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Laurent Poinsot

  1. Doubly Perfect Nonlinear Boolean Permutations.

    Fri, 03/26/2010 - 16:01 — SomNambul
    Authors: Laurent Poinsot
    Subjects: Discrete Mathematics
    Abstract

    Due to implementation constraints the XOR operation is widely used in order
    to combine plaintext and key bit-strings in secret-key block ciphers. This
    choice directly induces the classical version of the differential attack by the
    use of XOR-kind differences. While very natural, there are many alternatives to
    the XOR. Each of them inducing a new form for its corresponding differential
    attack (using the appropriate notion of difference) and therefore block-ciphers
    need to use S-boxes that are resistant against these nonstandard differential
    cryptanalysis.

  2. A formal calculus on the Riordan near algebra.

    Tue, 03/02/2010 - 16:02 — SomNambul
    Authors: Laurent Poinsot, Gérard Duchamp
    Subjects: Symbolic Computation
    Abstract

    The Riordan group is the semi-direct product of a multiplicative group of
    invertible series and a group, under substitution, of non units. The Riordan
    near algebra, as introduced in this paper, is the Cartesian product of the
    algebra of formal power series and its principal ideal of non units, equipped
    with a product that extends the multiplication of the Riordan group. The later
    is naturally embedded as a subgroup of units into the former. In this paper, we
    prove the existence of a formal calculus on the Riordan algebra.

  3. Partial monoids: associativity and confluence.

    Thu, 02/11/2010 - 16:02 — SomNambul
    Authors: Laurent Poinsot, Gérard Duchamp, Christophe Tollu
    Subjects: Discrete Mathematics
    Abstract

    A partial monoid $P$ is a set with a partial multiplication $\times$ (and
    total identity $1_P$) which satisfies some associativity axiom. The partial
    monoid $P$ may be embedded in a free monoid $P^*$ and the product $\star$ is
    simulated by a string rewriting system on $P^*$ that consists in evaluating the
    concatenation of two letters as a product in $P$, when it is defined, and a
    letter $1_P$ as the empty word $\epsilon$. In this paper we study the profound
    relations between confluence for such a system and associativity of the
    multiplication.

  4. M\"obius inversion formula for monoids with zero.

    Thu, 11/26/2009 - 16:01 — SomNambul
    Authors: Laurent Poinsot, Gérard Duchamp, Christophe Tollu
    Subjects: Combinatorics
    Abstract

    The M\"obius inversion formula, introduced during the 19th century in number
    theory, was generalized to a wide class of monoids called locally finite such
    as the free partially commutative, plactic and hypoplactic monoids for
    instance. In this contribution are developed and used some topological and
    algebraic notions for monoids with zero, similar to ordinary objects such as
    the (total) algebra of a monoid, the augmentation ideal or the star operation
    on proper series.

  5. Ladder Operators and Endomorphisms in Combinatorial Physics.

    Tue, 08/18/2009 - 11:53 — SomNambul
    Authors: Gérard Henry Edmond Duchamp, Laurent Poinsot, Allan I. Solomon, Karol A. Penson, Pawel Blasiak, Andrzej Horzela
    Subjects: Combinatorics
    Abstract

    Starting with the Heisenberg-Weyl algebra, fundamental to quantum physics, we
    first show how the ordering of the non-commuting operators intrinsic to that
    algebra gives rise to generalizations of the classical Stirling Numbers of
    Combinatorics. These may be expressed in terms of infinite, but {\em
    row-finite}, matrices, which may also be considered as endomorphisms of
    $\C[[x]]$. This leads us to consider endomorphisms in more general spaces, and
    these in turn may be expressed in terms of generalizations of the
    ladder-operators familiar in physics.

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