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Let R q (r , n) denote the r th order Reed-Muller code of length q n over F q. We consider two algebraic questions about the Reed-Muller code. Let H q (r , n) = R q (r , n) / R q (r − 1 , n). (1) When q = 2 , it is known that there is a "duality" between the actions of GL (n , F 2) on H 2 (r , n) and on H 2 (r ′ , n) , where r + r ′ = n. The result is false for a general q. However, we find that a slightly modified duality statement still holds when q is a prime or r < char F q. (2) Let F (F q n , F q) denote the F q -algebra of all functions from F q n to F q. It is known that when q is a prime, the Reed-Muller codes { 0 } = R q (− 1 , n) ⊂ R q (0 , n) ⊂ ⋯ ⊂ R q (n (q − 1) , n) = F (F q n , F q) are the only AGL (n , F q) -submodules of F (F q n , F q). In particular, H q (r , n) is an irreducible GL (n , F q) -module when q is a prime. For a general q , H q (r , n) is not necessarily irreducible. We determine all its submodules and the factors in its composition series. The factors of the composition series of H q (r , n) provide an explicit family of irreducible representations of GL (n , F q) over F q. [ABSTRACT FROM AUTHOR]

Permutation polynomials with low boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over 픽2n, we determine the boomerang uniformity of these permutation polynomials: (1) f1(x) = (x2m + x + δ)22m+1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 1; (2) f2(x) = (x2m + x + δ)22m−1+2m−1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 0; (3) f3(x) = (x2m + x + δ)23m−1+2m−1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 0. The results show that the boomerang uniformity of f1(x), f2(x) and f3(x) can attain 2n. [ABSTRACT FROM AUTHOR]

For each q of the form 4k, we determine all a Fq for which X + Xq + X2q-1 + aXq²-q+1 permutes Fq². We also construct a class of permutation trinomials over Fq² in case q = 1 (mod 3). [ABSTRACT FROM AUTHOR]

In this study, we proposed a novel method that integrates density functional theory (DFT) with the finite field method to accurately estimate the polarizability and dielectric constant of polymers. Our approach effectively accounts for the influence of electronic and geometric conformation changed on the dielectric constant. We validated our method using polyethylene (PE) and polytetrafluoroethylene (PTFE) as benchmark materials, and found that it reliably predicted their dielectric constants. Furthermore, we explored the impact of conformation variations in poly(vinylidene fluoride) (PVDF) on its dielectric constant and polarizability. The resulting dielectric constants of α- and γ-PVDF (3.0) showed excellent agreement with crystalline PVDF in experiments. Our findings illuminate the relationship between PVDF's structural properties and its electrical behavior, offering valuable insights for material design and applications. [ABSTRACT FROM AUTHOR]

Permutation polynomials over finite fields constitute an active research area and have important applications in many areas of science and engineering. In this paper, several classes of permutation polynomials with trace functions are presented over F p n (p = 2 , 3) by investigating the number of solutions to special equations. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q. In this paper, we investigate the permutation property of the quadrinomial f (x) = x + a 1 x q (q − 1) + 1 + a 2 x q + a 3 x 2 (q − 1) + 1 over F q 2 , where q is odd and a 1 , a 2 , a 3 ∈ F q * . More precisely, we obtain the necessary and sufficient conditions for f (x) to be a permutation polynomial over F q 2 . [ABSTRACT FROM AUTHOR]

Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for every positive even integer ℓ. In this paper, we study the case where $ x^m-1 $ x m − 1 has an arbitrary number of irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] ; in the previous studies, only some special cases where $ x^m-1 $ x m − 1 has exactly two or three irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for any even positive integer ℓ, where we require that $ q \equiv 1 \pmod {4} $ q ≡ 1 (mod 4) if the index $ \ell \ge 6 $ ℓ ≥ 6. By implementation of our method, we obtain a new optimal binary self-dual code $ [172, 86, 24] $ [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4. [ABSTRACT FROM AUTHOR]

Let \mathbb {F}_q denote the finite field of characteristic p and order q. Let \mathbb {Z}_q denote the unramified extension of the p-adic rational integers \mathbb {Z}_p with residue field \mathbb {F}_q. Given two positive integers m,n, define a box \mathcal B_m to be a subset of \mathbb {Z}_q^n with q^{nm} elements such that \mathcal B_m modulo p^m is equal to (\mathbb {Z}_q/p^m \mathbb {Z}_q)^n. For a collection of nonconstant polynomials f_1,\dots,f_s\in \mathbb {Z}_q[x_1,\ldots,x_n] and positive integers m_1,\dots,m_s, define the set of common zeros inside the box \mathcal B_m to be \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} It is an interesting problem to give the sharp estimates for the p-divisibility of |V|. This problem has been partially solved for the three cases: (i) m=m_1=\cdots =m_s=1, which is just the Ax-Katz theorem, (ii) m=m_1=\cdots =m_s>1, which was solved by Katz [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m=1, and m_1,\dots,m_s\geq 1, which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. Based on the multi-fold addition and multiplication of the finite Witt rings over \mathbb {F}_q, we investigate the remaining unconsidered case of m>1 and m\neq m_j for some 1\leq j\leq s, and finally provide a complete answer to this problem. [ABSTRACT FROM AUTHOR]

We investigate the rationality of Weil sums of binomials of the form W u K , s = ∑ x ∈ K ψ (x s − u x) , where K is a finite field whose canonical additive character is ψ , and where u is an element of K × and s is a positive integer relatively prime to | K × | , so that x ↦ x s is a permutation of K. The Weil spectrum for K and s , which is the family of values W u K , s as u runs through K × , is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if s is nondegenerate (i.e., if s is not a power of p modulo | K × | , where p is the characteristic of K). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where | K | = 5 and s ≡ 3 (mod 4). [ABSTRACT FROM AUTHOR]

By exploiting symmetries of finite fields, covering perfect hash families provide a succinct representation for covering arrays of index one. For certain parameters, this connection has led to both the best current asymptotic existence results and the best known efficient construction algorithms for covering arrays. The connection generalizes in a straightforward manner to arrays in which every t-way interaction is covered λ > 1 times, i.e., to covering arrays of index more than one. Using this framework, we focus on easily computed, explicit upper bounds on numbers of rows for various parameters with higher index. [ABSTRACT FROM AUTHOR]

Given a prime power q and a positive integer n, let 픽 q n {\mathbb{F}_{q^{n}}} represent a finite extension of degree n of the finite field 픽 q {{\mathbb{F}_{q}}} . In this article, we investigate the existence of m elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for n ≥ 6 {n\geq 6} , q = 3 k {q=3^{k}} , m = 2 {m=2} we establish that there are only 10 possible exceptions. [ABSTRACT FROM AUTHOR]

By further exploring some ideas in Ding and Yin, the results on preimage distributions of a class of zero-difference balanced functions are extended and the parameters of a class of partitioned difference families are determined by means of self-conjugate Gauss sums over finite fields. [ABSTRACT FROM AUTHOR]

This paper is aimed at using a lookup table method to improve the scalar multiplication performance of elliptic curve cryptography. The lookup table must be divided into two polynomials and requires two iterations of point doubling operation, for which negation operations are needed. It is well known that an inversion operation requires a lot of multiplication time. The advantage of this paper is that we are able to reduce one inverse element calculation for this problem and also improve the basic operations of finite fields through segmentation methods. If the normal basis method is used in the design of the inverse element operation, it must be converted to the normal basis through the standard basis. However, the conversion process requires a lot of matrix operations. Though the anti-element operation has good speed performance, it also increases the computational complexity. Using number theory and grouping methods will greatly improve the performance of inverse element operations. With application of the two-time point doubling operation in the hardware implementation, the developed approach reduces the computing time by 48% as compared with the conventional approach. The computational time of the scalar multiplication using the presented method is further improved by 67% over the traditional algorithm with only an area increase of 12%. Finally, the proposed lookup table-based technique can be utilized for software and hardware implementation, as the developed arithmetic operations are simple and are consistent in their execution. [ABSTRACT FROM AUTHOR]

For a finite field Fqn and a rational function f = f1/f2 ∈ Fqn(x), we present a sufficient condition for the existence of a primitive normal element α ∈ Fqn in such a way f(α) is also primitive in Fqn, where f(x) is a rational function in Fqn(x) of degree sum m (degree sum of f(x) = f1(x)/f2(x) is defined to be the sum of the degrees of f1(x) and f2(x)). Additionally, for rational functions of degree sum 4, we proved that there are only 37 and 16 exceptional values of (q, n) when q = 2k and q = 3k respectively. [ABSTRACT FROM AUTHOR]

The best known asymptotic bit complexity bound for factoring univariate polynomials over finite fields grows with d 1.5 + o (1) for input polynomials of degree d, and with the square of the bit size of the ground field. It relies on a variant of the Cantor–Zassenhaus algorithm which exploits fast modular composition. Using techniques by Kaltofen and Shoup, we prove a refinement of this bound when the finite field has a large extension degree over its prime field. We also present fast practical algorithms for the case when the extension degree is smooth. [ABSTRACT FROM AUTHOR]

In this paper, we establish some relations between irreducible polynomials over a finite field F q and its quadratic extension F q 2 . First we consider a relation between the numbers of irreducible polynomials of a fixed degree over F q and F q 2 , and some relations between self-reciprocal irreducible polynomials over F q and self-conjugate-reciprocal irreducible polynomials over F q 2 . We also obtain formulas for the number and the product of all self-conjugate-reciprocal irreducible monic (SCRIM) polynomials over F q 2 . [ABSTRACT FROM AUTHOR]

For each prime power q, we determine all a 휖 픽q2 for which X + Xq + X2q−1 + aXq2−q+1 permutes 픽q2. [ABSTRACT FROM AUTHOR]

Let K be a finite field of characteristic 2, and U T 2 : = U T 2 (K) be the Lie algebra of 2 × 2 upper triangular matrices over K with the multiplication x ∘ y = x y + y x = x y − y x. In this paper, we exhibit a finite basis of graded identities for the variety of Lie algebras generated by U T 2 for any grading and show that it has the Specht property. It is important to highlight that the technique used in order to solve the Specht problem is independent of the characteristic of the field and also of its cardinality. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative ring with unity and $ Z(R) $ Z (R) be its centre. The commuting graph of R denoted by $ \Gamma (R) $ Γ (R) is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let F be a finite field. In this paper, we show that if $ \Gamma (R)\cong \Gamma (M_3(F)) $ Γ (R) ≅ Γ (M 3 (F)) and $ Z(R) $ Z (R) is a field, then $ R\cong M_3(F) $ R ≅ M 3 (F). In particular, if $ \Gamma (R)\cong \Gamma (M_3(F_p)) $ Γ (R) ≅ Γ (M 3 (F p)) , then $ R\cong M_3(F_p) $ R ≅ M 3 (F p). [ABSTRACT FROM AUTHOR]

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C' are curves over a finite field K, with K-rational base points P and P', and let D and D' be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C,P) and (C',P') are doubly isogenous if Jac(C) and Jac(C') are isogenous over K and Jac(D) and Jac(D') are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon. [ABSTRACT FROM AUTHOR]

In Neri et al. (2022) [20] , the authors extended the family of 2-dimensional F q 2 t -linear MRD codes recently found in Longobardi et al. (2023) [13]. Also, for t ≥ 5 they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction t ≥ 5. Moreover, we prove that in the case when t = 4 , the linear sets of the projective line PG (1 , q 8) ensuing from codes in the relevant family, are not equivalent to any one known so far. [ABSTRACT FROM AUTHOR]

Let q be an odd prime power and suppose that $a,b\in \mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$. Denote by $\sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.029\,08$ and $\beta \approx 0.012\,59$ such that if $q\equiv 1 \bmod 4$ , then $\lim \sigma (q)/q^2 = \alpha $ , and if $q \equiv 3 \bmod 4$ , then $\lim \sigma (q)/q^2 = \beta $. [ABSTRACT FROM AUTHOR]

The nonlinearity of vectorial functions and of their restrictions to manifolds are defined as the Hamming distance to the set of affine mappings and of their restrictions to the manifold, respectively. Relations between the parameters of the nonlinearity of a vectorial function and their analogues for its coordinate functions and its restrictions to manifolds are established. An analogue of the Parseval identity for such parameters of vectorial functions is proved, which implies the upper bound (qk − 1)qn−k − qn/2−k for the nonlinearity of a mapping over a q-element field of n variables with k coordinates. Attainability conditions for this estimate are found, and a class of Boolean vectorial functions with high value of nonlinearity is constructed. Estimates characterizing the distribution of the nonlinearity of a vectorial function and of its restrictions to manifolds are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field F p. We apply symmetric (row and column) elementary transformations to the adjacency matrix over F p in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over F p for any odd prime p. Our key tool is quadratic residues of F p. [ABSTRACT FROM AUTHOR]

Let 픽 q be the finite field of odd characteristic p with q elements ( q = p n , n ∈ ℕ ) and let 픽 q * represent the set of nonzero elements of 픽 q . By making use of the Smith normal form of exponent matrices, we obtain an explicit formula for the number of rational points on the variety defined by the following system of equations over 픽 q : { ∑ i = 1 r a i (1) ⁢ x 1 e i ⁢ 1 (1) ⁢ ⋯ ⁢ x n e i ⁢ n (1) = b 1 , ∑ j ′ = 0 t - 1 ∑ i ′ = 1 r j ′ + 1 - r j ′ a r j ′ + i ′ (2) ⁢ x 1 e r j ′ + i ′ , 1 (2) ⁢ ⋯ ⁢ x n j ′ + 1 e r j ′ + i ′ , n j ′ + 1 (2) = b 2 , where b i ∈ 픽 q ( i = 1 , 2 ), t ∈ ℕ , 0 = n 0 < n 1 < n 2 < ⋯ < n t , n k - 1 < n ≤ n k for some 1 ≤ k ≤ t , 0 = r 0 < r 1 < r 2 < ⋯ < r t , a i (1) ∈ 픽 q * for i ∈ { 1 , ... , r } , a i ′ (2) ∈ 픽 q * for i ′ ∈ { 1 , ... , r t } , and the exponent of each variable is a positive integer. This generalizes the results obtained previously by Wolfmann, Sun, Cao, and others. Our result also gives a partial answer to an open problem raised by Hu, Hong and Zhao [S. N. Hu, S. F. Hong and W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 2015, 135–153]. [ABSTRACT FROM AUTHOR]

We determine the roots in F q 3 of the polynomial X 2 q ℓ + 1 + X + a for each positive integer ℓ and each a ∈ F q , where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more explicit than the previous results in this area. Our results resolve an open problem and a conjecture of Zheng, Kan, Zhang, Peng, and Li. [ABSTRACT FROM AUTHOR]

This paper constructs a novel Hopf algebra cf (UT •) on the class functions of the unipotent upper triangular groups UT n (F q) over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of GL n (F q) class functions. Furthermore, cf (UT •) contains a Hopf subalgebra which is isomorphic to a known combinatorial Hopf algebra, previously used to prove a conjecture about chromatic quasisymmetric functions. Some additional Hopf algebraic properties are also established. [ABSTRACT FROM AUTHOR]

We showed that isomorphism classes of idempotent evolution algebras are in bijection with the orbits of the semidirect product group of the symmetric group and the torus, considered the combinatoric problem of enumeration of isomorphism classes for these algebras over arbitrary finite fields, derived a general counting formula, and obtained explicit formulas for the numbers of isomorphism classes in dimensions 2, 3, and 4 over any finite field. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over a finite field Fq, q = pd, p > 3, and a prime number l> 3 such that q ≡1 (mod l) and ‘ l| #E(Fq). In this paper we study the possible factorisation patterns over Fq[x] of the ‘lk-division polynomials associated to E with k≥ 2, extending the work of Verdure [6] for k = 1. [ABSTRACT FROM AUTHOR]

Let G (R) be the unit graph associated with a ring R. Let p be a prime number and let R be a finite ring of order p or p 2 and be one of the rings Z p , Z p 2 , G F (p 2) , Z p (+) Z p or Z p × Z p . We determine the geodetic number g (G (R)) associated with each such ring. [ABSTRACT FROM AUTHOR]

We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials. [ABSTRACT FROM AUTHOR]

Let q denote the finite field with q elements. In this paper, we investigate the trinomial f (x) = x 4 q + 1 + α x 5 q + x q + 4 over the finite field 5 2 k , where α ∈ 5 k * with k being a positive integer. We prove that the trinomial f (x) permutes 5 2 k if and only if α = − 1 and k is even. This work is a continuation of the previous work of Bai and Xia [A new class of permutation trinomials constructed from Niho exponents, Cryptogr. Commun. 10 (2018) 1023–1036]. [ABSTRACT FROM AUTHOR]

In an embedded and resource-constrained environment, Elliptic Curve Cryptography (ECC) has been noted as an efficient and suitable methodology for achieving information security via public-key cryptography. However, the drawback of ECC is its lack of unifiedness in point operation that makes it prone to side-channel attack. Also, ECC does not satisfy the completeness property due to which the addition formula is not defined for all the pairs of input points. Edward curve, with its unified addition law and completeness property, proved to be the answer to aforementioned flaws. High throughput while maintaining low resource is a key issue for elliptic curve cryptography (ECC) hardware implementations in many applications. This paper presents the implementation of a Binary Edward curve Crypto processor over G F (2 233) for FPGA platforms. The architecture is modified to perform scalar multiplication in a parallel manner using two hybrid Karatsuba field multipliers. Field inversion being one of the most tedious operations while reconversion, is also performed in a parallel manner using an efficient Hex Itoh-Tsujii inversion algorithm. The hardware resources are shared for performing point operations and inversion. Exploiting parallelism in point and inversion operations has resulted in reduction of the clock cycles consumed and the resultant architecture is more efficient in terms of throughput over area. The design takes 0.038 ms on Xilinx Virtex-4 and 0.031 ms on Virtex-7 FPGA platforms to perform a 233-bit point multiplication operation. It takes 73.57%, 13.71%, 14.76% and 48.76% more efficient than existing scalar multiplication with BEC. This proposed scalable, side-channel attack resilient design outperforms the existing techniques with respect to throughput over area. • The parallelism in BEC is done with two modular hybrid Karatsuba multiplier. • Inversion in reconversion is done with new Hex-Itoh Tsujii inversion algorithm (HITA). • Point operation and HITA uses the same resources with reduced clock cycles. • This architecture is designed for FPGA platforms yielding high throughput and area-time performance. [ABSTRACT FROM AUTHOR]

Group action on the projective space PG(3,q)is a method which can be used to construct some geometric objects for example cap. We constructed new caps in PG(3,13) of degrees 2, 3, 4, 7,14 and sizes 2, 4, 5, 7, 10, 14, 17, 20, 28, 34, 35, 68, 70, 85, 119, 140, 170, 238, 340, 476, 595, 1190. Then the incomplete caps are extended to complete caps. [ABSTRACT FROM AUTHOR]

The nonlinearity and additive nonlinearity of a function are defined as the Hamming distances, respectively, to the set of all affine mappings and to the set of all mappings having nontrivial additive translators. On the basis of the revealed relation between the nonlinearities and the Fourier coefficients of the characters of a function, convenient formulas for nonlinearity evaluation for practically important classes of functions over an arbitrary finite field are found. In the case of a field of even characteristic, similar results were obtained for the additive nonlinearity in terms of the autocorrelation coefficients. The formulas obtained made it possible to present specific classes of functions with maximal possible and high nonlinearity and additive nonlinearity. [ABSTRACT FROM AUTHOR]

Given F = 픽pt, a field with pt elements, where p is a prime power and t ≥ 7 is a positive integer. Let n ∈ ℕ and f = f1/f2 is a rational function, where f1 and f2 are distinct irreducible polynomials with deg(f1) +deg(f2) = n ≤ pt in F[x]. We construct a sufficient condition on (p,t) which guarantees primitive pairing (휖,f(휖)) exists in F such that Tr픽 pt/픽p(휖) = a and Tr픽 pt/픽p(f(휖)) = b for any prescribed a,b ∈ 픽p. Further, we demonstrate for any positive integer n ≤ pt, such a pair definitely exists for large t. The scenario when n = 2 is handled separately and we verified that such a pair exists for all (p,t) except from possible 71 values of (p,t). A result for the case n = 3 is given as well. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the Wedderburn decompositions of the semisimple group algebras of all groups up to order 120. More precisely, we explicitly compute the Wedderburn decompositions of the semisimple group algebras of 26 non-metabelian groups. [ABSTRACT FROM AUTHOR]

We determine the minimal bi-degree(s) of an irreducible filling curve over F q for ℙ 1 × ℙ 1 by constructing examples. It is (q + 1 , q + 1) if q ≠ 2 , and they are (4, 3) and (3, 4) if q = 2. [ABSTRACT FROM AUTHOR]

–Let F be a finite field of characteristic 2. In this article, we have looked into the existence of normal complement of G in V (F G) , where G is either the alternating group A4 or the dihedral group D 4 m of order 4m, for an odd integer m ≥ 3 . Also, we have explicitly found a normal complement of the symmetric group S4 in V (F S 4) over the field F containing 2 elements. [ABSTRACT FROM AUTHOR]

An n-place function over a field with q elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even n, the maximally nonlinear functions are bent functions, but, for q > 2, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found. [ABSTRACT FROM AUTHOR]

An n-place function over a field F q with q elements is called maximally nonlinear if it has the largest nonlinearity among all q-valued n-place functions. We show that, for even n=2, a function is maximally nonlinear if and only if its nonlinearity is q n − 1 (q − 1) − q n 2 − 1 ; for n=1, the corresponding criterion for maximal nonlinearity is q − 2. For q > 2 and even n=2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions. [ABSTRACT FROM AUTHOR]

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra A h generated by elements x , y , which satisfy the relation y x − x y = h for some 0 ≠ h ∈ F [ x ]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of A h over an infinite field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

We prove that for any prime power $q\notin \{3,4,5\}$ , the cubic extension $\mathbb {F}_{q^{3}}$ of the finite field $\mathbb {F}_{q}$ contains a primitive element $\xi $ such that $\xi +\xi ^{-1}$ is also primitive, and $\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi)=a$ for any prescribed $a\in \mathbb {F}_{q}$. This completes the proof of a conjecture of Gupta et al. ['Primitive element pairs with one prescribed trace over a finite field', Finite Fields Appl. 54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree $n\ge 3$. [ABSTRACT FROM AUTHOR]

Digital time was defined KSR-PP [2] with three two-digit positions as h2h1 : m2m1 : s2s1. It was identified with appropriate restricted place values on the hours (H), minutes (M) and seconds (S) shown to be 86400-element cyclic Time Group, TG. Here it is shown to be a finite time field, TF. A palindromic sequence of 119-elements and its sub-sequences are shown to be consequences of TF. [ABSTRACT FROM AUTHOR]

Recently, Kim and Son [9] have obtained an infinite sequence of irreducible polynomials over the finite field Fq of characteristic p by using the transformation. Here, it is shown that one can also build this infinite sequence of irreducible polynomials over the field Fq of characteristic p by using a different transformation. Moreover, this transformation is helpful to establish the sequence of normal polynomials of degree npr in Fq[x] by using an appropriate initial normal polynomial of degree n. [ABSTRACT FROM AUTHOR]

For a field E of characteristic different from 2 and cohomological 2-dimension one, quadratic forms over the rational function field E (X) are studied. A characterisation in terms of polynomials in E [ X ] is obtained for having that quadratic forms over E (X) satisfy a local-global principle with respect to discrete valuations that are trivial on E. In this way new elementary proofs for the local-global principle are achieved in the cases where E is finite or pseudo-algebraically closed. The study is complemented by various examples. [ABSTRACT FROM AUTHOR]

The Elliptic curve cryptosystem is a public-key cryptosystem that receives more focus in recent years due to its higher security with smaller key size when compared to RSA. Smartcards and other applications have highlighted the importance of security in resource-constrained situations. To meet the increasing need for speed in today's applications, hardware acceleration with cryptographic algorithms is required. In this paper, we present a novel parallel architecture for elliptic curve scalar multiplication based on a modified Lopez-Dahab–Montgomery(LDM) algorithm, to reduce the total time delay for computing scalar multiplication. It comprises three main steps: affine to projective conversion, point addition and doubling in the main loop followed by reconversion to affine coordinate. The modified parallel algorithm with new inversion in the reconversion yields lesser clock cycle and total time delay compared to existing techniques in the literature for the National Institute of Standards and Technology recommended trinomial G F (2 233) . Our proposed architecture implemented on Virtex4 and Virtex7 FPGA technologies, respectively, achieved a lesser clock cycle of 956, which yields a lesser delay of 20.025 and 8.22 μs. Compared with the state-of-the-art of existing techniques, two multiplications are reduced in the reconstruction process and our processor yields 18.29% and 27.21% increase in area-time performance in Virtex 4 and Virtex 7 devices, respectively. [ABSTRACT FROM AUTHOR]

Bateman and Duquette have initiated the study of Salem elements in positive characteristic. Later, Kerada introduced and studied j-Salem numbers (j ≥ 2). To complete this circle of ideas we introduce 2-Salem elements over the field of formal Laurent series over a finite field q . We extend the above-cited studies and we provide some analog results. The main result of this note is Theorem 1.2 which characterizes the set T 2 ′ ∗ of 2-Salem series when q ≠ 2 r . [ABSTRACT FROM AUTHOR]

For a prime p and positive integers m, n, let F q be a finite field with q = p m elements and F q n be an extension of F q. Let h(x) be a polynomial over F q n satisfying the following conditions: (i) Tr m nm (x) ∘ h (x) = τ (x) ∘ Tr m nm (x) ; (ii) For any s ∈ F q , h(x) is injective on Tr m nm (x) - 1 (s) , where τ (x) is a polynomial over F q. For b , c ∈ F q , δ ∈ F q n , and positive integers i, j, d with q ≡ ± 1 (mod d) , we propose a class of permutation polynomials of the form b (Tr m nm (x) + δ) 1 + i (q n - 1) d + c (Tr m nm (x) + δ) 1 + j (q n - 1) d + h (x) over F q n by employing the Akbary–Ghioca–Wang (AGW) criterion in this paper. Accordingly, we also present the permutation polynomials of the form b (Tr m nm (x) + δ) 1 + i (q n - 1) d + h (x) by letting c = 0 and choosing some special i, which covered some known results of this form. [ABSTRACT FROM AUTHOR]

Let x3 + ax + b be an irreducible cubic polynomial over p. Then by using transformations x → x + j and x → j–1x, where j ∈ F*p, we can generate new irreducible cubic polynomials. In this paper, we use the elliptic curve equation y2 = x3 + ax + b and some transformations to generate an irreducible cubic polynomial, say Q. We generate the public and private keys in the elliptic curve digital signature algorithm (ECDSA), by defining the isomorphism between.Finally, we discuss the security of keys and the validity of signature. [ABSTRACT FROM AUTHOR]