The topological dual of the space of formal series with any number, even infinite, of noncommutative variables over an Hausdorff topological field, under the product topology, is the space of polynomials. It implies that continuous endomorphisms on series are exactly the infinite but row-finite matrices. Because their totality is a Fréchet algebra, a basic integral and differential calculus can be defined for infinite triangular matrices; such a calculus is futhermore developped in a general way and leads to an exponential-logarithm Lie-like correspondence. These analytic tools are then successfully applied on the first Weyl algebra faithfully represented as an algebra of continuous operators on the space of formal power series (in one variable). Afterwards we prove that every endomorphism of an infinite (countable) dimensional vector space may be explicitly obtained as the sum of a summable family of elementary operators, called ladder operators (generalizing the Weyl algebra) in a way similar to Jacobson's density theorem. By (topological) duality we obtain the same result for continuous operators on a space of infinite linear combinations. Besides we introduce the total (contracted) algebra of a monoid with a zero (as a completion of the usual contracted algebra) which is used to compute new Moebius inversion formulae along with some Hilbert series.
Some informations: In November 8th, 2011, I defended an "habilitation à diriger des recherches" in two fields: Mathematics and Computer Science.Notions of perfect nonlinearity and bent functions are particularly relevant in cryptography because they formalize maximal resistances against the very efficient differential and linear attacks. This thesis is then dedicated to the study of these cryptographic objects. We naturally interpret these notions in a more abstract and theoretical framework essentially by the substitution of the translations which occur in the definition of perfect nonlinearity by any group action. The properties of these actions as fidelity and regularity allow to decline this new concept into several alternatives. We develop as well its dual characterization using the Fourier transform that leads to an adapted notion of bentness. In particular in the case of a non Abelian group action, we use the linear representations theory to establish a dual matrix version. Furthermore, following the same principle, we generalize those combinatorics objects called difference sets which characterize perfect nonlinearity of functions with values in the finite field with two elements. This allows us to exhibit some constructions of functions which satisfy our generalized criteria, in particular in those cases where bent functions in the usual sense do not exist.
Some
informations: I
defended a PhD in Mathematics, with Professor Sami Harari as
supervisor, at the University of South Toulon-Var, in
September
12th, 2005.
Its French title is "Non linéarité parfaite
généralisée au sens des actions de
groupe,
contribution aux fondements de la solidité cryptographique",
which can be translated into "Group-action based generalized perfect
nonlinearity, contribution to the foundations of cryptography".
Here are the slides
of the defense, and a poster.