Jean-Pierre Flori; GÃ©rard Cohen

In this note, we give a simple proof of the combinatorial conjecture proposed by Tang, Carlet and Tang, based on which they constructed two classes of Boolean functions with many good cryptographic properties. We also give more general properties about the generalisation of the conjecture they propose.]]>

Steven T. Dougherty; Jon-Lark Kim; Buket Ozkaya; Lin Sok; Patrick SolÃ©

Linear complementary dual (LCD) codes are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings R<SUB align="right">k. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.]]>

Olivier Hudry; Antoine Lobstein

Given a graph G = (V, E) and an integer r ≥ 1, we call 'r-dominating code' any subset C of V such that every vertex in V is at distance at most r from at least one vertex in C. We investigate and locate in the complexity classes of the polynomial hierarchy, several problems linked with domination in graphs, such as, given r and G, the existence of, or search for, optimal r-dominating codes in G, or optimal r-dominating codes in G containing a subset of vertices X ⊂ V .]]>

Tuvi Etzion

In this paper, we raise a variant of a classic problem in an extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers d > 2, n > 2 and k, what is the minimum possible number of vertices in a d-regular undirected graph whose subgraphs with n vertices contain at most k edges? The goal of this paper is to give the exact number of vertices for each instance of the problem and to provide some bounds for general values of n, d and k. A few general bounds with some exact values, for this Turán-type problem, are given. We present an almost complete solution for 2 < n < 6.]]>

Hugues Randriambololona

This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gérard Cohen influenced important parts of my research. The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalising the notion of separating systems in the context of metric convexity. The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry - sets of points forming only acute angles - who seem not to be aware of each other, while they are working on precisely the same problem! Of course, these three themes are closely intertwined.]]>

Pascale Charpin; Gohar M. Kyureghyan

In this paper, we study computational aspects for determining the differential uniformity of mappings on finite fields of characteristic 2. In particular, we show: (1) A mapping has differential uniformity 2 (i.e. it is almost perfect non-linear) if and only if its difference mappings defined by the elements of a fixed hyperplane are 2-to-1. (2) For a large family of mappings of a special shape, it is enough to consider difference mappings defined by the elements from a suitable multiplicative subgroup.]]>

Hans Georg Schaathun; Minoru Kuribayashi

Digital fingerprinting has been much studied in the literature for more than 20 years, motivated by applications in copyright protection. A popular and practical approach is to use spread spectrum watermarking to embed fingerprints in multimedia objects. Solutions are normally only validated against known attacks in simulation. In this paper, we review known attacks on spread spectrum fingerprinting and give a mathematical argument for the efficacy of the so-called moderated minority extreme (MMX) attack. We also provide a new, mathematical description the obfuscation technique which we proposed at ICIP 2015, identifying some key properties which are necessary to resist the MMX and other attacks.]]>