As a core component of Substitution-Permutation Networks, diffusion layer is mainly introduced by matrices from maximum distance separable (MDS) codes. Surprisingly, up to now, most constructions of MDS matrices require to perform an equivalent or even exhaustive search. Especially, not many MDS proposals are known that obtain an excellent hardware efficiency and simultaneously guarantee a rema...

Traditionally, communication networks are composed of routing nodes, which relay and duplicate data. Work in recent years has shown that for the case of multicast, an improvement in both rate and code-construction complexity can be gained by replacing these routing nodes by linear coding nodes. These nodes transmit linear combinations of the inputs transmitted to them. In this work, we deal wit...

This work considers the problem of constructing efficient MDS matrices over the field F2m . Efficiency is measured by the metric XOR count which was introduced by Khoo et al. in CHES 2014. Recently Sarkar and Syed (ToSC Vol. 1, 2016) have shown the existence of 4 × 4 Toeplitz MDS matrices with optimal XOR counts. In this paper, we present some characterizations of Toeplitz matrices in light of ...

We investigate in this paper the dependence relation between the space-time periodic coefficients A, q and μ of the reaction-diffusion equation ∂tu−∇ · (A(t, x)∇u) + q(t, x) · ∇u = μ(t, x)u(1− u), and the spreading speed of the solutions of the Cauchy problem associated with this equation and compactly supported initial data. We prove in particular that (1) taking the spatial or temporal averag...

The radiation (reaction, Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary, however, the relation between the reaction constant in the Robin boundary condition and t...

In this article, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size ε, and its asymptotic behavior as the parameter ε vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the ‘trivial’ eigenvalues 0 and 1, and of a subset which ...

We study the existence and the uniqueness of the solution to a class of Fokker-Planck type equations with irregular coefficients, more precisely with coefficients in Sobolev spaces W . Our arguments are based upon the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations [6]. The present work extends the results of our previous article [17], where on...

We study the persistence and propagation (or blocking) phenomena for a species in periodically hostile environments. The problem is described by a reaction-diffusion equation with zero Dirichlet boundary condition. We first derive the existence of a minimal nonnegative nontrivial stationary solution and study the large-time behavior of the solution of the initial boundary value problem. To the ...

Ciphers get their strength from the mathematical functions of confusion and diffusion, also known as substitution and permutation. These were the basics of classical cryptography and they are still the basic part of modern ciphers. In block ciphers diffusion is achieved by the use of Maximum Distance Separable (MDS) matrices. In this paper we present some methods for constructing dynamic (and r...

We describe Fugue, a hash function supporting inputs of length upto 2 − 1 bits and hash outputs of length upto 512 bits. Notably, Fugue is not based on a compression function. Rather, it is directly a hash function that supports variable-length inputs. The starting point for Fugue is the hash function Grindahl, but it extends that design to protect against the kind of attacks that were develope...