Papers by Amela Muratović-Ribić
arXiv (Cornell University), Mar 25, 2015
For any given polynomial f over the finite field Fq with degree at most q − 1, we associate it wi... more For any given polynomial f over the finite field Fq with degree at most q − 1, we associate it with a q × q matrix A(f) = (a ik) consisting of coefficients of its powers (f (x)) k = q−1 i=0 a ik x i modulo x q − x for k = 0, 1,. .. , q − 1. This matrix has some interesting properties such as A(g • f) = A(f)A(g) where (g • f)(x) = g(f (x)) is the composition of the polynomial g with the polynomial f. In particular, A(f (k)) = (A(f)) k for any k-th composition f (k) of f modulo x q − x with k ≥ 0. As a consequence, we prove that the rank of A(f) gives the cardinality of the value set of f. Moreover, if f is a permutation polynomial then the matrix associated with its inverse A(f (−1)) = A(f) −1 = P A(f)P where P is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequenceā = {a 0 , a 1 , a 2 , ...} generated by an = f (n) (a 0) with initial value a 0 , in terms of the order of the associated matrix. Finally we show that A(f) is diagonalizable in some extension field of Fq when f is a permutation polynomial over Fq. v f = (a 0 , a 1 ,. .. , a q−1) T .
arXiv (Cornell University), Jan 25, 2018
We show the existence of many infinite classes of permutations over finite fields and bent functi... more We show the existence of many infinite classes of permutations over finite fields and bent functions by extending the notion of linear translators, introduced by Kyureghyan [12]. We call these translators Frobenius translators since the derivatives of f : F p n → F p k , where n = rk, are of the form f (x + uγ) − f (x) = u p i b, for a fixed b ∈ F p k and all u ∈ F p k , rather than considering the standard case corresponding to i = 0. This considerably extends a rather rare family {f } admitting linear translators of the above form. Furthermore, we solve a few open problems in the recent article [4] concerning the existence and an exact specification of f admitting classical linear translators, and an open problem introduced in [9] of finding a triple of bent functions f 1 , f 2 , f 3 such that their sum f 4 is bent and that the sum of their duals f * 1 +f * 2 +f * 3 +f * 4 = 1. Finally, we also specify two huge families of permutations over F p n related to the condition that G(y) = −L(y)+(y+δ) s −(y+δ) p k s permutes the set S = {β ∈ F p n : T r n k (β) = 0}, where n = 2k and p > 2. Finally, we offer generalizations of constructions of bent functions from [16] and described some new bent families using the permutations found in [4].
For any given polynomial f over the finite field Fq with degree at most q − 1, we associate it wi... more For any given polynomial f over the finite field Fq with degree at most q − 1, we associate it with a q × q matrix A(f) = (a ik) consisting of coefficients of its powers (f (x)) k = q−1 i=0 a ik x i modulo x q − x for k = 0, 1,. .. , q − 1. This matrix has some interesting properties such as A(g • f) = A(f)A(g) where (g • f)(x) = g(f (x)) is the composition of the polynomial g with the polynomial f. In particular, A(f (k)) = (A(f)) k for any k-th composition f (k) of f modulo x q − x with k ≥ 0. As a consequence, we prove that the rank of A(f) gives the cardinality of the value set of f. Moreover, if f is a permutation polynomial then the matrix associated with its inverse A(f (−1)) = A(f) −1 = P A(f)P where P is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequenceā = {a 0 , a 1 , a 2 , ...} generated by an = f (n) (a 0) with initial value a 0 , in terms of the order of the associated matrix. Finally we show that A(f) is diagonalizable in some extension field of Fq when f is a permutation polynomial over Fq. v f = (a 0 , a 1 ,. .. , a q−1) T .
Contemporary mathematics, 2015
In this paper, we present a characterization of a semi-multiplicative analogue of planar function... more In this paper, we present a characterization of a semi-multiplicative analogue of planar functions over finite fields. When the field is a prime field, these functions are equivalent to a variant of a doubly-periodic Costas array and so we call these functions Costas. We prove an equivalent conjecture of Golomb and Moreno that any Costas polynomial over a prime field is a monomial. Moreover, we give a class of Costas polynomials over extension fields and conjecture that this class represents all Costas polynomials. This conjecture is equivalent to the conjecture that there are no non-Desarguesian planes of a given type with prime power order.
Finite Fields and Their Applications, 2014
A recursive construction of complete mappings over finite fields is provided in this work. These ... more A recursive construction of complete mappings over finite fields is provided in this work. These permutation polynomials, characterized by the property that both f (x) ∈ F q [x] and its associated mapping f (x) + x are permutations, have an important application in cryptography in the construction of bent-negabent functions which actually leads to some new classes of these functions. Furthermore, we also provide a recursive construction of mappings over finite fields of odd characteristic, having an interesting property that both f (x) and f (x + c) + f (x) are permutations for every c ∈ F q. Both the multivariate and univariate representations are treated and some results concerning fixed points and the cycle structure of these permutations are given. Finally, we utilize our main result for the construction of so-called negabent functions and bent functions over finite fields.
IEEE Transactions on Information Theory, 2018
In this paper, we show that the maximum number of bent component functions of a vectorial functio... more In this paper, we show that the maximum number of bent component functions of a vectorial function F : G F(2) n → G F(2) n is 2 n − 2 n/2. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ G F(2 n)[x], where F has only a few terms. The only known power functions having such a large number of bent components are x d , where d = 2 n/2 + 1. In this paper, we show that the binomials F i (x) = x 2 i (x + x 2 n/2) also have such a large number of bent components, and these binomials are inequivalent to the monomials x 2 n/2 +1 if 0 < i < n/2. In addition, the functions F i have differential properties much better than x 2 n/2 +1. We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1. Index Terms-Cryptography, Boolean functions, bent functions, vectorial bent functions, trace functions, equivalence of functions. I. INTRODUCTION B ENT functions are extremal combinatorial objects with several areas of application, such as coding theory, maximum length sequences, cryptography, the theory of difference sets to name a few. The term bent Boolean function was introduced by Rothaus [42]; another early thorough investigation of bent functions is [19]. For a recent survey article, see [12] and the two books [35], [43], see also [17] for a more general discussion of functions on finite fields. Among other equivalent characterizations of bent functions, the one that is most often used is a characterization of bent functions as a class of Boolean functions having so-called flat Walsh-Hadamard spectra. It means that for any bent function over G F(2) n , its Hamming distance to any affine function in n variables is constant including the distance to the all-zero function (or all-one function).
IEEE Transactions on Information Theory, Feb 1, 2014
ABSTRACT In this paper, we provide necessary and sufficient conditions for a function of the form... more ABSTRACT In this paper, we provide necessary and sufficient conditions for a function of the form $F(x)=Tr_{k}^{2k}(sum_{i=1}^{t}a_{i}x^{r_{i}(2^{k}-1)})$ to be bent. Three equivalent statements, all of them providing both the necessary and sufficient conditions, are derived. In particular, one characterization provides an interesting link between the bentness and the evaluation of $F$ on the cyclic group of the $(2^{k}+1)$th primitive roots of unity in $GF(2^{2k})$. More precisely, for this group of cardinality $2^{k}+1$ given by ${cal U}={uin GF(2^{2k}):u^{2^{k}+1}=1}$, it is shown that the property of being vectorial bent implies that $Im(F)=GF(2^{k})cup{0}$, if $F$ is evaluated on ${cal U}$, that is, $F(u)$ takes all possible values of $GF(2^{k})^{ast}$ exactly once and the zero value is taken twice when $u$ ranges over ${cal U}$. This condition is then reformulated in terms of the evaluation of certain elementary symmetric polynomials related to $F$, which in turn gives some necessary conditions on the coefficients - tex Notation=&quot;TeX&quot;&gt;$a_{i}$ (for binomial trace functions) that can be stated explicitly. Finally, we show that a bent trace monomial of Dillon&#39;s type $Tr_{k}^{2k}(lambda x^{r(2^{k}-1)})$ is never a vectorial bent function.
To identify and specify trace bent functions of the form T r n 1 (P (x)), where P (x) ∈ GF (2 n)[... more To identify and specify trace bent functions of the form T r n 1 (P (x)), where P (x) ∈ GF (2 n)[x], has been an important research topic lately. We show that an infinite class of quadratic vectorial bent functions can be specified in the univariate polynomial form as F (x) = T r^n_k (αx^2^i (x + x^k)), where n = 2k, i = 0,n-1, and α \notin GF(2^k). Most notably apart from the cases i \in {0,k} for which the polynomial x^2^i (x+x^2^k) is affinely inequivalent to the monomial x^{2^k+1}, for the remaining indices i the function x^2^i (x+x^2^k) seems to be affinely inequivalent to x^2^k+1, as confi rmed by computer simulations for small n. It is well-known that Tr^n_1( x^2^k+1) is Boolean bent for exactly 2^{2k}-2^k values (this is at the same time the maximum cardinality possible) of α \in GF(2n) and the same is true for our class of quadratic bent functions of the form T r^n_k (αx^2^i (x + x^k)) though for i > 0 the associated functions F : GF(2^n) -> GF(2^n) are in general C...
WSEAS Transactions on Systems and Control archive, 2018
To prove that a fuzzy dependency follows from a set of fuzzy dependences can be a very demanding ... more To prove that a fuzzy dependency follows from a set of fuzzy dependences can be a very demanding task. As far as we know, an algorithm or an application that generally and automatically solves the problem, does not exist. The main goal of this paper is to offer such an algorithm. In order to achieve our goal we consider fuzzy dependences as fuzzy formulas. In particular, we fix fuzzy logic operators: conjunction, disjunction and implication, and allow only these operators to appear within fuzzy formulas. Ultimately, we prove that a fuzzy dependency follows from a set of fuzzy dependences if and only if the corresponding fuzzy formula is a logical consequence of the corresponding set of fuzzy formulas. To prove an implication of the last type, one usually uses the resolution principle, i.e., the steps that can be fully automated. Our methodology assumes the use of soundness and completeness of fuzzy dependences inference rules as well as the extensive use of active fuzzy multivalued ...
2015 XXV International Conference on Information, Communication and Automation Technologies (ICAT), 2015
Integer linear programming is a popular method of generating school timetables. Although computat... more Integer linear programming is a popular method of generating school timetables. Although computationally simpler, school timetabling is less developed area than university timetabling, because the models which resolve timetabling problems proposed thus far have been adjusted to individual cases differing from country to country. A proposed model meets most of constraints appeared in different school timetabling systems.
IEEE Transactions on Information Theory, 2014
Discrete Applied Mathematics, 2017
In this note, using rather elementary technique and the derived formula that relates the coeffici... more In this note, using rather elementary technique and the derived formula that relates the coefficients of a polynomial over a finite field and its derivative, we deduce many interesting results related to derivatives of Boolean functions and derivatives of mappings over finite fields. For instance, we easily identify several infinite classes of polynomials which cannot possess linear structures. The same technique can be applied for deducing a nontrivial upper bound on the degree of so-called planar mappings.
Discrete Applied Mathematics, 2018
Given are necessary conditions for a permutation polynomial to be the derivative of a planar mapp... more Given are necessary conditions for a permutation polynomial to be the derivative of a planar mapping. These conditions are not sufficient and there might exist permutation polynomials which are not derivatives of some planar mapping satisfying these conditions. For the first time we show that there is a close connection between two seemingly unrelated structures, namely planar and complete mappings. It is shown that any planar mapping induces a sequence of complete mappings having some additional interesting properties. Furthermore, a class of almost planar mappings over extension fields is introduced having the property that its derivatives are permutations in most of the cases. This class of functions then induces many infinite classes of complete mappings (permutations) as well.
For any given polynomial f over the finite field F_q with degree at most q-1, we associate it wit... more For any given polynomial f over the finite field F_q with degree at most q-1, we associate it with a q× q matrix A(f)=(a_ik) consisting of coefficients of its powers (f(x))^k=∑_i=0^q-1a_ik x^i modulo x^q -x for k=0,1,...,q-1. This matrix has some interesting properties such as A(g∘ f)=A(f)A(g) where (g∘ f)(x) = g(f(x)) is the composition of the polynomial g with the polynomial f. In particular, A(f^(k))=(A(f))^k for any k-th composition f^(k) of f with k ≥ 0. As a consequence, we prove that the rank of A(f) gives the cardinality of the value set of f. Moreover, if f is a permutation polynomial then the matrix associated with its inverse A(f^(-1))=A(f)^-1=PA(f)P where P is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequence a̅ = {a_0, a_1, a_2, ... } generated by a_n = f^(n)(a_0) with initial value a_0, in terms of the order of the associated matrix. Finally we show that A(f) is diagonalizable in some extension ...
Ars Mathematica Contemporanea, 2015
In this note, we give the explicit formula for the number of multisubsets of a finite abelian gro... more In this note, we give the explicit formula for the number of multisubsets of a finite abelian group G with any given size such that the sum is equal to a given element g ∈ G. This also gives the number of partitions of g into a given number of parts over a finite abelian group. An inclusion-exclusion formula for the number of multisubsets of a subset of G with a given size and a given sum is also obtained.
The Electronic Journal of Combinatorics, 2013
In this paper we find an exact formula for the number of partitions of an element $z$ into $m$ ... more In this paper we find an exact formula for the number of partitions of an element $z$ into $m$ parts over a finite field, i.e. we find the number of nonzero solutions of the equation $x_1+x_2+\cdots +x_m=z$ over a finite field when the order of terms does not matter. This is equivalent to counting the number of $m$-multi-subsets whose sum is $z$. When the order of the terms in a solution does matter, such a solution is called a composition of $z$. The number of compositions is useful in the study of zeta functions of toric hypersurfaces over finite fields. We also give an application in the study of polynomials of prescribed ranges over finite fields.
Cryptography and Communications, 2019
We show the existence of many infinite classes of permutations over finite fields and bent functi... more We show the existence of many infinite classes of permutations over finite fields and bent functions by extending the notion of linear translators, introduced by Kyureghyan [12]. We call these translators Frobenius translators since the derivatives of f : F p n → F p k , where n = rk, are of the form f (x + uγ) − f (x) = u p i b, for a fixed b ∈ F p k and all u ∈ F p k , rather than considering the standard case corresponding to i = 0. This considerably extends a rather rare family {f } admitting linear translators of the above form. Furthermore, we solve a few open problems in the recent article [4] concerning the existence and an exact specification of f admitting classical linear translators, and an open problem introduced in [9] of finding a triple of bent functions f 1 , f 2 , f 3 such that their sum f 4 is bent and that the sum of their duals f * 1 +f * 2 +f * 3 +f * 4 = 1. Finally, we also specify two huge families of permutations over F p n related to the condition that G(y) = −L(y)+(y+δ) s −(y+δ) p k s permutes the set S = {β ∈ F p n : T r n k (β) = 0}, where n = 2k and p > 2. Finally, we offer generalizations of constructions of bent functions from [16] and described some new bent families using the permutations found in [4].
IEEE Transactions on Information Theory, 2017
In this paper, we show that the maximum number of bent component functions of a vectorial functio... more In this paper, we show that the maximum number of bent component functions of a vectorial function F : G F(2) n → G F(2) n is 2 n − 2 n/2. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ G F(2 n)[x], where F has only a few terms. The only known power functions having such a large number of bent components are x d , where d = 2 n/2 + 1. In this paper, we show that the binomials F i (x) = x 2 i (x + x 2 n/2) also have such a large number of bent components, and these binomials are inequivalent to the monomials x 2 n/2 +1 if 0 < i < n/2. In addition, the functions F i have differential properties much better than x 2 n/2 +1. We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1. Index Terms-Cryptography, Boolean functions, bent functions, vectorial bent functions, trace functions, equivalence of functions. I. INTRODUCTION B ENT functions are extremal combinatorial objects with several areas of application, such as coding theory, maximum length sequences, cryptography, the theory of difference sets to name a few. The term bent Boolean function was introduced by Rothaus [42]; another early thorough investigation of bent functions is [19]. For a recent survey article, see [12] and the two books [35], [43], see also [17] for a more general discussion of functions on finite fields. Among other equivalent characterizations of bent functions, the one that is most often used is a characterization of bent functions as a class of Boolean functions having so-called flat Walsh-Hadamard spectra. It means that for any bent function over G F(2) n , its Hamming distance to any affine function in n variables is constant including the distance to the all-zero function (or all-one function).
Contemporary Developments in Finite Fields and Applications, 2016
For any given polynomial f over the finite field Fq with degree at most q − 1, we associate it wi... more For any given polynomial f over the finite field Fq with degree at most q − 1, we associate it with a q × q matrix A(f) = (a ik) consisting of coefficients of its powers (f (x)) k = q−1 i=0 a ik x i modulo x q − x for k = 0, 1,. .. , q − 1. This matrix has some interesting properties such as A(g • f) = A(f)A(g) where (g • f)(x) = g(f (x)) is the composition of the polynomial g with the polynomial f. In particular, A(f (k)) = (A(f)) k for any k-th composition f (k) of f modulo x q − x with k ≥ 0. As a consequence, we prove that the rank of A(f) gives the cardinality of the value set of f. Moreover, if f is a permutation polynomial then the matrix associated with its inverse A(f (−1)) = A(f) −1 = P A(f)P where P is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequenceā = {a 0 , a 1 , a 2 , ...} generated by an = f (n) (a 0) with initial value a 0 , in terms of the order of the associated matrix. Finally we show that A(f) is diagonalizable in some extension field of Fq when f is a permutation polynomial over Fq. v f = (a 0 , a 1 ,. .. , a q−1) T .
Contemporary Mathematics, 2015
In this paper, we present a characterization of a semi-multiplicative analogue of planar function... more In this paper, we present a characterization of a semi-multiplicative analogue of planar functions over finite fields. When the field is a prime field, these functions are equivalent to a variant of a doubly-periodic Costas array and so we call these functions Costas. We prove an equivalent conjecture of Golomb and Moreno that any Costas polynomial over a prime field is a monomial. Moreover, we give a class of Costas polynomials over extension fields and conjecture that this class represents all Costas polynomials. This conjecture is equivalent to the conjecture that there are no non-Desarguesian planes of a given type with prime power order.
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Papers by Amela Muratović-Ribić