arXiv subjects
Publications and source records attributed to Bruno Martin.
We describe the average behaviour of the Brjuno function in the neighbourhood of any given point of the unit interval. In particular, we show that its Lebesgue set is the set of Brjuno numbers and we fi nd the asymptotic behaviour of the modulus of continuity of its integral.
In this paper we prove that the avalanche problem for Kadanoff sandpile model (KSPM) is P-complete for two-dimensions. Our proof is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with configurations in KSPM. The proof is also related to the known prediction problem for sandpile which is in NC for one-dimensional sandpiles and is P-complete for dimension 3 or greater. The computational complexity of the prediction problem remains open for two-dimensional sandpiles.
A number-conserving cellular automaton is a cellular automaton whose states are integers and whose transition function keeps the sum of all cells constant throughout its evolution. It can be seen as a kind of modelization of the physical conservation laws of mass or energy. In this paper, we first propose a necessary condition for triangular and hexagonal cellular automata to be number-conserving. The local transition function is expressed by the sum of arity two functions which can be regarded as 'flows' of numbers. The sufficiency is obtained through general results on number-conserving cellular automata. Then, using the previous flow functions, we can construct effective number-conserving simulations between hexagonal cellular automata and triangular cellular automata.
Generation of pseudo random sequences by cellular automata, as well as by hybrid cellular automata is surveyed. An application to the fast evaluation and FPGA implementation of some classes of boolean functions is sketched out.
We consider the enciphering of a data stream while being compressed by a LZ algorithm. This has to be compared to the classical encryption after compression methods used in security protocols. Actually, most cryptanalysis techniques exploit patterns found in the plaintext to crack the cipher; compression techniques reduce these attacks. Our scheme is based on a LZ compression in which a Vernam cipher has been added. We make some security remarks by trying to measure its randomness with statistical tests. Such a scheme could be employed to increase the speed of security protocols and to decrease the computing power for mobile devices.
This paper considers interactions between cellular automata and cryptology. It is known that non-linear elementary rule which is correlation-immune don't exist. This results limits the use of cellular automata as pseudo-random generators suitable for cryptographic applications. In addition, for this kind of pseudo-random generators, a successful cryptanalysis was proposed by Meier and Staffelbach. However, other ways to design cellular automata capable to generate good pseudo-random sequences remain and will be discussed in the end of this article.