Ary Functions Research Articles

On a class of bent, near-bent, and 2-plateaued functions over finite fields of odd characteristic

<abstract><p>The main purpose of this paper is to study a class of the $ p $-ary functions $ f_{\lambda, u, v}(x) = Tr_1^k(\lambda x^{p^k+1})+Tr^n_1(ux)Tr_1^n(vx) $ for any odd prime $ p $ and $ n = 2k, \lambda\in GF(p^k)^*, u, v\in GF(p^n)^*. $ With the help of Fourier transforms, we are able to subdivide the class of all $ f_{\lambda, u, v} $ into sublcasses of bent, near-bent and 2-plateaued functions. It is shown that the choice of $ \lambda, u $ and $ v $, ensuring that $ f $ is bent, 2-plateaued or near-bent, is directly related to finding the subset $ A\subset GF(p)^3 $. The efficient method for defining the set $ A\subset GF(p)^3 $ is described in detail.</p></abstract>

Secondary constructions of (non)weakly regular plateaued functions over finite fields

Plateaued (vectorial) functions over finite fields have diverse applications in symmetric cryptography, coding theory, and sequence theory. Constructing these functions is an attractive research topic in the literature. We can distinguish two kinds of constructions of plateaued functions: secondary constructions and primary constructions. The first method uses already known functions to obtain new functions while the latter do not need to use previously constructed functions to obtain new functions. In this work, the first secondary constructions of (non)weakly regular plateaued (vectorial) functions are presented over the finite fields of odd characteristics. We also introduce some recursive constructions of (non)weakly regular plateaued $p$-ary functions by using already known such functions. We obtain nontrivial plateaued functions from the previously known trivial plateaued (partially bent) functions in the proposed construction methods.

On the Number of Affine Equivalence Classes of Boolean Functions and q-Ary Functions

Let R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (r,n) be the rth order q-ary Reed-Muller code of length q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , which is the set of functions from \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> to \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> represented by polynomials of degree ≤ r in \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> [X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , ... ,X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ]. The affine linear group AGL(n,\mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> ) acts naturally on R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (r,n). We derive two formulas concerning the number of orbits of this action: (i) an explicit formula for the number of AGL orbits of R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n(q-1),n), and (ii) an asymptotic formula for the number of AGL orbits of R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n,n)/R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (1,n). The number of AGL orbits of R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n,n) has been numerically computed by several authors for n ≤ 31; the binary case of result (i) is a theoretic solution to the question. Result (ii) answers a question by MacWilliams and Sloane.

The quest for Diophantine finite-fold-ness

The Davis-Putnam-Robinson theorem showed that every partially computable $m$-ary function f(a1, ..., am) = c on the natural numbers can be specified by means of an exponential Diophantine formula involving, along with parameters a1, ... am, c, some number k of existentially quantified variables. Yuri Matiyasevich improved this theorem in two ways: on the one hand, he proved that the same goal can be achieved with no recourse to exponentiation and, thereby, he provided a negative answer to Hilbert's 10th problem; on the other hand, he showed how to construct an exponential Diophantine equation specifying f which, once a1, ... am have been fixed, is solved by at most one tuple of values for the remaining variables. This latter property is called single-foldness. Whether there exists a single- (or, at worst, finite-) fold polynomial Diophantine representation of any partially computable function on the natural numbers is as yet an open problem. This work surveys relevant results on this subject and tries to draw a route towards a hoped-for positive answer to the finite-fold-ness issue.

Some Properties of Cryptographic Functions Employed in Wireless Communication Systems

The wireless transmission of different type of data like images, texts, videos etc. through an open network is considered as prone with respect to several security threats and passive attacks. CDMA communication systems used for various wireless communications requires improved cryptographic algorithms and air interface security checks. The -ary cryptographic function with suitable properties are key elements in providing secure transmission of data through wireless network. In this paper, we use the Walsh-Hadamard spectrum as a fundamental tool for analysis of some properties of generalized -ary functions. We generalize some existing results obtained for Boolean functions to the -ary functions and obtain some new characterization of -ary functions based on spectral analysis.

On m-Junctive Predicates on a Finite Set

We consider predicates on finite sets. The predicates invariant under some (m + 1)-ary near-unanimity function are called m-junctive. We propose to represent the predicates on a finite set in generalized conjunctive normal forms (GCNFs). The properties for GCNFs of m-junctive predicates are obtained. We prove that each m-junctive predicate can be represented by a strongly consistent GCNF in which every conjunct contains at most m variables. This representation of an m-junctive predicate is called reduced. Some fast algorithm is proposed for finding a reduced representation for an m-junctive predicate. It is shown how the obtained properties of GCNFs of m-junctive predicates can be applied for constructing a fast algorithm for the generalized S-satisfiability problem in the case that S contains only the predicates invariant under a common near unanimity function.

Characterization of &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$p$ &lt;/tex-math&gt; &lt;/inline-formula&gt;-ary Bent Functions in Terms of Strongly Regular Graphs

A $p$ -ary function $f$ in $n$ variables is an $l$ - form if $f({\mathrm{ tu}})=t^{l}f(u)$ for any nonzero $t$ in $\mathbb {Z}_{p}$ and $u$ in $\mathbb {Z}^{n}_{p}$ . Let $n$ be a positive even integer, $p$ an odd prime, and $l$ an element of $\{1,2,\ldots ,p-1\}$ provided that $l \neq p-1$ if $p>3$ . Let $f$ be a $p$ -ary bent function in $n$ variables of $l$ -form with $f(0)=0$ and $\gcd (l-1,p-1)=1$ , and let $H_{l}=\{t^{l}:t\in \mathbb {Z}^{*}_{p}\}$ . We denote by $G_{f,l}$ the Cayley graph ${\mathrm {Cay}} (\mathbb {Z}^{n}_{p},\cup _{s\in H_{l}}f^{-1}(s))$ . Our main results are as follows: 1) if there is weakly regular $p$ -ary bent $f$ which is not regular, then $l$ is 2; 2) if $l=2$ , then $f$ is weakly regular $p$ -ary bent if and only if the Cayley graph $G_{f,l}$ is strongly regular; 3) if $l\neq 2$ , then $f$ is regular $p$ -ary bent if and only if the Cayley graph $G_{f,l}$ is strongly regular; 4) $G_{f,l}$ can be replaced by ${\mathrm {Cay}} (\mathbb {Z}^{n}_{p},f^{-1}(0)\setminus \{0\})$ in 2) and 3); and 5) amorphic association schemes are derived by using 2) and 3). We prove our main results by computing at most four distinct restricted eigenvalues of $G_{f,l}$ .

Combination of Video Change Detection Algorithms by Genetic Programming

Within the field of computer vision, change detection algorithms aim at automatically detecting significant changes occurring in a scene by analyzing the sequence of frames in a video stream. In this paper we investigate how state-of-the-art change detection algorithms can be combined and used to create a more robust algorithm leveraging their individual peculiarities. We exploited genetic programming (GP) to automatically select the best algorithms, combine them in different ways, and perform the most suitable post-processing operations on the outputs of the algorithms. In particular, algorithms’ combination and post-processing operations are achieved with unary, binary and ${n}$ -ary functions embedded into the GP framework. Using different experimental settings for combining existing algorithms we obtained different GP solutions that we termed In Unity There Is Strength . These solutions are then compared against state-of-the-art change detection algorithms on the video sequences and ground truth annotations of the ChangeDetection.net 2014 challenge. Results demonstrate that using GP, our solutions are able to outperform all the considered single state-of-the-art change detection algorithms, as well as other combination strategies. The performance of our algorithm are significantly different from those of the other state-of-the-art algorithms. This fact is supported by the statistical significance analysis conducted with the Friedman test and Wilcoxon rank sum post-hoc tests.

Generalized Plateaued Functions and Admissible (Plateaued) Functions

Plateaued functions are very important crypto- graphic functions due to their various desirable cryptographic characteristics. We point out that plateaued functions are more general than bent functions (that is, functions with maximum nonlinearity). Some Boolean plateaued functions have large nonlinearity, which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers, and contributes, when they are the component functions of the substitution boxes in block ciphers, to protection against linear cryptanalysis. P-ary plateaued functions have attracted recently some attention in the literature, and many activities on generalized $p$ -ary functions have been carried out. This paper increases our knowledge on plateaued functions in the general context of generalized $p$ -ary functions. We first introduce two new versions of plateaued functions, which we shall call generalized plateaued functions and admissible plateaued functions. The generalized plateaued functions extend the standard notion of plateaued $p$ -ary functions to those whose outputs are in the ring $\mathbb {Z}_{p^{k}}$ . Next, we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components. Finally, we provide for the first time two constructions of generalized plateaued functions. In particular, we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes.

Decompositions of functions based on arity gap

We study the arity gap of functions of several variables defined on an arbitrary set A and valued in another set B . The arity gap of such a function is the minimum decrease in the number of essential variables when variables are identified. We establish a complete classification of functions according to their arity gap, extending existing results for finite functions. This classification is refined when the codomain B has a group structure, by providing unique decompositions into sums of functions of a prescribed form. As an application of the unique decompositions, in the case of finite sets we count, for each n and p , the number of n -ary functions that depend on all of their variables and have arity gap p .

Superpositions of continuous functions defined on a Baire space

We consider uniformly continuous functions on a Baire space and introduce the notion of a continuity modulus of a function. We formulate a condition on the growth of the continuity modulus ? guaranteeing that superpositions of n-ary functions with continuity modulus ? do not exhaust all (n + 1)-ary functions with continuity modulus ? for any n. Moreover, negating this property leads to the inverse effect.

Covering $\mathbb R^{n+1}$ by graphs of $n$-ary functions and long linear orderings of Turing degrees

A point (x 0 ,...,x n ) ∈ X n+1 is covered by a function f: X n → X iff there is a permutation σ of n + 1 such that x σ(0) = f(x σ(1) ,...,x σ(n) ). By a theorem of Kuratowski, for every infinite cardinal κ exactly κ n-ary functions are needed to cover all of (K +n ) n+1 . We show that for arbitrarily large uncountable κ it is consistent that the size of the continuum is C +n and R n+1 is covered by κ n-ary continuous functions. We study other cardinal invariants of the σ-ideal on R n+1 generated by continuous n-ary functions and finally relate the question of how many continuous functions are necessary to cover R 2 to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.

On an infinite sequence of improving Boolean bases

We consider complexity of formulas for Boolean functions in finite complete bases. It is shown that, as regards complexity, the basis of all ( k+1)-ary functions is essentially better than the basis of all k-ary functions for all k⩾2.

$1$-ary functions and the F. C. P.

$1$-ary functions and the F. C. P.

Classification theory for a $1$-ary function

The classification problem, namely the problem of characterizing classifiable theories, was (almost) thoroughly solved by S. Shelah. It involves an obvious algebraic question, namely to find, given an elementary class K of 1st order structures, under which conditions a structure M∈K satisfies «Th(M) classifiable». This is the question we deal with for the class K of all structures M with a 1-ary (total) function