Combinatorics on words in information security: Unavoidable regularities in the construction of multicollision attacks on iterated hash functions

Classically in combinatorics on words one studies unavoidable regularities that appear in sufficiently long strings of symbols over a fixed size alphabet. In this paper we take another viewpoint and focus on combinatorial properties of long words in which the number of occurrences of any symbol is restritced by a fixed constant. We then demonstrate the connection of these properties to constructing multicollision attacks on so called generalized iterated hash functions.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Learning in evolutionary environments

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Learning in Evolutionary Environments

The purpose of this work is to present a sort of short selective guide to an enormous and diverse literature on learning processes in economics. We argue that learning is an ubiquitous characteristic of most economic and social systems but it acquires even greater importance in explicitly evolutionary environments where: a) heterogeneous agents systematically display various forms of "bounded rationality"; b) there is a persistent appearance of novelties, both as exogenous shocks and as the result of technological, behavioural and organisational innovations by the agents themselves; c) markets (and other interaction arrangements) perform as selection mechanisms; d) aggregate regularities are primarily emergent properties stemming from out-of-equilibrium interactions. We present, by means of examples, the most important classes of learning models, trying to show their links and differences, and setting them against a sort of ideal framework of "what one would like to understand about learning...". We put a signifiphasis on learning models in their bare-bone formal structure, but we also refer to the (generally richer) non-formal theorising about the same objects. This allows us to provide an easier mapping of a wide and largely unexplored research agenda.Learning, Evolutionary Environments, Economic Theory, Rationality

Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case

We consider a model case for a strictly convex domain of dimension $d\geq 2$ with smooth boundary and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.Comment: 53 pages, 4 figures, to appear in Annals of Math. Fixed typos, added remark