Marco Calderini

Bounding the Optimal Rate of the ICSI and ICCSI problem

In this work we study both the index coding with side information (ICSI) problem introduced by Birk and Kol in 1998 and the more general problem of index coding with coded side information (ICCSI), described by Shum et al in 2012. We estimate the optimal rate of an instance of the index coding problem. In the ICSI problem case, we characterize those digraphs having min-rank one less than their order and we give an upper bound on the min-rank of a hypergraph whose incidence matrix can be associated with that of a 2-design. Security aspects are discussed in the particular case when the design is a projective plane. For the coded side information case, we extend the graph theoretic upper bounds given by Shanmugam et al in 2014 on the optimal rate of index code.

A note on APN permutations in even dimension

APN permutations in even dimension are vectorial Boolean functions that play a special role in the design of block ciphers. We study their properties, providing some general results and some applications to the low-dimension cases. In particular, we prove that none of their components can be quadratic. For an APN vectorial Boolean function (in even dimension) with all cubic components we prove the existence of a component having a large number of balanced derivatives. Using these restrictions, we obtain the first theoretical proof of the non-existence of APN permutations in dimension 4. Moreover, we derive some contraints on APN permutations in dimension 6.

On weak differential uniformity of vectorial Boolean functions as a cryptographic criterion

We study the relation between weakly differential uniformity and other security parameters for Boolean functions. In particular, we focus on both power functions and 4-bit S-Boxes.We study the relation among some security parameters for vectorial Boolean functions which prevent attacks on the related block cipher. We focus our study on a recently-introduced security criterion, called weak differential uniformity, which prevents the existence of an undetectable trapdoor based on imprimitive group action. We present some properties of functions with low weak differential uniformity, especially for the case of power functions and 4-bit S-Boxes.

On Differential Uniformity of Maps that May Hide an Algebraic Trapdoor

We investigate some differential properties for permutations in the affine group, of a vector space V over the binary field, with respect to a new group operation

Generalized Algebraic Geometric Codes From Maximal Curves

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Index Coding, Network Coding and Broadcast with Side-Information

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Primitivity of PRESENT and other lightweight ciphers

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Generalized AG Codes as Evaluation Codes

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Elementary abelian regular subgroups as hidden sums for cryptographic trapdoors

Some new results on Generalized Algebraic Geometric (GAG) codes are obtained. First, we provide some constructions which significantly improve the general lower bounds on the minimum distance of a GAG code. GAG codes associated to specific maximal curves over finite fields are then investigated. As a result, 2895 improvements on MinTs tables are obtained. Finally, we construct asymptotically good GAG codes with better parameters with respect to those constructed by Spera in 2005. Maximal curves play a role in this context as well.

Error Correction for Index Coding With Coded Side Information

Index coding, the problem of efficient broadcast to many receivers with side-infomation, is a rich and active research area. It has applications to a range of multi-user broadcast scenarios such as video-on-demand and satellite communications. It has attracted significant theoretical interest both as a hard problem in its own right and due to its connections to other network capacity problems. The central problem of index coding, that of determining the optimal rate of an index code, is still open. We describe recent advances on the index coding problem and its generalizations in the context of broadcast with side-information. The two main approaches to bounding the optimal rate of an index code, namely rank-minimization methods and linear programming models, are discussed in detail. The latter of these, based on graph-theoretical ideas in the classical case, can be extended even to generalizations of the index coding problem for which there is no associated side-information hypergraph. We discuss error-correction in the index coding problem, the corresponding bounds on the optimal transmission rate and decoding algorithms. We also illustrate the connections to network coding, interference alignment and coded caching.