70,220 research outputs found
On the generalized linear equivalence of functions over finite fields
In this paper we introduce the concept of generalized linear equivalence between functions defined over finite fields; this can be seen as an extension of the classical criterion of linear equivalence, and it is obtained by means of a particular geometric representation of the functions. After giving the basic definitions, we prove that the known equivalence relations can be seen as particular cases of the proposed generalized relationship and that there exist functions that are generally linearly equivalent but are not such in the classical theory. We also prove that the distributions of values in the Difference Distribution Table (DDT) and in the Linear Approximation Table (LAT) are invariants of the new transformation; this gives us the possibility to find some Almost Perfect Nonlinear (APN) functions that are not linearly equivalent (in the classical sense) to power functions, and to treat them accordingly to the new formulation of the equivalence criterion
Addendum to ``On the Generalized Linear Equivalence of Functions over Finite Fields\u27\u27
In this paper we discuss the example of APN permutation introduced in the paper ``On the Generalized Linear Equivalence of Functions over Finite Fields\u27\u27, presented at Asiacrypt 2004. We show that the permutation given there is indeed classically linearly equivalent to a power monomial. More in general, we show that no new class of APN functions can be discovered starting from permutation polynomials of the type used in the paper, and applied on the APN monomial
On Gaussian multiplicative chaos
We propose a new definition of the Gaussian multiplicative chaos (GMC) and an
approach based on the relation of subcritical GMC to randomized shifts of a
Gaussian measure. Using this relation we prove general uniqueness and
convergence results for subcritical GMC that hold for Gaussian fields with
arbitrary covariance kernels.Comment: 34 pages. Major revision of the original version. Numeration has
changed significantl
Finite automata and algebraic extensions of function fields
We give an automata-theoretic description of the algebraic closure of the
rational function field F_q(t) over a finite field, generalizing a result of
Christol. The description takes place within the Hahn-Mal'cev-Neumann field of
"generalized power series" over F_q. Our approach includes a characterization
of well-ordered sets of rational numbers whose base p expansions are generated
by a finite automaton, as well as some techniques for computing in the
algebraic closure; these include an adaptation to positive characteristic of
Newton's algorithm for finding local expansions of plane curves. We also
conjecture a generalization of our results to several variables.Comment: 40 pages; expanded version of math.AC/0110089; v2: refereed version,
includes minor edit
About Hrushovski and Loeser's work on the homotopy type of Berkovich spaces
Those are the notes of the two talks I gave in april 2013 in St-John (US
Virgin Islands) during the Simons Symposium on non-Archimedean and tropical
geometry. They essentially consist of a survey of Hrushovski and Loeser's work
on the homotopy type of Berkovich spaces; the last section explains how the
author has used their work for studying pre-image of skeleta.Comment: 31 pages. This text will appear in the Proceedings Book of the Simons
Symposium on non-Archimedean and tropical geometry (april 2013, US Virgin
Islands). I've taken into account the remarks and suggestion of the referee
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