Covering Radius of the (n−3)(n-3)-rd Order Reed-Muller Code in the Set of Resilient Functions

In this paper, we continue the study of the covering radius in the set of resilient functions, which has been defined by Kurosawa. This new concept is meaningful to cryptography especially in the context of the new class of algebraic attacks on stream ciphers proposed by Courtois and Meier at Eurocrypt 2003 and Courtois at Crypto 2003. In order to resist such attacks the combining Boolean function should be at high distance from lower degree functions. Using a result from coding theory on the covering radius of $(n-3)$-rd Reed-Muller codes, we establish exact values of the the covering radius of $RM(n-3,n)$ in the set of $1$-resilient Boolean functions of $n$ variables, when $\lfloor n/2 \rfloor = 1 \mod\;2$. We also improve the lower bounds for covering radius of the Reed-Muller codes $RM(r,n)$ in the set of $t$-resilient functions, where $\lceil r/2 \rceil = 0 \mod\;2$, $t \leq n-r-2$ and $n\geq r+3$

A Construction of Bent Functions of n + 2 Variables from a Bent Function of n Variables and Its Cyclic Shifts

We present a method to iteratively construct new bent functions of n + 2 variables from a bent function of n variables and its cyclic shift permutations using minterms of n variables and minterms of 2 variables. In addition, we provide the number of bent functions of n + 2 variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions.The work of the first author was partially supported by Spanish Grant MTM2011-24858 of the Ministerio de Economía y Competitividad of the Gobierno de España

The degree of a Boolean function and some algebraic properties of its support

In this paper, the support of a Boolean function is used to establish some algebraic properties. These properties allow the degree of a Boolean function to be obtained without having to calculate its algebraic normal form. Furthermore, some algorithms are derived and the average time computed to obtain the degree of some Boolean functions from its support.Partially supported by Spanish grant MTM2011-24858 of the Ministerio de Economía y Competitividad of the Gobierno de España and by the research project UMH-Bancaja with reference IPZS01

The complexity of Boolean functions from cryptographic viewpoint

Cryptographic Boolean functions must be complex to satisfy Shannon\u27s principle of confusion. But the cryptographic viewpoint on complexity is not the same as in circuit complexity. The two main criteria evaluating the cryptographic complexity of Boolean functions on $F_2^n$ are the nonlinearity (and more generally the $r$-th order nonlinearity, for every positive $r< n$) and the algebraic degree. Two other criteria have also been considered: the algebraic thickness and the non-normality. After recalling the definitions of these criteria and why, asymptotically, almost all Boolean functions are deeply non-normal and have high algebraic degrees, high ($r$-th order) nonlinearities and high algebraic thicknesses, we study the relationship between the $r$-th order nonlinearity and a recent cryptographic criterion called the algebraic immunity. This relationship strengthens the reasons why the algebraic immunity can be considered as a further cryptographic complexity criterion