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We introduce a class of permutation polynomial over F q n that can be written in the form L (x) x q + 1 or L (x q + 1) x for some q -linear polynomial L over F q n . Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form. [ABSTRACT FROM AUTHOR]

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form Tr (a x q + 1) + ℓ (x 2) on an extension of the finite field F q n / F q. Specifically, we investigate those functions on F q 2 / F q and construct several typical kinds of planar functions. We also completely characterize them on F q 3 / F q. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions. [ABSTRACT FROM AUTHOR]

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over F q 2 , whose numerators are so-called q -quadratic polynomials. To this end, we will first determine the exact number of zeros of a special q -quadratic polynomial in F q 2 , by calculating character sums related to quadratic forms of F q 2 / F q. Then given some rational function, we can demonstrate whether it induces a permutation of F q 2 . [ABSTRACT FROM AUTHOR]

2-to-1 mappings over finite fields play an important role in symmetric cryptography, particularly in the constructions of APN functions, bent functions, and semi-bent functions. Very recently, Mesnager and Qu [IEEE Trans. Inf. Theory 65 (12): 7884-7895] provided a systematic study of 2-to-1 mappings over finite fields. In particular, they determined all 2-to-1 mappings of degree at most 4 over any finite field. Besides, another research direction is to consider 2-to-1 polynomials with few terms. Some results about 2-to-1 monomials and binomials have been obtained in [IEEE Trans. Inf. Theory 65 (12): 7884-7895]. Motivated by their work, in this present paper, we push further the study of 2-to-1 mappings, particularly over finite fields with characteristic 2 (binary case being the most interesting for applications). Firstly, we completely determine 2-to-1 polynomials with degree 5 over F2n using the well-known Hasse-Weil bound. Besides, we consider 2-to-1 mappings with few terms, mainly trinomials and quadrinomials. Using the multivariate method and the resultant of two polynomials, we present two classes of 2-to-1 trinomials, which explain all the examples of 2-to-1 trinomials of the form xk + β xℓ + α ∈ F2n[x] with n ≤ 7. We derive twelve classes of 2-to-1 quadrinomials with trivial coefficients over F2n. [ABSTRACT FROM AUTHOR]

Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of $x^{r}h(x^{(q-1)/d})$ , those from permutation polynomials, from linear translators and from APN functions. Then we present 2-to-1 polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of 2-to-1 mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results. [ABSTRACT FROM AUTHOR]

In recent literature, a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality) is presented. The key ingredient for constructing such optimal linear LRC codes is the so-called $r$ -good polynomials, where r$ is equal to the locality of the LRC code. However, given a prime p$ , known constructions of r exist only for some special integers r$ , and the problem of constructing optimal LRC codes over small field for any given locality is still open. In this paper, by using function composition, we present two general methods of designing good polynomials, which lead to three new constructions of r$ -good polynomials. Such polynomials bring new constructions of optimal LRC codes. In particular, our constructed polynomials as well as the power functions yield optimal (n,k,r) for all positive integers $r$ as localities, where $q$ is near the code length $n$ . [ABSTRACT FROM AUTHOR]

In the present article, we provide a complete characterization of permutations on the finite field F 4 m of shape f ϵ _ (X) : = ϵ 1 X ‾ q + 1 + ϵ 2 X ‾ q X + ϵ 3 X ‾ X q + ϵ 4 X q + 1 , definitively (where q = 2 k , Q = 2 m , m is odd, (m , k) = 1 , X ‾ = X Q). Our achievement firstly extends the very recent literature (a long series of interesting recent (2019-2022) results derived in (at least) seven articles) about the bijectivity on F Q 2 of f ϵ _ . It also gives a complete proof of the conjecture (Conjecture 19) posed in the recent literature by Li et al. (2021) [24] about the characterization of permutations f ϵ _ with 4-uniform BCT (which can be viewed, thanks to their nice cryptographic properties, as promising candidates as S-boxes to be used in designing secure block ciphers in symmetric cryptography). To derive our results about the quadrinomials f ϵ _ , we shall keep the essence of our successful approach initiated by Kim et al. in [23] by relating the problem of characterization of the bijectivity of f ϵ _ over F Q 2 to the equation X q + 1 + X + a = 0 but also by performing algebraic developments with new nature by considering some rational functions over finite fields. Besides, we shall employ novel algebraic techniques and derive new auxiliary results (some of them have their independent interest) going beyond the recent results given in [23]. Notably, we present a complete proof of the bijectivity of f ϵ _ over F Q 2 without any restriction. Specifically, our results are valid when the coefficients ϵ i (i ∈ { 1 , 2 , 3 , 4 }) lie in F Q 2 and for any value of k ≥ 1 (at the opposite of the results in [23] valid with the condition that the coefficients ϵ i (i ∈ { 1 , 2 , 3 , 4 }) lie in the finite field F Q while covering all values of k ≥ 1). Finally, we emphasize that very recently, Göloğlu has presented in (Göloğlu, 2022 [13]), an alternative novel approach providing a complete description of permutations of the form mentioned above using biprojective polynomials. [ABSTRACT FROM AUTHOR]

Bent functions are optimal combinatorial objects. Since their introduction, substantial efforts have been directed toward their study in the last three decades. A complete classification of bent functions is elusive and looks hopeless today, therefore, not only their characterization, but also their generation are challenging problems. This paper is devoted to the construction of bent functions. First, we provide several new effective constructions of bent functions, self-dual bent functions, and antiself-dual bent functions. Second, we provide seven new infinite families of bent functions by explicitly calculating their dual. [ABSTRACT FROM AUTHOR]

The connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, Lisoněk exploited it to reformulate the Charpin-Gong characterization of a large class of hyper-bent functions in terms of numbers of points on hyperelliptic curves. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin-Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by Mesnager. Doing so, a polynomial time and space test for the hyper-bentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong. [ABSTRACT FROM AUTHOR]

We show that any Boolean function, in even dimension, equal to the sum of a Boolean function g which is constant on each element of a spread and of a Boolean function h whose restrictions to these elements are all linear, is semibent if and only if g and h are both bent. We deduce a large number of infinite classes of semibent functions in explicit bivariate (respectively, univariate) polynomial form. [ABSTRACT FROM PUBLISHER]

Kloosterman sums have recently become the focus of much research, most notably due to their applications in cryptography and coding theory. In this paper, we extensively investigate the link between the semibentness property of functions in univariate forms obtained via Dillon and Niho functions and Kloosterman sums. In particular, we show that zeros and the value four of binary Kloosterman sums give rise to semibent functions in even dimension with maximum degree. Moreover, we study the semibentness property of functions in polynomial forms with multiple trace terms and exhibit criteria involving Dickson polynomials. [ABSTRACT FROM AUTHOR]

Abstract: One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well-known Maiorana–McFarland class. We first notice that H can be extended to a slightly larger class that we denote by . We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. and to Leander and Kholosha, are the univariate form of the functions of class . Their restrictions to the vector spaces , , are linear. We also characterize the bent functions whose restrictions to the ʼs are affine. We answer the open question raised by Dobbertin et al. (2006) in on whether the duals of the Niho bent functions introduced in the paper are affinely equivalent to them, by explicitly calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana–McFarland class, which brings us back to the problem of knowing whether H (or ) is a subclass of the Maiorana–McFarland completed class. We then show that the condition for a function in bivariate form to belong to class is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial (also called oval polynomial, a notion from finite geometry). Thanks to the existence in the literature of 8 classes of nonlinear o-polynomials, we deduce a large number of new cases of bent functions in , which are potentially affinely inequivalent to known bent functions (in particular, to Maiorana–McFarlandʼs functions). [Copyright &y& Elsevier]

Oblivious polynomial evaluation (OPE) is a secure two-party cryptographic protocol that enables a receiver to obliviously retrieve polynomial value on its private input for a private polynomial of a sender. This paper deals with the problem of OPE in the quantum domain. Herein, we propose the construction of the first quantum secure OPE, namely QuOPE. Our design is a constant round protocol that only requires quantum communication of one quantum superposition state that achieves all the stipulated security requirements of an OPE. In particular, privacy and security of the sender and receiver's private data are ensured. In addition, we extend the idea of QuOPE to quantum rational polynomial evaluation (QuORPE). Furthermore, we explore the possibility of utilizing QuOPE as a building block in the automated medical diagnosis platform. [ABSTRACT FROM AUTHOR]

This paper consists of two main contributions. First, the Niho bent function consisting of 2^r exponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed Maiorana–McFarland class \cal M. In particular, it is proven that two families do not belong to the completed class \cal M. The latter result gives a positive answer to an open problem whether the class H of bent functions introduced by Dillon in his thesis of 1974 differs from the completed class \cal M. [ABSTRACT FROM PUBLISHER]