Generalization of a class of APN binomials to Gold-like functions

In 2008 Budaghyan, Carlet and Leander generalized a known instance of an APN function over the finite field F212 and constructed two new infinite families of APN binomials over the finite field F2n , one for n divisible by 3, and one for n divisible by 4. By relaxing conditions, the family of APN binomials for n divisible by 3 was generalized to a family of differentially 2t -uniform functions in 2012 by Bracken, Tan and Tan; in this sense, the binomials behave in the same way as the Gold functions. In this paper, we show that when relaxing conditions on the APN binomials for n divisible by 4, they also behave in the same way as the Gold function x2s+1 (with s and n not necessarily coprime). As a counterexample, we also show that a family of APN quadrinomials obtained as a generalization of a known APN instance over F210 cannot be generalized to functions with 2t -to-1 derivatives by relaxing conditions in a similar way.acceptedVersio

Classification of all DO planar polynomials with prime field coefficients over GF(3^n) for n up to 7

We describe how any function over a finite field $\mathbb{F}_{p^n}$ can be represented in terms of the values of its derivatives. In particular, we observe that a function of algebraic degree $d$ can be represented uniquely through the values of its derivatives of order $(d-1)$ up to the addition of terms of algebraic degree strictly less than $d$. We identify a set of elements of the finite field, which we call the degree $d$ extension of the basis, which has the property that for any choice of values for the elements in this set, there exists a function of algebraic degree $d$ whose values match the given ones. We discuss how to reconstruct a function from the values of its derivatives, and discuss the complexity involved in transitioning between the truth table of the function and its derivative representation. We then specialize to the case of quadratic functions, and show how to directly convert between the univariate and derivative representation without going through the truth table. We thus generalize the matrix representation of qaudratic vectorial Boolean functions due to Yu et al. to the case of arbitrary characteristic. We also show how to characterize quadratic planar functions using the derivative representation. Based on this, we adapt the method of Yu et al. for searching for quadratic APN functions with prime field coefficients to the case of planar DO functions. We use this method to find all such functions (up to CCZ-equivalence) over $\mathbb{F}_{3^n}$ for $n \le 7$. We conclude that the currently known planar DO polynomials cover all possible cases for $n \le 7$. We find representatives simpler than the known ones for the Zhou-Pott, Dickson, and Lunardon-Marino-Polverino-Trombetti-Bierbrauer families for $n = 6$. Finally, we discuss the computational resources that would be needed to push this search to higher dimensions

On Two Fundamental Problems on APN Power Functions

The six infinite families of power APN functions are among the oldest known instances of APN functions, and it has been conjectured in 2000 that they exhaust all possible power APN functions. Another long-standing open problem is that of the Walsh spectrum of the Dobbertin power family, which is still unknown. Those of Kasami, Niho and Welch functions are known, but not the precise values of their Walsh transform, with rare exceptions. One promising approach that could lead to the resolution of these problems is to consider alternative representations of the functions in questions. We derive alternative representations for the infinite APN monomial families. We show how the Niho, Welch, and Dobbertin functions can be represented as the composition xi∘x1/j of two power functions, and prove that our representations are optimal, i.e. no two power functions of lesser algebraic degree can be used to represent the functions in this way. We investigate compositions xi∘L∘x1/j for a linear polynomial L , show how the Kasami functions in odd dimension can be expressed in this way with i=j being a Gold exponent and compute all APN functions of this form for n≤9 and for L with binary coefficients, thereby showing that our theoretical constructions exhaust all possible cases. We present observations and data on power functions with exponent ∑k−1i=122ni−1 which generalize the inverse and Dobbertin families. We present data on the Walsh spectrum of the Dobbertin function for n≤35 , and conjecture its exact form. As an application of our results, we determine the exact values of the Walsh transform of the Kasami function at all points of a special form. Computations performed for n≤21 show that these points cover about 2/3 of the field.acceptedVersio

Relation between o-equivalence and EA-equivalence for Niho bent functions

Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence.publishedVersio

Mitochondrial physiology

As the knowledge base and importance of mitochondrial physiology to evolution, health and disease expands, the necessity for harmonizing the terminology concerning mitochondrial respiratory states and rates has become increasingly apparent. The chemiosmotic theory establishes the mechanism of energy transformation and coupling in oxidative phosphorylation. The unifying concept of the protonmotive force provides the framework for developing a consistent theoretical foundation of mitochondrial physiology and bioenergetics. We follow the latest SI guidelines and those of the International Union of Pure and Applied Chemistry (IUPAC) on terminology in physical chemistry, extended by considerations of open systems and thermodynamics of irreversible processes. The concept-driven constructive terminology incorporates the meaning of each quantity and aligns concepts and symbols with the nomenclature of classical bioenergetics. We endeavour to provide a balanced view of mitochondrial respiratory control and a critical discussion on reporting data of mitochondrial respiration in terms of metabolic flows and fluxes. Uniform standards for evaluation of respiratory states and rates will ultimately contribute to reproducibility between laboratories and thus support the development of data repositories of mitochondrial respiratory function in species, tissues, and cells. Clarity of concept and consistency of nomenclature facilitate effective transdisciplinary communication, education, and ultimately further discovery

Mitochondrial physiology

As the knowledge base and importance of mitochondrial physiology to evolution, health and disease expands, the necessity for harmonizing the terminology concerning mitochondrial respiratory states and rates has become increasingly apparent. The chemiosmotic theory establishes the mechanism of energy transformation and coupling in oxidative phosphorylation. The unifying concept of the protonmotive force provides the framework for developing a consistent theoretical foundation of mitochondrial physiology and bioenergetics. We follow the latest SI guidelines and those of the International Union of Pure and Applied Chemistry (IUPAC) on terminology in physical chemistry, extended by considerations of open systems and thermodynamics of irreversible processes. The concept-driven constructive terminology incorporates the meaning of each quantity and aligns concepts and symbols with the nomenclature of classical bioenergetics. We endeavour to provide a balanced view of mitochondrial respiratory control and a critical discussion on reporting data of mitochondrial respiration in terms of metabolic flows and fluxes. Uniform standards for evaluation of respiratory states and rates will ultimately contribute to reproducibility between laboratories and thus support the development of data repositories of mitochondrial respiratory function in species, tissues, and cells. Clarity of concept and consistency of nomenclature facilitate effective transdisciplinary communication, education, and ultimately further discovery

On properties of bent and almost perfect nonlinear functions

(Vectorial) Boolean functions play an important role in all domains related to computer science, and in particular, in cryptography. The safety of a cryptosystem is quantified via some characteristics of (vectorial) Boolean functions implemented in it. The nonlinearity and differential uniformity are among the most important characteristics of cryptographic Boolean functions. Thus, bent and almost perfect nonlinear functions, which have the best possible nonlinearity and differential uniformity, respectively, are optimal cryptographic objects. This thesis is devoted to the investigation of the propertiesof these functions and is based on published articles. In Paper I, a special subclass of bent Boolean functions, Niho bent functions, is studded. Boolean functions, and bent functions in particular, are considered up to the so-called EA-equivalence, which is the most general known equivalence relation preserving bentness. However, for Niho bent functions, there is a more general equivalence relation called o-equivalence, which is induced from the equivalence of o-polynomials (a special type of permutation polynomials). In this paper we study a group of transformations which generates all possible o-equivalent Niho bent functions from a given o-polynomial, and we exclude all transformations that never produce EA-inequivalent functions. We identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions in a same o-equivalence class. For all known o-monomials, we identify the exact form of transformations which always lead to EA-inequivalent Niho bent functions. For o-polynomials, which are not monomials, we identify the exact form of transformations which can potentially lead to EA-inequivalent functions. Paper II is devoted to the study of two long-standing open problems about APN power functions. The six infinite families of APN power functions are among the oldest known instances of APN functions. It was conjectured in 2000 that there does not exist any APN power function inequivalent to the known ones. This is the first long term open problem we study in Paper II. The functions affine equivalent to a power function have the form the composition of two linear transformations and the power function in between. This gives an idea to examine the composition of power functions with a linear function in between. So, we investigate such compositions , and show that some of the known APN power functions can be obtained from other known APN power functions through this construction. Moreover, we compute all APN functions of this form for n less or equal than 9 and for linear functions with binary coefficients, thereby confirming that our theoretical constructions exhaust all possible cases of known APN power functions. In addition, we present observations and data on power functions with exponents of the special form defined over the field of the dimension mk which generalize the inverse and the Dobbertin families of APN power functions. Another long-standing open problem is the Walsh spectrum of the Dobbertin APN power family. In Paper II, we derive alternative representations for some of the known families of APN monomials. We show that the Niho and Dobbertin functions can be represented as the composition of two power functions, and prove that our representations are optimal, i.e. no two power functions of lesser algebraic degree can produce the same composition. We show as well that the exponents of the Welch functions are optimal in this sense. Based on a computational data performed for n less or equal than 35, we present a conjecture depending on the parity of n, which wholly describes the Walsh spectrum of the Dobbertin functions. In Paper III, we generalize an infinite family of APN binomials, for n divisible by 4 behaves exactly as the Gold function in respect to their differential uniformity, the size of the image space and being permutations

Generalization of a class of APN binomials to Gold-like functions

In 2008 Budaghyan, Carlet and Leander generalized a known instance of an APN function over the finite field F212 and constructed two new infinite families of APN binomials over the finite field F2n , one for n divisible by 3, and one for n divisible by 4. By relaxing conditions, the family of APN binomials for n divisible by 3 was generalized to a family of differentially 2t -uniform functions in 2012 by Bracken, Tan and Tan; in this sense, the binomials behave in the same way as the Gold functions. In this paper, we show that when relaxing conditions on the APN binomials for n divisible by 4, they also behave in the same way as the Gold function x2s+1 (with s and n not necessarily coprime). As a counterexample, we also show that a family of APN quadrinomials obtained as a generalization of a known APN instance over F210 cannot be generalized to functions with 2t -to-1 derivatives by relaxing conditions in a similar way

On Two Fundamental Problems on APN Power Functions

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