Dembowski-Ostrom Polynomials Research Articles

Dembowski-Ostrom polynomials and Dickson polynomials

<p style='text-indent:20px;'>We give a complete classification of Dembowski-Ostrom polynomials from the composition of Dickson polynomials of arbitrary kind and monomials over finite fields. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of the obtained Dembowski-Ostrom polynomials.</p>

Commutative quasigroups and semifields from planar functions

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : ( N μ ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q 1 and Q 2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form ( α , α , γ ) .

Bazı Dembowski-Ostrom Polinomlarının Planaritesi Üzerine

Planar mappings, defined by Dembowski and Ostrom, are identified as a means to construct projective planes. Then, many important applications of planar mappings appear in different fields such as cryptography and coding theory. In this paper, we provide sufficient and necessary conditions for the planarity of certain Dembowski-Ostrom polynomials over the finite field extension of degree three with odd characteristic. In particular, we completely determine the coefficients of the given Dembowski-Ostrom polynomials to be planar.

Dembowski–Ostrom polynomials and reversed Dickson polynomials

We discuss the problem of classifying Dembowski–Ostrom polynomials from the composition of reversed Dickson polynomials of arbitrary kind and monomials over finite fields of odd characteristic. Moreover, by using a variant of the Weil bound for the number of points of affine algebraic curves over finite fields, we discuss the planarity of all such Dembowski–Ostrom polynomials. Planar Dembowski–Ostrom polynomials have applications in many areas including cryptography and coding theory.

Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application

For a function $f$ from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n}$ , the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an “intermediate” concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it conveys a detailed structural information about $f$ . We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.

Quadratic permutations, complete mappings and mutually orthogonal latin squares

Abstract We investigate the permutation behavior of a special class of Dembowski-Ostrom polynomials over a finite field of characteristic 2 of the form P(X) = L 1(X)(L 2(X)+L 1(X)L 3(X)) where L 1, L 2, L 3 are linearized polynomials. To our knowledge, the given class has not been studied previously in the literature. We identify several new types of permutation polynomials of this class. While most of the newly identified polynomials are linearly equivalent to permutation monomials, we show that there exist subclasses that are not affine equivalent to monomials, and we describe their forms. One of the newly identified classes contains a subclass of complete mappings. We use these complete mappings to define new sets of mutually orthogonal Latin squares, as well as new vectorial bent functions from the Maiorana-McFarland class. Moreover, the quasigroup polynomials obtained in the process are different and inequivalent to the previously known ones.

A note on commutative semifield planes

Abstract Let q be an odd prime power. We prove that a planar function f from 𝔽 q to itself can be written as an affine Dembowski–Ostrom polynomial if and only if the projective plane derived from f is a commutative semifield plane.

Dembowski-Ostrom polynomials from reversed Dickson polynomials

This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2. The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form \({x^{1 + {2^\alpha }}}\) is almost perfect nonlinear. It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.

Classes of weak Dembowski–Ostrom polynomials for multivariate quadratic cryptosystems

Abstract T. Harayama and D. K. Friesen [J. Math. Cryptol. 1 (2007), 79–104] proposed the linearized binomial attack for multivariate quadratic cryptosystems and introduced weak Dembowski–Ostrom (DO) polynomials in this framework over the finite field 𝔽2. We extend the linearized binomial attack to multivariate quadratic cryptosystems over 𝔽 p for any prime p and redefine the weak DO polynomials for general case. We identify infinite classes of weak DO polynomials for these systems by considering highly degenerate quadratic forms over algebraic function fields and Artin–Schreier type curves to achieve our results. This gives a general answer to the conjecture stated by Harayama and Friesen and also a partial enumeration of weak DO polynomials over finite fields.

Solution to An Isotopism Question Concerning Rank 2 Semifields

AbstractIn [8] Dempwolff gives a construction of three classes of rank two semifields of order , with q and n odd, using Dembowski–Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for , in particular we prove that two of these classes, labeled and , are new for , whereas presemifields in family are isotopic to Generalized Twisted Fields for each .

Ambiguity and Deficiency of Permutations Over Finite Fields With Linearized Difference Map

The concepts of ambiguity and deficiency for a bijection on a finite Abelian group were recently introduced. In this paper, we present some further fundamental results on the ambiguity and deficiency of functions; in particular, we note that they are invariant under the well-known Carlet-Charpin-Zinoviev-equivalence, we obtain upper and lower bounds on the ambiguity and deficiency of differentially k-uniform functions, and we give a lower bound on the nonlinearity of functions that achieve the lower bound of ambiguity and deficiency. In addition, we provide an explicit formula in terms of the ranks of matrices on the ambiguity and deficiency of a Dembowski-Ostrom (DO) polynomial, and using this technique, we find exact values for known cases of DO permutations with few terms. We also derive exact values for the ambiguities and deficiencies of DO permutations obtained from trace functions. The key relationship between the above polynomials is that they all have linearized difference map.

More translation planes and semifields from Dembowski–Ostrom polynomials

In this article we use pairs of Dembowski–Ostrom polynomials with special properties (see (P1)–(P3) in the introduction below) to construct translation planes of order q n which admit cyclic groups of order q n −1 having orbits of lengths 1, 1, (q n −1)/2, (q n −1)/2 on the line at infinity. The same pairs also define semifields of order q 2n . We discuss the properties of these translation planes and semifields. These constructions extend the related construction in Dempwolff and Muller, Osaka J Math [5].

Paley type group schemes and planar Dembowski–Ostrom polynomials

In this paper we give some necessary and sufficient conditions for Dembowski–Ostrom polynomials to be planar. These conditions give a simple explanation of the Coulter–Matthews and Ding–Yin commutative semifields and enable us to obtain permutation polynomials from some of the Zha–Kyureghyan–Wang commutative semifields. We then give a generalization of Feng’s construction of Paley type group schemes in extra-special p -groups of exponent p and construct a family of Paley type group schemes in what we call the flag groups of finite fields. We also determine the strong multiplier groups of these group schemes. In the last section of this paper, we give a straightforward generalization of the twin prime power construction of difference sets to a construction of Hadamard designs from twin Paley type association schemes.

On the number of distinct values of a class of functions over a finite field

Several authors have recently shown that a planar function over a finite field of order q must have at least ( q + 1 ) / 2 distinct values. In this note this result is extended by weakening the hypothesis significantly and strengthening the conclusion. We also give an algorithm for determining whether a given bivariate polynomial ϕ ( X , Y ) can be written as f ( X + Y ) − f ( X ) − f ( Y ) for some polynomial f. Using the ideas of the algorithm, we then show a Dembowski–Ostrom polynomial is planar over a finite field of order q if and only if it yields exactly ( q + 1 ) / 2 distinct values under evaluation; that is, it meets the lower bound of the image size of a planar function.

Dembowski–Ostrom polynomials from Dickson polynomials

Motivated by several recent results, we determine precisely when F k ( X d , a ) − F k ( 0 , a ) is a Dembowski–Ostrom polynomial, where F k ( X , a ) is a Dickson polynomial of the first or second kind. As a consequence, we obtain a classification of all such polynomials which are also planar; all examples found are equivalent to previously known examples.

Commutative presemifields and semifields

Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski–Ostrom polynomial and conversely, any planar Dembowski–Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert's commutative twisted fields. A classification of all planar Dembowski–Ostrom polynomials over any finite field of order p 3 , p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X 10 + a X 6 − a 2 X 2 with a ≠ 0 describes precisely two new commutative presemifields of order 3 e for each odd e ⩾ 5 .

Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums SigmaxisinF(q)zetap Tr(af(x)+bx)(a,bisinFq,q=pm,pges3) for all perfect nonlinear functions f which is a Dembowski-Ostrom polynomial or p = 3,f=x(3(k)+1)/2 where k is odd and (k,m)=1. As applications, we determine (1) the correlation distribution of the m-sequence {alambda=Tr(gammalambda)}(lambda=0,1,...) and the sequence {blambda=Tr(f(gammalambda))}(lambda=0,1,...) over Fp where gamma is a primitive element of Fq and (2) the weight distributions of the linear codes over Fp defined by f.

Planar polynomials for commutative semifields with specified nuclei

We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 38 with left nucleus of order 3 and middle nucleus of order 32.

Weil sum for birthday attack in multivariate quadratic cryptosystem

We propose a new cryptanalytic application of a number theoretic tool Weil sum to birthday attack against multivariate quadratic trapdoor function. This new customization of birthday attack is developed by evaluating the explicit Weil sum of the underlying univariate polynomial and the exact number of solutions of the associated bivariate equation. We designed and implemented a new algorithm for computing Weil sum values so that we could explicitly identify some class of weak Dembowski-Ostrom polynomials and their equivalent forms in multivariate quadratic trapdoor function. Our customized attack can be also regarded as an equation solving algorithm for the system of some special quadratic equations over finite fields, and it is fundamentally different from the Gröbner basis methods. Both theoretical observation and experiment show that the required computational complexity of the attack on these weak polynomial instances can be asymptotically less than the square root complexity of the common birthday attack by factor of as large as 2n/8 in terms of the extension degree n of . We also suggest a few open problems that any MQ-based short signature scheme must explicitly take into account for the basic design principles.

On the Evaluation of Weil Sums of Dembowski–Ostrom Polynomials

Let Fq denote the finite field of q elements, q=pe odd, let χ1 denote the canonical additive character of Fq where χ1(c)=e2πiTr(c)/p for all c∈Fq, and let Tr represent the trace function from Fq to Fp. We are interested in evaluating Weil sums of the form S=S(a1, …, an)=∑x∈Fqχ1(D(x)) where D(x)=∑ni=1aixpαi+pβi, αi⩾βi for each i, is known as a Dembowski–Ostrom polynomial (or as a D-O polynomial). Coulter has determined the value of S when D(x)=axpα+1; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of Flq-solutions to the diagonal-type equation ∑li=1xpγ+1i+(xi+λ)pγ+1=0, where l is even, e/gcd(γ, e) is odd, and h (X)=λpe−γXpe−γ+λpγX is a permutation polynomial over Fq.