Search

With the increasing use of digital images, there is a growing need for secure and efficient image encryption algorithms to ensure their confidentiality during transmission. In this paper, we propose an image encryption scheme for grayscale and color images that is based on primitive polynomial and Kronecker product of two invertible Suslin matrices. In the proposed scheme, primitive polynomial to map the pixel values of plain image to the elements of finite field is used, which results in pixel permutation. Then, the shuffled pixels are further diffused using a matrix obtained from the Kronecker product of two invertible Suslin matrices. To enhance security, the proposed scheme incorporates the logistic chaotic map. Further, we add authentication with the help of digital signature. Finally, the proposed scheme is implemented using Python and analyzed using various security parameters. The proposed methodology outperforms existing techniques in several critical aspects, including resistance to noise and data loss attacks, key space size, time complexity, correlation coefficient of adjacent pixels and entropy. Notably, the proposed scheme showcases a significant improvement in performance with PSNR values approaching ideal levels for noisy images and a vast 256-bit key space, bolstering security. These advancements along with the disruption of pixel correlations and increased randomness collectively makes encryption scheme a promising and secure choice for safeguarding grayscale and color images during transmission. [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]

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]

Let 픽 q d be the d-dimensional vector space over the finite field 픽 q with q elements. For each non-zero r in 픽 q and E ⊂ 픽 q d , we define W ⁢ (r) as the number of quadruples (x , y , z , w) ∈ E 4 such that Q ⁢ (x - y) Q ⁢ (z - w) = r , where Q is a non-degenerate quadratic form in d variables over 픽 q . When Q ⁢ (α) = ∑ i = 1 d α i 2 with α = (α 1 , ... , α d) ∈ 픽 q d , Pham (2022) recently used the machinery of group actions and proved that if E ⊂ 픽 q 2 with q ≡ 3 ⁢ (mod ⁡ 4) and | E | ≥ C ⁢ q , then we have W ⁢ (r) ≥ c ⁢ | E | 4 q for any non-zero square number r ∈ 픽 q , where C is a sufficiently large constant, c is some number between 0 and 1, and | E | denotes the cardinality of the set E. In this article, we improve and extend Pham's result in two dimensions to arbitrary dimensions with general non-degenerate quadratic distances. As a corollary of our results, we also generalize the sharp results on the Falconer-type problem for the quotient set of distance set due to the first two authors and Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. The key new ingredient is to relate the estimate of the W ⁢ (r) to a quadratic homogeneous variety in 2 ⁢ d -dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums which are more handleable than the Kloosterman sums appearing in the standard distance-type problems. [ABSTRACT FROM AUTHOR]

We consider polynomials f(x1, ..., xn) over a finite filed that satisfy the following condition: the set of solutions of the equation f(x1, ..., xn) = b, where b is some element of the field, coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with the right-hand side b (or with respect to b). We describe a number of properties of multiaffine polynomials. Then on the basis of these properties we propose a polynomial algorithm that takes a polynomial over a finite field and an element of the field as an input and decides whether the polynomial is multiaffine with respect to this element. In case of the positive answer the algorithm also outputs a system of linear equations that corresponds to this polynomial. The complexity of the proposed algorithm is the smallest in comparison with other known algorithms that solve this problem. [ABSTRACT FROM AUTHOR]

Let a prime p ≡ 1(mod3) and z be non-cubic in Fp. Gauss proved that the number of solutions of equation... In 1978, Chowla, Cowles, and Cowles determined the sign of d for the case of 2 being non-cubic in Fp. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field Fq with q = pk, p ≡ 1(mod3), and determined the sign of d for the case of 3 being non-cubic. [ABSTRACT FROM AUTHOR]

In this paper, the approximate synchronization of leader-follower multiagent systems (MASs) over finite fields is studied in regard to local and global synchronization. First, the approximately synchronous state set (ASSS) is obtained. Second, combined with ASSS and transient periods, some criteria for the local and global approximate synchronization of systems are given. Moreover, the algorithms for calculating the maximum approximately synchronous basin (MASB) and the maximum control invariant set (MCIS) are presented. Third, the global approximate synchronization of the system is achieved by designing the state feedback control, and a design algorithm of the controller using the truth matrix method is proposed. Moreover, the results of approximate synchronization are degenerated to complete synchronization. Last, two examples are shown to demonstrate the results of this paper. [ABSTRACT FROM AUTHOR]

Wan and Zhang (2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields, and proposed a problem whether their bound can be improved. In this paper, the author improves Wan-Zhang's bound from three aspects. The proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric polynomial. And the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound. [ABSTRACT FROM AUTHOR]

Codebooks with small inner-product correlations are desirable in many fields, including compressed sensing, direct spread code division multiple access (CDMA) systems, and space-time codes. The objective of this paper is to present a class of codebooks and explore their applications in strongly regular Cayley graphs. The obtained codebooks are nearly optimal in the sense that their maximum cross-correlation amplitude nearly meets the Welch bound. As far as we know, this construction of codebooks provides new parameters. [ABSTRACT FROM AUTHOR]

Purpose – The author studies forms over finite fields obtained as the determinant of Hermitian matrices and use these determinatal forms to define and study the base polynomial of a square matrix over a finite field. Design/methodology/approach – The authors give full proofs for the new results, quoting previous works by other authors in the proofs. In the introduction, the authors quoted related references. Findings – The authors get a few theorems, mainly describing some monic polynomial arising as a base polynomial of a square matrix. Originality/value – As far as the author knows, all the results are new, and the approach is also new.

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]

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]

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 rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Xinjiang University (Natural Science Edition) is the property of Xinjiang University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

An elementary and self-contained argument for the complete determination of maximal plane curves of degree 3 over F4 will be given, which complements Hirschfeld-Storme-Thas-Voloch’s theorem on a characterization of Hermitian curves in P². This complementary part should be understood as the classification of Sziklai’s example of maximal plane curves of degree q − 1 over Fq. Although two maximal plane curves of degree 3 over F4 up to projective equivalence over F4 appear, they are birationally equivalent over F4 each other. [ABSTRACT FROM AUTHOR]

The rapid development of information technology puts forward new demands for signal sampling technology. Compressed sensing is a novel signal sampling theory that can realize signal sampling and compression simultaneously. The measurement matrix is the decisive factor affecting the reconstruction ability of compressed sensing. Therefore, the construction of measurement matrices is the key problem of compressed sensing theory. Taking full advantage of the finite field structure of the polynomial measurement matrix, a novel method for the deterministic measurement matrix based on two finite fields is proposed. In this paper, for more available choices of polynomial matrix and flexible signal length, a new expansion of polynomial sensing matrix is proposed by using two finite field. Experiments show that, the improved matrix offers more options of sensing matrix and overcome the shortcoming of fixed length measurement. Further comparisons to the block-polynomial method, show that the proposed two finite field polynomial matrix can improve the PSNR of reconstructed image by 10% ~ 15%. [ABSTRACT FROM AUTHOR]

Inspired by the Arnold chaotic map, we design a new generation formula for constructing S-boxes, which effectively integrates the characteristics and advantages of chaos map and algebraic model and lays a good foundation for producing strong S-boxes. Then, using this generation formula as the core component, we propose a scheme for constructing dynamic S-boxes. The proposed scheme is driven by a chaotic map and can efficiently generate a large number of S-boxes. Taking advantage of the ergodicity of chaotic system, our scheme can ensure that the generated S-boxes have both excellent comprehensive performance and good diversity. The experimental results in this paper also confirm that the dynamic S-boxes produced by our method not only meet the requirements of designing encryption algorithms but also provide stronger security for image encryption. Moreover, our scheme also has high efficiency in generating S-boxes. Therefore, our scheme has good application potential in cryptography, such as providing high-performance S-boxes for image security and designing encryption algorithms with dynamic S-boxes. [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]

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 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]

The classical n-variable Kloosterman sums over finite fields are well understood by Deligne’s theorem from complex point of view and by Sperber’s theorem from p-adic point of view. In this paper, we study the complex and p-adic estimates of invertedn-variable Kloosterman sums, addressing a question of Katz (Finite Fields Appl 1(3):395–398, 1995). We shall give two complex estimates. The first one is elementary based on Gauss sums. The second estimate is deeper, depending on the cohomological results of Adolphson–Sperber, Denef–Loeser and Fu for twisted toric exponential sums. This deeper result assumes that the characteristic p does not divide n + 1. Combining with Dwork’s p-adic theory, we also determine the exact p-adic valuations for zeros and poles of the L-function associated to invertedn-variable Kloosterman sums in the case p ≡ 1 mod (n + 1). As we shall see, the invertedn-variable Kloosterman sum is more complicated than the classical n-variable Kloosterman sum in all aspects in the sense that our understanding is less complete, partly because the Hodge numbers are now mostly 2 instead of 1. [ABSTRACT FROM AUTHOR]

Let Fq be the finite field of order q = pm where p is a prime, and m is a positive integer. This work introduces FqR-linear (Θ, ΔΘ)-cyclic codes where R = Fq + uFq with u² = u, i.e., (Θ, ΔΘ)-cyclic codes over FqR. With the help of the decomposition method, we study the structural properties and determine the generator polynomials of FqR-linear (Θ, ΔΘ)-cyclic codes. Further, we define the Gray map over FqR and find the Gray images of FqR-linear (Θ, ΔΘ)-cyclic codes over Fq. Finally, with the help of our established results, we have constructed some new codes corresponding to FqR-linear (Θ, ΔΘ)-cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we determine the structure of the unit groups of the semisimple group algebras of non-metabelian groups of order 144. Up to isomorphism, there are 197 non-isomorphic groups of order 144, and only 28 are non-metabelian. Mittal and Sharma [19] studied the unit groups of the semisimple group algebras of non-metabelian groups of order 144 that have exponent either 36 or 72. In this work, we characterize the unit groups of the group algebras of non-metabelian groups of order 144 having exponent 12 and 24. This paper completes the study of the unit groups of the group algebras of non-metabelian groups up to order 144. [ABSTRACT FROM AUTHOR]

Various applications use asymmetric cryptography to secure communications between both parties, and it raises the main issue of generating vast amounts of computation and storage. Thus, elliptic curve cryptography (ECC) is a methodology that emerged to overcome this issue using its low computation and generation of small keys with its strong encryption strategy. ECC is becoming mandatory and used mostly for public key encryption protocols. ECC has expanded cumulative acceptance in practice due to the reduced bit magnitude of operands compared to RSA for safety level. Previously, protocols designed for ECC suggested calculation of scalar development and it was accomplished in finite fields as projective, affine, and Jacobian simulations of coordinates. Arithmetic operations in a limited area establish the core benefits of the ECC algorithm. Even though ECC generated an issue of complex key generation using its curve formation, to overcome this issue a hyperelliptic curve cryptosystems (HECC) is proposed in this study. HECC perform ECC in the Public Key Cryptography (PKC) domain. This study presented an optimization-based key generation and made a random selection of integers for encrypting the message. Selecting a prime number as the private key and multiplying it to the encrypted message to generate a public key is done. This encrypted message is mapped to the curve to check whether it satisfies the curve equation or not. Once an encrypted message is obtained, it is then sent to a second party for pursuing the message. On the side of the second party, a reverse process called decryption takes place. Thus, a secured transmission of data communication takes place. Implementing this algorithm in MATLAB resulted in 94% accuracy and an error of 6%, which was a higher performance ratio than previous methods.

Let $ \mathbb{F}_q $ be the finite field of order $ q $ where $ q = p^{k} $, $ k $ is a positive integer and $ p $ is an odd prime. Let $ \mathbb{F}_q^* $ represent the nonzero elements of $ \mathbb{F}_{q} $. For $ f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, \cdots, x_n] $, we used $ N\big(f(x_1, \cdots, x_n) = 0\big) $ to denote the number of $ \mathbb{F}_q $-rational points of the affine hypersurface $ f(x_1, \cdots, x_n) = 0 $. In 2020, Zhao et al. obtained the explicit formulae for $ N(x_1^4+x_2^4 = c) $, $ N(x_1^4+x_2^4+x_3^4 = c) $ and $ N(x_1^4+x_2^4+x_3^4+x_4^4 = c) $ over $ \mathbb{F}_q $, with $ c\in\mathbb{F}_q^* $. In this paper, by using Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for $ N(a_1x_1^4+a_2x_2^4 = c) $ and $ N(b_1x_1^4+b_2x_2^4+b_3x_3^4 = c) $ with $ a_i, b_j\in \mathbb{F}_q^* (1\leq i \leq 2, 1 \leq j \leq 3) $ and $ c\in \mathbb{F}_q $. Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface $ a_1x_1^4+a_2x_2^4+\cdots+a_nx_n^4 = c $ of $ n\geq 4 $ variables with $ a_i\in\mathbb{F}_q^* $ $ (1\leq i\leq n) $, $ c\in\mathbb{F}_q $ and $ p\equiv1({\rm{mod}} \ 4) $, can also be deduced. These extended and improved earlier results.

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]

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]

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]

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]

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]

In this paper, using the classification of degree 7 permutations over F q , we classify certain sparse PPs of the form P (x) = x r f (x q n - 1 q - 1 ) of F q n for n = 2 and 3. In particular, we give necessary and sufficient conditions for the polynomial f a , b (x) : = x (x 2 (q 2 + q + 1) + a x q 2 + q + 1 + b) in F q 3 [ x ] to be a permutation polynomial over F q 3 , where q > 409 is a prime power. [ABSTRACT FROM AUTHOR]

In this paper, we construct three classes of permutation quadrinomials with Niho exponents of the form f (x) = α 0 x r + α 1 x s 1 (p m - 1) + r + α 2 x s 2 (p m - 1) + r + α 3 x s 3 (p m - 1) + r ∈ F p n [ x ] , where p is an odd prime, n = 2 m is a positive even integer, and (r , s 1 , s 2 , s 3) = (1 , - 1 p k - 2 , 1 , p k - 1 p k - 2) , (1 , p k + 1 p k + 2 , 1 , 1 p k + 2) and (3, 1, 2, 3), respectively. The exponents of the first two classes are considered for the first time, and the third class covers all the permutation polynomials proposed by Gupta (Designs Codes and Cryptography 88, 1–17, 2020). [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]

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]

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]

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]

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]

In this paper, we present the construction of two Hadamard matrix forms over GF(2m) to generate 4x4 and 8x8 involutory MDS (IMDS) matrices. The first form provides a straightforward way to generate 4x4 IMDS matrices, while the second is an efficient way to generate 8x8 IMDS matrices using a hybrid (combination of search-based methods and direct construction) approach. In addition, we propose an algorithm for computing the branch number of any non-singular matrix over GF(2m) and improve its computational complexity for Hadamard matrices. Using this algorithm and the proposed Hadamard matrix form, we obtain 2kx2k lightweight involutory and non-involutory Hadamard MDS matrices with low XOR counts for k=2,3. Finally, we carry out a comparative study based on the XOR count to demonstrate that MDS matrices created using our Hadamard matrix forms have lower XOR counts than MDS matrices available in the literature as of today. [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]

Copyright of Journal of South-Central Minzu University (Natural Science Edition) is the property of Journal of South-Central Minzu University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Wedderburn and Hasse theorems have always fascinated many algebraists as there are just very few versions of their proofs and that are somehow not easy to follow because they contain other results inside. In this note, we aim to look at their proofs again but nowwith more simplified versions that include sufficient details in order to let them be understandable for most readers in the subjects of finite fields and elliptic curves. In the proof of Wedderburn’s theorem, we succeed to show how it is equivalent to define either a stabilizer or centralizer of non-central elements and that helps to consider the equation on the order of group by using interchangeably the index of its subgroup and the order of the orbit. As for Hasse’s theorem, we provide a short proof in just two paragraphs with a recall of the most important results that have been used, and then, we state three of its version with an explanation of how this all leads in the end to the need of an algorithm like the one of René Schoof in 1985. [ABSTRACT FROM AUTHOR]