Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): Robert W. Fitzgerald, Yasanthi Kottegoda We complete A. Klapper's work on the invariant of a one-term trace form over a finite field of odd characteristic. We apply this to computing the probability of a successful impersonation attack on an authentication code proposed by C. Ding et al. (2005).

Abstract: Publication date: November 2017 Source:Finite Fields and Their Applications, Volume 48 Author(s): José María Grau, Antonio M. Oller-Marcén In this paper we compute the sum of the k-th powers of all the elements of a finite commutative unital ring, thus generalizing known results for finite fields, the rings of integers modulo n or the ring of Gaussian integers modulo n. As an application, we focus on quotient rings of the form ( Z / n Z ) [ x ] / ( f ( x ) ) for a polynomial f ∈ Z [ x ] .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Daniel Panario, Daniel Santana, Qiang Wang We obtain exact formulas for the differential spectrum, deficiency and ambiguity of all normalized permutation polynomials of degree up to six over finite fields.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Everett W. Howe The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil–Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5, 6, and 7 with small defect. Our aim is to be able to produce, in a reasonable amount of time, curves that can be used to populate the online table of curves with many points found at manypoints.org .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Weijun Fang, Jiejing Wen, Fang-Wei Fu As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q-polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Cem Güneri, Ferruh Özbudak, Buket Özkaya, Elif Saçıkara, Zahra Sepasdar, Patrick Solé Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder Theorem yields a decomposition of GQC codes into a sum of concatenated codes. This decomposition leads to a trace formula, a minimum distance bound, and to a criteria for the GQC code to be self-dual or to be linear complementary dual (LCD). Explicit long GQC codes that are LCD, but not QC, are exhibited.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Alan Adolphson, Steven Sperber The Hasse–Witt matrix of a hypersurface in P n over a finite field of characteristic p gives essentially complete mod p information about the zeta function of the hypersurface. But if the degree d of the hypersurface is ≤n, the zeta function is trivial mod p and the Hasse–Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse–Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all d. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Sara D. Cardell, Amparo Fúster-Sabater In this work, different decimation-based sequence generators for cryptographic purposes have been analyzed in detail. In fact, the modified self-shrinking generator was first introduced as an improved version of the self-shrinking generator. However, it is here proven that the sequences produced by both generators belong to the same family of sequences, that is the class of the generalized self-shrinking sequences. Thus, both sequences have the same properties as well as the same weaknesses. Moreover, such sequences can be generated by linear structures based on one-dimensional cellular automata. The linearity inherent to the cellular automata-based models can be used to launch a cryptanalytic attack against such non-linear generators.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): U. Caner Cengiz, Paul Pollack, Enrique Treviño Let A ∈ F 2 [ T ] . We say A is perfect if A coincides with the sum of all of its divisors in F 2 [ T ] . We prove that the number of perfect polynomials A with A ≤ x is O ϵ ( x 1 / 12 + ϵ ) for all ϵ > 0 , where A = 2 deg A . We also prove that every perfect polynomial A with 1 < A ≤ 1.6 × 10 60 is divisible by T or T + 1 ; that is, there are no small “odd” perfect polynomials.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Xiangyong Zeng, Xishun Zhu, Nian Li, Xianping Liu Permutation polynomials having the form ( x 2 i + x + δ ) s 1 + ( x 2 i + x + δ ) s 2 + x over finite fields of even characteristic are investigated in this paper. Eight classes of such polynomials are proved to be permutations.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Robert C. Valentini Let v be the number of distinct values of the polynomial f ( x ) = x 4 + a x 2 + b x , where a and b are elements of the finite field k of size q, where q is odd. When b is 0, an exact formula for v can be given. When b is not 0, as a consequence of the Tchebotarev Density Theorem, v = ( 5 / 8 ) q + O ( q ) . When the Galois closure of k ( x ) / k ( f ( x ) ) is a rational function field, v can be determined exactly. In this work we determine those f ( x ) 's for which this occurs and give the formula for v.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Antonio Cossidente, Giuseppe Marino, Francesco Pavese The maximal cliques of the graph NU ( 4 , q 2 ) related to the Hermitian surface of PG ( 3 , q 2 ) and of the graph NO ± ( 2 n + 2 , q ) , q even, n ≥ 1 , are classified.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Thomas Blackford This paper studies the Galois images of constacyclic codes over F q m of length relatively prime to q, and determines when those images are equal and when they intersect only at the zero codeword. The subfield subcodes and trace codes of constacyclic codes are also determined.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Yuwei Xu, Yongqiang Li, Chuankun Wu, Feng Liu An involution is a permutation whose compositional inverse is itself. Differentially 4-uniform involutions with high algebraic degree and high nonlinearity are important for the design of block ciphers because they possess good cryptographic properties and the same component can be used in both encryption circuit and decryption circuit. A well known differentially 4-uniform involution is the multiplicative inverse function, which is used as the S-boxes in AES and Camellia. Although many differential 4-uniform permutations have been constructed, there are only a few classes of differentially 4-uniform involutions. Recently, Charpin et al. presented an idea of constructing involutions, which is called piece by piece construction. With this method, we construct a family of differentially 4-uniform involutions with optimal algebraic degree and high nonlinearity. It is also shown with the help of computer that such involutions which are CCZ-inequivalent to the known differentially 4-uniform permutations in small number of even dimensions are constructed. Furthermore, the number of CCZ-inequivalent classes of our involutions over F 2 n increases exponentially when n increases.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Theodoulos Garefalakis, Giorgos Kapetanakis We consider the problem of enumerating polynomials over F q , that have certain coefficients prescribed to given values and permute certain substructures of F q . In particular, we are interested in the group of N-th roots of unity and in the submodules of F q . We employ the techniques of Konyagin and Pappalardi to obtain results that are similar to their results in Konyagin and Pappalardi (2006) [8]. As a consequence, we prove conditions that ensure the existence of low-degree permutation polynomials of the mentioned substructures of F q .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Kanat Abdukhalikov In this paper we study those bent functions which are linear on elements of spreads, their connections with ovals and line ovals, and we give descriptions of their dual bent functions. In particular, we give a geometric characterization of Niho bent functions and of their duals, we give explicit formula for the dual bent function and present direct connections with ovals and line ovals. We also show that bent functions which are linear on elements of inequivalent spreads can be EA-equivalent.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Nicholas M. Katz We find some new one-parameter families of exponential sums in every odd characteristic whose geometric and arithmetic monodromy groups are G 2 .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Satoshi Yoshiara It is shown that the Kasami function defined on F 2 n with n even is plateaued. This generalizes a result [3, Theorem 11], where the restriction ( n , 3 ) = 1 is assumed. The result is used to establish the CCZ-inequivalence of the Kasami function defined on F 2 n with n even to the other known monomial APN functions [4].

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): F.E. Brochero Martínez, Theodoulos Garefalakis, Lucas Reis, Eleni Tzanaki In this paper, we find a lower bound for the order of the group 〈 θ + α 〉 ⊂ F ‾ q ⁎ , where α ∈ F q , θ is a generic root of the polynomial F A , r ( X ) = b X q r + 1 − a X q r + d X − c ∈ F q [ X ] and a d − b c ≠ 0 .

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): A. Tomasi, A. Meneghetti, M. Sala We quantify precisely the distribution of the output of a binary random number generator (RNG) after conditioning with a binary linear code generator matrix by showing the connection between the Walsh spectrum of the resulting random variable and the weight distribution of the code. Previously known bounds on the performance of linear binary codes as entropy extractors can be derived by considering generator matrices as a selector of a subset of that spectrum. We also extend this framework to the case of non-binary codes.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Masamichi Kuroda, Shuhei Tsujie Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However, APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties.

Abstract: Publication date: September 2017 Source:Finite Fields and Their Applications, Volume 47 Author(s): Nian Li Permutation polynomials with a few terms attract researchers' interest in recent years due to their simple algebraic form and some additional extraordinary properties. In this paper, by analyzing the quadratic factors of a fifth-degree polynomial and a seventh-degree polynomial over the finite field F 3 2 k , two conjectures on permutation trinomials over F 3 2 k proposed recently by Li, Qu, Li and Fu are settled, where k is a positive integer.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Xueying Shi, Qin Yue, Xiaomeng Zhu Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper, we mainly apply a new method of classical Hermitian self-orthogonal codes to construct three classes of new quantum MDS codes, and these quantum MDS codes provide large minimum distance.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Youngho Jang, Sangtae Jeong, Chunlan Li As a follow-up to our previous work [10], we characterize the measure-preservation of 1-Lipschitz functions on F q [ [ T ] ] in terms of the three well-known bases: Carlitz polynomials, digit derivatives, and digit shifts.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Siyu Liu, Felice Manganiello, Frank R. Kschischang Over a finite field F q m , the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew polynomials to the familiar concept of linear independence for vector spaces. This relation allows for the definition of a representable matroid called the F q m [ x ; σ ] -matroid, with rank function that makes it a metric space. Specific submatroids of this matroid are individually bijectively isometric to the projective geometry of F q m equipped with the subspace metric. This isometry allows one to use the F q m [ x ; σ ] -matroid in a matroidal network coding application.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Chufeng Nien This paper relates n × 1 local gamma factors over finite fields with Gauss sums and characterizes F q × -distinguished characters of F q 2 × in terms of special values of twisted local gamma factors.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Abidin Kaya Recently, Karadeniz and Yildiz introduced an efficient method to search for self-dual codes. It is called lifting method and can be applied to some alphabets. In this work, by considering R 2 -lifts of binary self-dual codes of length 16 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the first time in the literature. The results are tabulated.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Anju, R.K. Sharma In this article, we establish a sufficient condition for the existence of a primitive element α ∈ F q such that for any matrix ( a b c 0 d e ) ∈ M 2 × 3 ( F q ) of rank 2, the element ( a α 2 + b α + c ) / ( d α + e ) is a primitive element of F q , where q = 2 k for some positive integer k. We also give a sufficient condition for the existence of a primitive normal element α ∈ F q n over F q such that ( a α 2 + b α + c ) / ( d α + e ) is a primitive element of F q n for every matrix ( a b c 0 d e ) ∈ M 2 × 3 ( F q n ) of rank 2.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Nurdagül Anbar, Wilfried Meidl For c ∈ F 2 n , a c-bent4 function f from the finite field F 2 n to F 2 is a function with a flat spectrum with respect to the unitary transform V f c , which is designed to describe the component functions of modified planar functions. For c = 0 the transform V f c reduces to the conventional Walsh transform, and hence a 0-bent4 function is bent. In this article we generalize the concept of partially bent functions to the transforms V f c . We show that every quadratic function is partially bent, and hence it is plateaued with respect to any of the transforms V f c . In detail we analyse two quadratic monomials. The first has values as small as possible in its spectra with respect to all transforms V f c , and the second has a flat spectrum for a large number of c. Moreover, we show that every quadratic function is c-bent4 for at least three distinct c. In the last part we analyse a cubic monomial. We show that it is c-bent4 only for c = 1 , the function is then called negabent, which shows that non-quadratic functions exhibit a different behaviour.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Zhi-Wei Sun In this paper we study some sophisticated supercongruences involving dual sequences. For n = 0 , 1 , 2 , … define d n ( x ) = ∑ k = 0 n ( n k ) ( x k ) 2 k and s n ( x ) = ∑ k = 0 n ( n k ) ( x k ) ( x + k k ) = ∑ k = 0 n ( n k ) ( − 1 ) k ( x k ) ( − 1 − x k ) . For any odd prime p and p-adic integer x, we determine ∑ k = 0 p − 1 ( ± 1 ) k d k ( x ) 2 and ∑ k = 0 p − 1 ( 2 k + 1 ) d k ( x ) 2 modulo p 2 ; for example, we establish the new p-adic congruence ∑ k = 0 p − 1 ( − 1 ) k d k ( x ) 2 ≡ ( − 1 ) 〈 x 〉 p ( mod p 2 ) , where 〈 x 〉 p denotes the least nonnegative integer r with x ≡ r ( mod p ) . For any prime p > 3 and p-adic integer x, we determine PubDate: 2017-04-23T07:06:14Z

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Elif Segah Oztas, Bahattin Yildiz, Irfan Siap In this work we introduce a novel approach to find reversible codes over different alphabets, using so-called coterm polynomials and a module-construction. We obtain many optimal reversible codes with these constructions. In an attempt to apply the constructions to the DNA, we identify k-bases of DNA with elements in the ring R 2 k = F 2 [ u ] / ( u 2 k − 1 ) , and by using a form of coterm polynomials, we are able to solve the reversibility and complement problems in DNA codes over this ring. With a freedom on the choice of k we are able to embed any DNA code in a suitable ring, giving an algebraic structure to the DNA codes. We are also able to find reversible and reversible-complement codes that are not necessarily linear cyclic codes.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Abhishek Bhowmick, Thái Hoàng Lê, Yu-Ru Liu We prove a character sum estimate in F q [ t ] and answer a question of Shparlinski.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Mariusz Kwiatkowski, Mark Pankov Let Γ k ( V ) be the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the finite field F q consisting of q elements and 1 < k < n − 1 . Denote by Γ ( n , k ) q the restriction of the Grassmann graph to the set of all non-degenerate linear [ n , k ] q codes. We describe maximal cliques of the graph Γ ( n , k ) q and show that every automorphism of this graph is induced by a monomial semilinear automorphism of V.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): B.K. Dass, Namita Sharma, Rashmi Verma The paper begins by giving a counter example to show that the algorithm for construction of new perfect poset codes from a given perfect poset code by removal of a coordinate as given by Lee (2004) [11] does not hold. The algorithm has been improved and generalized to obtain new perfect poset block codes from a given perfect poset block code. The modified necessary and sufficient conditions for the construction of new perfect poset codes have been derived as a particular case. A bound has been obtained on the height of poset P s that turns a given π-code into r-perfect ( P s , π ) -code. We show that there does not exist a poset which admits the binary Simplex code of order 3 to be a 2-perfect poset code. Also, all the poset structures which admit the extended ternary Golay code to be a 3-perfect poset code have been classified.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Ilaria Cardinali, Luca Giuzzi A k-polar Grassmannian is a geometry having as pointset the set of all k-dimensional subspaces of a vector space V which are totally isotropic for a given non-degenerate bilinear form μ defined on V. Hence it can be regarded as a subgeometry of the ordinary k-Grassmannian. In this paper we deal with orthogonal line Grassmannians and with symplectic line Grassmannians, i.e. we assume k = 2 and μ to be a non-degenerate symmetric or alternating form. We will provide a method to efficiently enumerate the pointsets of both orthogonal and symplectic line Grassmannians. This has several nice applications; among them, we shall discuss an efficient encoding/decoding/error correction strategy for line polar Grassmann codes of either type.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): M. Calderini, M. Sala, I. Villa APN permutations in even dimension are vectorial Boolean functions that play a special role in the design of block ciphers. We study their properties, providing some general results and some applications to the low-dimension cases. In particular, we prove that none of their components can be quadratic. For an APN vectorial Boolean function (in even dimension) with all cubic components we prove the existence of a component having a large number of balanced derivatives. Using these restrictions, we obtain the first theoretical proof of the non-existence of APN permutations in dimension 4. Moreover, we derive some constraints on APN permutations in dimension 6.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Yongbo Xia, Chunlei Li Based on a generic construction, two classes of ternary three-weight linear codes are obtained from a family of power functions, including some APN power functions. The weight distributions of these linear codes are determined by studying the properties of some exponential sums related to the power functions.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Danyao Wu, Pingzhi Yuan, Cunsheng Ding, Yuzhen Ma Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over F 2 m in Zieve's paper [30]. We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over F 2 m . Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Liqing Xu, Hao Chen Binary constant weight codes have important applications in various topics and have been studied for many years. Optimal or near-optimal binary constant weight codes of small lengths have been determined. In this paper we propose an improvement of the Ericson–Zinoviev construction of binary constant weight codes from q-ary codes. By applying this improvement to Reed–Solomon codes, some new or optimal binary constant weight codes are presented. In particular new binary constant weight codes A ( 64 , 10 , 8 ) ≥ 4112 and A ( 64 , 12 , 8 ) ≥ 522 are constructed. We also give explicitly constructed binary constant weight codes which improve the Gilbert and the Graham–Sloane lower bounds asymptotically in a small range of parameters. Some new binary constant weight codes constructed from algebraic-geometric codes by applying our this improvement are also presented.

Abstract: Publication date: July 2017 Source:Finite Fields and Their Applications, Volume 46 Author(s): Hengzhou Xu, Baoming Bai, Dan Feng, Cheng Sun Motivated by the works on the girth of Tanner ( 3 , 5 ) and ( 3 , 7 ) quasi-cyclic (QC) low-density parity-check (LDPC) codes done by S. Kim et al. and M. Gholami et al., respectively, we analyze the cycles of Tanner ( 3 , 11 ) QC LDPC codes and present the sufficient and necessary conditions for the existence of cycles of lengths 4, 6, 8, and 10 in Tanner ( 3 , 11 ) QC LDPC codes. By checking these conditions, the girth values of Tanner ( 3 , 11 ) QC LDPC codes are derived.