Publications

Using $P_{\tau}$ property for designing bent functions provably outside the completed Maiorana-McFarland class

Published in Design, Codes and Cryptography, 2024

Abstract: In this article, we identify certain instances of bent functions, constructed using the so-called $P_{\tau}$ property, that are provably outside the completed Maiorana-McFarland ($\mathcal{MM}^{\#}$) class.This also partially answers an open problem in posed by Kan et al. (IEEE Trans. Inf. Theory, doi: 10.1109/TIT.2022.3140180. (2022)). We show that this design framework (using the $P_{\tau}$ property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes $\mathcal{MM}^{\#}$, $\mathcal{C},\mathcal{D}$ and $\mathcal{D}_{0}$, where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form $h(y,z)=Tr(y\pi(z)) + G_1(Tr_1^m(\alpha_1y),\ldots,Tr_1^m(\alpha_ky))G_2(Tr_1^m(\beta_{k+1}z),\ldots,Tr_1^m(\beta_{\tau}z))+ G_3(Tr_1^m(\alpha_1y),\ldots,Tr_1^m(\alpha_ky))$ to be bent and outside $\mathcal{MM}^{\#}$ is specified without a strong assumption that the components of the permutation $\pi$ do not admit linear structures.

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Constructions of several special classes of cubic bent functions outside the completed Maiorana-McFarland class

Published in Information and Computation, 2024

Abstract: Two main secondary constructions of bent functions are the direct and indirect sum methods. We show that the direct sum, under more relaxed conditions compared to those of Polujan and Pott (2020), can generate bent functions provably outside the completed Maiorana-McFarland class (). We also show that the indirect sum method of generating bent functions, by imposing certain conditions (which are completely absent if only the bentness of the resulting function is required) on the initial bent functions, can be employed in the design of bent functions outside . Furthermore, applying this method to suitably chosen bent functions we construct several generic classes of homogeneous cubic bent functions (considered as a difficult problem) that might possess additional properties (namely without affine derivatives and/or outside ). Our results significantly improve upon the best known instances of this type of bent functions given by Polujan and Pott (2020), and additionally we provide a solution to an open problem presented in their paper.

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Explicit infinite families of bent functions outside the completed Maiorana–McFarland class

Published in Design, Codes and Cryptography, 2023

Abstract: During the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a “new” family belongs to the completed Maiorana–McFarland ($\mathcal{M}^{\#}$) class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion in $\mathcal{M}^{\#}$ and eventually we obtain many infinite families of bent functions that are provably outside $\mathcal{M}^{\#}$. The fact that a bent function f is in/outside $\mathcal{M}^{\#}$ if and only if its dual is in/outside $\mathcal{M}^{\#}$ is employed in the so-called 4-decomposition of a bent function on $\mathbb{F}_2^n$, which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004–2019, 2003) in terms of the second-order derivatives and later reformulated in (Hodžić et al. in IEEE Trans Inf Theory 65(11):7554–7565, 2019) in terms of the duals of its restrictions to the cosets of an $(n−2)$-dimensional subspace V. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside $\mathcal{M}^{\#}$. For instance, for the elementary case of defining a bent function $h(x,y_1,y_2)=f(x)\oplus y_1y_2$ on $\mathbb{F}_2^{n+2}$ using a bent function $f$ on $\mathbb{F}_2^n$, we show that $h$ is outside $\mathcal{M}^{\#}$ if and only if $f$ is outside $\mathcal{M}^{\#}$. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $f_1\parallel f_1\parallel f_2\parallel (1\oplus f_2)$ also gives bent functions outside $\mathcal{M}^{\#}$ if $f_1$ or $f_2$ is outside $\mathcal{M}^{\#}$. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outside $\mathcal{M}^{\#}$ are provided.

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Zbirka rešenih nalog iz teorije kolobarjev in končnih polj

Published in Study material - UP FAMNIT, 2023

Abstract: This textbook is based on tutorials delivered for the course Algebra IV - Algebraic Structures, which is a part of the study program Mathematics in Slovene Language, 1st Bologna Cycle, at the University of Primorska. Topics contained in this textbook are rings, fraction fields, factorisation of polynomials over fields, integral domains, ideals, factor rings, extension fields, finite fields, field automorphisms.

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Vectorial Boolean functions with the maximum number of bent components beyond the Nyberg’s bound

Published in Design, Codes and Cryptography, 2023

Abstract: Recently, several interesting constructions of vectorial Boolean functions with the maximum number of bent components (MNBC functions, for short) were proposed. However, many of them have component functions from the completed Maiorana-McFarland class $\mathcal{M}^{\#}$. Moreover, no examples of MNBC functions containing component functions provably outside $\mathcal{M}^{\#}$ are known. In this paper, we classify all MNBC functions in six variables. Based on the analysis of the obtained equivalence classes, we propose several infinite families of MNBC functions with component functions outside the M # class. In particular, two of our new constructions are solutions to the open problem [Bapić et al (eds) Proceedings of the twelfth international workshop on coding and cryptography, 2022, Item 1., p. 9].

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Quadratic almost bent functions - Their partial characterization and design in the spectral domain

Published in Discrete Applied Mathematics, 2022

Abstract: Quadratic almost bent (AB) functions are characterized by the property that the duals of their component functions are bent functions. We prove that these duals are also quadratic and illustrate that these bent duals may give rise to vectorial bent functions (in certain cases having a maximal output dimension). A necessary and sufficient condition for ensuring bentness of the linear combinations of quadratic bent duals is provided. Moreover, we provide a rather detailed analysis related to the structure of quadratic AB functions in the spectral domain, more precisely with respect to their Walsh supports, their intersection and restrictions of these bent duals to suitable subspaces. In particular, we completely determine the intersection of Walsh supports of the coordinate (semi-bent) functions for Gold AB mappings. We also provide the design of quadratic AB functions in the spectral domain by identifying (using computer simulations) suitable sets of bent dual functions. For instance, when n=7, this approach provides several AB functions which are not CCZ-equivalent to Gold functions.

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Constructions of (vectorial) bent functions outside the completed Maiorana–McFarland class

Published in Discrete Applied Mathematics, 2022

Abstract: Two new classes of bent functions derived from the Maiorana–McFarland ($\mathcal{M}$) class, so-called $\mathcal{C}$ and $\mathcal{D}$, were introduced by Carlet (1993) almost three decades ago. In Zhang (2020) sufficient conditions for specifying bent functions in $\mathcal{C}$ and $\mathcal{D}$ which are outside the completed $\mathcal{M}$ class, denoted by $\mathcal{M}^{\#}$, were given. Furthermore in Pasalic et al. (2021) the notion of vectorial bent functions which are weakly or strongly outside $\mathcal{M}^{\#}$, referring respectively to the case whether some or all nonzero linear combinations (called components) of its coordinate functions are in class $\mathcal{C}$ (or $\mathcal{D}$) but provably outside $\mathcal{M}^{\#}$, was introduced. In this article we continue the work of finding new instances of vectorial bent functions weakly/strongly outside $\mathcal{M}^{\#}$ using a different approach. Namely, a generic method for the construction of vectorial bent $(n,t)$-functions of the form $F(x,y)=G(x,y)+H(x,y)$, $n=2m$, $t$ divides $m$, was recently proposed in Bapić (2021), where $G$ is a given bent $(n,t)-$function satisfying certain properties and $H$ is an arbitrary $(t,t)$-function having certain form. We introduce a new superclass of bent functions $\mathcal{SC}$ which contains the classes $\mathcal{D}_0$ and $\mathcal{C}$ whose members are provably outside $\mathcal{M}^{\#}$. Most notably, using indicators of the form $\mathbf{1}_{L^{\perp}}(x,y)+\delta_0(x)$ to define members of this class leads for the first time to modifications of the $\mathcal{M}$ class performed on sets rather than on affine subspaces. We also show that for suitable choices of $H$, the function $F$ is a vectorial bent function weakly/strongly outside the class $\mathcal{M}^{\#}$. In this context, a new concept of being almost strongly outside $\mathcal{M}^{\#}$ is introduced and some families of vectorial bent functions with this property are given. Furthermore, we provide two new families of vectorial bent functions strongly outside $\mathcal{M}^{\#}$ (considered to be an intrinsically hard problem) whose output dimension is greater than $2$, thus giving first examples of such functions in the literature.

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Constructing new superclasses of bent functions from known ones

Published in Cryptography and Communications, 2022

Abstract: Some recent research articles (Zhang et al. in Lecture Notes in Computer Science, 10194, 298-313. (2017), Zhang et al. in Discret. Appl. Math. 285(1), 458-472. (2020)) addressed an explicit specification of indicators that specify bent functions in the so-called $\mathcal{C}$ and $\mathcal{D}$ classes, derived from the Maiorana-McFarland ($\mathcal{M}$) class by C. Carlet in 1994 (Carlet in In Lecture Notes in Computer Science 765, 77–101. (1993)). Many of these bent functions that belong to $\mathcal{C}$ or $\mathcal{D}$ are provably outside the completed $\mathcal{M}$ class. Nevertheless, these modifications are performed on affine subspaces, whereas modifying bent functions on suitable subsets may provide us with further classes of bent functions. In this article, we exactly specify new families of bent functions obtained by adding together indicators typical for the $\mathcal{C}$ and $\mathcal{D}$ class, thus essentially modifying bent functions in $\mathcal{M}$ on suitable subsets instead of subspaces. It is shown that the modification of certain bent functions in $\mathcal{M}$ gives rise to new bent functions which are provably outside the completed $\mathcal{M}$ class. Moreover, we consider the so-called 4-bent concatenation (using four different bent functions on the same variable space) of the (non)modified bent functions in $\mathcal{M}$ and show that we can generate new bent functions in this way which do not belong to the completed $\mathcal{M}$ class either. This result is obtained by specifying explicitly the duals of four constituent bent functions used in the concatenation. The question whether these bent functions are also excluded from the completed versions of $\mathcal{PS}$, $\mathcal{C}$ or $\mathcal{D}$ remains open and is considered difficult due to the lack of membership indicators for these classes.

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Exercises and Solutions in Rings and Fields (Notebook for Algebra IV - Algebraic Structures)

Published in Study material - UP FAMNIT, 2021

Abstract: This textbook is based on tutorials delivered for the course Algebra IV - Algebraic Structures, which is a part of the study program Mathematics, 1st Bologna Cycle, at the University of Primorska. Topics contained in this textbook are rings, fraction fields, factorisation of polynomials over fields, integral domains, ideals, factor rings, extension fields, finite fields, field automorphisms.

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A new method for secondary constructions of vectorial bent functions

Published in Design, Codes and Cryptography, 2021

Abstract: In 2017, Tang et al. have introduced a generic construction for bent functions of the form $f(x)=g(x)+h(x)$, where $g$ is a bent function satisfying some conditions and $h$ is a Boolean function. Recently, Zheng et al. generalized this result to construct large classes of bent vectorial Boolean functions from known ones in the form $F(x)=G(x)+h(X)$, where $G$ is a vectorial bent and $h$ is a Boolean function. In this paper, we further generalize this construction to obtain vectorial bent functions of the form $F(x)=G(x)+\mathbf{H}(X)$, where $\mathbf{H}$ is also a vectorial Boolean function. This allows us to construct new infinite families of vectorial bent functions, EA-inequivalent to $G$, which was used in the construction. Most notably, specifying $\mathbf{H} (x)=\mathbf{h} (Tr_1^n(u_1x),\ldots,Tr_1^n(u_tx))$, the function $\mathbf{h}$ can be chosen arbitrarily, which gives a relatively large class of different functions for a fixed function $G$. We also propose a method of constructing vectorial $(n,n)$-functions having maximal number of bent components.

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Analiza II - Infinitezimalni račun: zbirka rešenih nalog

Published in Study material - UP FAMNIT, 2020

Abstract: This textbook is based on tutorials delivered for the course Analysis II - Infinitezimal Calculus, which is a part of the study programs Mathematics, Bioinformatics and Mathematics in Economy and Finances, 1st Bologna Cycle, at the University of Primorska. Topics contained in this textbook are differentiability and integrability of functions of one variable.

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Selected Topics in Numerical Mathematics: Lecture Notes

Published in Study material - UP FAMNIT, 2019

Abstract: This textbook is based on lectures delivered for the course Selected Topics in Numerical Mathematics, which is a part of the study program Mathematical Sciences, 2nd Bologna Cycle, at the University of Primorska. Topics contained in this textbook are Approximation, Ordinary Differential Equations and Partial Differential Equations.

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dr. Amar Bapić

Publications