Couceiro, Miguel Lehtonen, Erkko

In this paper we address the question "How many properties of Boolean functions can be defined by means of linear equations?" It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related...

Amarilli, Antoine Capelli, Florent Monet, Mikaël Senellart, Pierre
Published in
Theory of Computing Systems

The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width ...

Hao, Xuexuan Zhang, Fengrong Xia, Shixiong Zhou, Yong
Published in
Quantum Information Processing

Quantum algorithms for the analysis of Boolean functions have received a lot of attention over the last few years. The algebraic normal form (ANF) of a linear Boolean function can be recovered by using the Bernstein–Vazirani (BV) algorithm. No research has been carried out on quantum algorithms for learning the ANF of general Boolean functions. In ...

Logachev, Oleg A. Fedorov, Sergey N. Yashchenko, Valerii V.
Published in
Discrete Mathematics and Applications

A new equivalence relation on the set of Boolean functions is introduced: functions are declared to be Δ-equivalent if their autocorrelation functions are equal. It turns out that this classification agrees well with the cryptographic properties of Boolean functions: for functions belonging to the same Δ-equivalence class a number of their cryptogr...

Couceiro, Miguel Lehtonen, Erkko Mercuriali, Pierre Péchoux, Romain

A normal form system (NFS) for representing Boolean functions is thought of as a set of stratified terms over a fixed set of connectives. For a fixed NFS A, the complexity of a Boolean function f with respect to A is the minimum of the sizes of terms in A that represent f. This induces a preordering of NFSs: an NFS A is polynomially as efficient as...

Redkin, Nikolay P.
Published in
Discrete Mathematics and Applications

We study generalized (in terms of bases) complexity of implementation of linear Boolean functions by Boolean circuits in arbitrary functionally complete bases; the complexity of a circuit is defined as the number of gates. Let L*(n) be the minimal number of gates sufficient for implementation of an arbitrary linear Boolean function of n variables i...

Beierle, Christof Biryukov, Alex Udovenko, Aleksei

A set ⊆ 2 is called degree-d zero-sum if the sum ∑ ∈ ( ) vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean functions of degree at most n − d − 1. We prove some results on the existence of degree-d zero-sum sets of full rank, i.e., those that contain n linearly independ...

Charpin, Pascale Peng, Jie

The associated codes of almost perfect nonlinear (APN) functions have been widely studied. In this paper we consider more generally the codes associated with functions that have differential uniformity at least 4. We emphasize, for such a function F , the role of codewords of weight 3 and 4 and of some cosets of its associated code C F. We give som...

Canteaut, Anne Perrin, Léo Tian, Shizhu

Whether there exist Almost Perfect Non-linear permutations (APN) operating on an even number of bit is the so-called Big APN Problem. It has been solved in the 6-bit case by Dillon et al. in 2009 but, since then, the general case has remained an open problem. In 2016, Perrin et al. discovered the butterfly structure which contains Dillon et al.'s p...

Charpin, Pascale

Crooked permutations were defined twenty years ago. It was firstly shown that they can be used to construct interesting objects in graph theory. The field of applications was extended later, since crooked functions, bijective or not, correspond to APN functions and to some optimal codes. We adopt an unified presentation, of crooked functions, expla...