Jean-Baptiste Yunès

An intrinsically non minimal-time Minsky-like 6-states solution to the Firing Squad synchronization problem

Here is presented a 6-states non minimal-time solution which is intrinsically Minsky-like and solves the three following problems: unrestricted version on a line, with one initiator at each end of a line and the problem on a ring. We also give a complete proof of correctness of our solution, which was never done in a publication for Minskys solutions.

A FAMILY OF SMALLEST SYMMETRICAL FOUR-STATE FIRING SQUAD SYNCHRONIZATION PROTOCOLS FOR RING ARRAYS

An existence or non-existence of five-state firing squad synchronization protocol has been a longstanding and famous open problem for a long time. In this paper, we answer partially to this problem by proposing a family of smallest four-state firing squad synchronization protocols that can synchronize any one-dimensional ring cellular arrays of length n = 2k for any positive integer k. The number four is the smallest one in the class of synchronization protocols proposed so far.

Simple new algorithms which solve the firing squad synchronization problem: a 7-states 4n-steps solution

We present a new family of solutions to the firing squad synchronization problem. All these solutions are built with a few finite number of signals, which lead to simple implementations with 7 or 8 internal states. Using one of these schemes we are able to built a 7-states 4n+O(log n)-steps solution to the firing squad synchronization problem. These solutions not only solves the unrestricted problem (initiator at one of the two ends), but also the problem with initiators at both ends and the problem on a ring.

About 4-States Solutions to the Firing Squad Synchronization Problem

We present some elements of a new family of time-optimal solutions to a less restrictive firing squad synchronization problem. These solutions are all built on top of some elementary algebraic cellular automata. Thus, this gives a very new insight on the problem and a more general way of computing on cellular automata.

On maximal QROBDD’s of boolean functions

We investigate the structure of ``worst-case quasi reduced ordered decision diagrams and Boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of ``hard Boolean functions as functions whose QROBDD are ``worst-case ones. So we exhibit the relation between hard functions and the Storage Access function (also known as Multiplexer).

HFE and BDDs: A Practical Attempt at Cryptanalysis

HFE (Hidden Field Equations) is a public key cryptosystem using univariate polynomials over finite fields. It was proposed by J. Patarin in 1996. Well chosen parameters during the construction produce a system of quadratic multivariate polynomials over ({mathbb{F}_2}) as the public key. An enclosed trapdoor is used to decrypt messages. We propose a ciphertext-only attack which mainly consists in satisfying a boolean formula. Our algorithm is based on BDDs (Binary Decision Diagrams), introduced by Bryant in 1986, which allow to represent and manipulate, possibly efficiently, boolean functions. This paper is devoted to some experimental results we obtained while trying to solve the Patarin’s challenge. This approach was not successful, nevertheless it provided some interesting information about the security of HFE cryptosystem.

A Note on Synchronization Steps in Firing Squad Synchronization Problem

The firing squad synchronization problem on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed. In this paper, we propose two synchronization algorithms and their implementations, each having O(n^2) and O(2^n) synchronization steps for n cells, respectively.

A Two-Dimensional Optimum-Time Firing Squad Synchronization Algorithm and Its Implementation

The firing squad synchronization problem on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed for not only one-dimensional arrays but two-dimensional arrays. In the present paper, we propose a new and simpler optimum-time synchronization algorithm that can synchronize any rectangle array of size m ×n with a general at one corner in m + n + max (m, n) − 3 steps. An implementation for the algorithm in terms of local transition rules is also given.