509 research outputs found
Recursions associated to trapezoid, symmetric and rotation symmetric functions over Galois fields
This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordRotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of some rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of F2, an elementary method is used to obtain explicit recurrences for exponential sums of some of these functions. The concept of trapezoid Boolean function is also introduced and it is showed that the linear recurrences that exponential sums of trapezoid Boolean functions satisfy are the same as the ones satisfied by exponential sums of the corresponding rotations symmetric Boolean functions. Finally, it is proved that exponential sums of trapezoid and symmetric polynomials also satisfy linear recurrences with integer coefficients over any Galois field F2. Moreover, the Discrete Fourier Transform matrix and some Complex Hadamard matrices appear as examples in some of our explicit formulas of these recurrences
Boolean delay equations on networks: An application to economic damage propagation
We introduce economic models based on Boolean Delay Equations: this formalism
makes easier to take into account the complexity of the interactions between
firms and is particularly appropriate for studying the propagation of an
initial damage due to a catastrophe. Here we concentrate on simple cases, which
allow to understand the effects of multiple concurrent production paths as well
as the presence of stochasticity in the path time lengths or in the network
structure.
In absence of flexibility, the shortening of production of a single firm in
an isolated network with multiple connections usually ends up by attaining a
finite fraction of the firms or the whole economy, whereas the interactions
with the outside allow a partial recovering of the activity, giving rise to
periodic solutions with waves of damage which propagate across the structure.
The damage propagation speed is strongly dependent upon the topology. The
existence of multiple concurrent production paths does not necessarily imply a
slowing down of the propagation, which can be as fast as the shortest path.Comment: Latex, 52 pages with 22 eps figure
Counting packings of generic subsets in finite groups
A packing of subsets in a group is a
sequence such that are
disjoint subsets of . We give a formula for the number of packings if the
group is finite and if the subsets satisfy
a genericity condition. This formula can be seen as a generalization of the
falling factorials which encode the number of packings in the case where all
the sets are singletons
Quantum linearization attacks
Recent works have shown that quantum period-finding can be used to break many popular constructions (some block ciphers such as Even-Mansour, multiple MACs and AEs...) in the superposition query model. So far, all the constructions broken exhibited a strong algebraic structure, which enables to craft a periodic function of a single input block. Recoverin
A friendly introduction to Fourier analysis on polytopes
This book is an introduction to the nascent field of Fourier analysis on
polytopes, and cones. There is a rapidly growing number of applications of
these methods, so it is appropriate to invite students, as well as
professionals, to the field. We assume a familiarity with Linear Algebra, and
some Calculus. Of the many applications, we have chosen to focus on: (a)
formulations for the Fourier transform of a polytope, (b) Minkowski and
Siegel's theorems in the geometry of numbers, (c) tilings and multi-tilings of
Euclidean space by translations of a polytope, (d) Computing discrete volumes
of polytopes, which are combinatorial approximations to the continuous volume,
(e) Optimizing sphere packings and their densities, and (f) use iterations of
the divergence theorem to give new formulations for the Fourier transform of a
polytope, with an application. Throughout, we give many examples and exercises,
so that this book is also appropriate for a course, or for self-study.Comment: 204 pages, 46 figure
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