/, the composition of the automorphism group, and the structure of idempotents over Z/sub 4/. We also derive a square root bound on the minimum Lee weight, and explore the connections with the finite Fourier transform. Certain self-dual codes over Z/sub 4/ are shown to determine even unimodular lattices, including the extended quadratic residue code of length q+1, where q/spl equiv/-1(mod8) is a prime power. When q=23, the quaternary Golay code determines the Leech lattice in this way. This is perhaps the simplest construction for this remarkable lattice that is known. >

Type II codes over Z/sub 4/

The conditions satisfied by the weight enumerator of self-dual codes, defined over the ring of integers module four, have been studied by Klemm (1989), then by Conway and Sloane (1993). The MacWilliams (1977) transform determines a group of substitutions, each of which fixes the weight enumerator of a self-dual code. This weight enumerator belongs to the ring of polynomials fixed by the group of substitutions, called the ring R of invariants. Among all of the quaternary self-dual codes, some have the property that all euclidean weights are multiples of 8. These codes are called type II codes by analogy with the binary case. An upper bound on their minimum euclidean weight is given, thereby leading to a natural notion of extremality akin to similar concepts for type II binary codes and type II lattices. The most interesting examples of type II codes are perhaps the extended quaternary quadratic residue codes. This class of codes includes the octacode [8, 4, 6] and the lifted Golay [24, 12, 12]. Other classes of interest comprise a multilevel construction from binary Reed-Muller and lifted double circulant codes.

3-Colored 5-Designs and Z4-Codes

New 5-designs on 24 points were constructed recently by Harada by the consideration of Z_4-codes. We use Jacobi polynomials as a theoretical tool to explain their existence as resulting of properties of the symmetrized weight enumerator (swe) of the code. We introduce the notion of a colored t-design and we show that the words of any given Lee composition, in any of the 13 Lee-optimal self-dual codes of length 24 over Z_4, form a colored 5-design. New colored 3-designs on 16 points are also constructed in that way.

Translates of linear codes over Z/sub 4/

We give a method to compute the complete weight distribution of translates of linear codes over Z/sub 4/. The method follows known ideas that have already been used successfully by others for Hamming weight distributions. For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for the Preparata codes are both equal to ten, independent of the code length.

Secure time-stamping schemes: a distributed point of view

Time-stamping is a technique used to prove the existence of a digital document prior to a specific point in time. Today, implemented schemes rely on a centralized server model that has to be trusted. We point out the drawbacks of these schemes, showing that the unique serveur represent a weak point for the system. We propose an alternative scheme which is based on a network of servers managed by administratively independent entities. This scheme appears to be a trusted and reliable distributed time-stamping scheme.RésuméL’horodatage électronique est une technique qui permet de prouver l’existence d’un document avant un instant précis. Actuellement, les schémas implantés adoptent une architecture centralisée basée sur un serveur jouant le rôle de tiers de confiance. Dans de tels schémas, le serveur d’horodatage représente une faiblesse pour le système. Nous proposons un système basé sur un réseau de serveurs gérés par des entités administrativement indépendantes. Nous montrons que ce schéma distribué est robuste et sûr.

Multisignatures as Secure as the Diffie-Hellman Problem in the Plain Public-Key Model

A multisignature scheme allows a group of signers to cooperate to generate a compact signature on a common document. The length of the multisignature depends only on the security parameters of the signature schemes and not on the number of signers involved. The existing state-of-the-art multisignature schemes suffer either from impractical key setup assumptions, from loose security reductions, or from inefficient signature verification. In this paper, we present two new multisignature schemes that address all of these issues, i.e., they have efficient signature verification, they are provably secure in the plain public-key model, and their security is tightly related to the computation and decisional Diffie-Hellman problems in the random oracle model. Our construction derives from variants of EDL signatures.

Quaternary constructions of formally self-dual binary codes and unimodular lattices

Quaternary codes have been studied recently in connection with the construction of sequences with low correlation, lattices and good non linear codes (Kerdock, Preparata). In this paper, we construct formally self-dual binary codes and unimodular lattices using quaternary codes. Two different processes are studied: constructions using Hensel lifting and (u¦u+v) construction. We give a number of examples of formally self-dual binary codes of length n≤64. We obtain a new construction of the Leech lattice, and two new constructions of the Gosset lattice.

Jacobi Polynomials, Type II Codes, and Designs

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices. They are useful to compute coset weight enumerators, and weight enumerators of children. We determine them in most interesting cases in length at most 32, and in some cases in length 72. We use them to construct group divisible designs, packing designs, covering designs, and (t,r)-designs in the sense of Calderbank-Delsarte. A major tool is invariant theory of finite groups, in particular simultaneous invariants in the sense of Schur, polarization, and bivariate Molien series. A combinatorial interpretation of the Aronhold polarization operator is given. New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced and studied here.

On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in O(n(2q)^log * q (n)) where n corresponds to the extension degree, and log * q (n) is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field F 3^57. Index Terms Multiplication algorithm, bilinear complexity, elliptic function field, interpolation on algebraic curve, finite field.