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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either 36 or 72 and deduce the unit group $U(\mathbb{F}_qG)$ of semisimple group algebra $\mathbb{F}_qG$. Here, $q$ denotes the power of a prime, i.e., $q=p^r$ for $p$ prime and a positive integer $r$. Up to isomorphism, there are $6$ groups of order 144 that have exponent either 36 or 72. Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order 144 that are a direct product of two nontrivial groups. In all, this paper covers the unit groups of semisimple group algebras of 17 non-metabelian groups.\looseness-1

Let $ p $ stand for an odd prime and let $ \eta\in \mathbb Z^+ $ (the set of positive integers). Let $ \mathbb F_q $ denote the finite field having $ q = p^\eta $ elements and $ \mathbb F_q^* = \mathbb F_q\setminus \{0\} $. In this paper, when the determinants of exponent matrices are coprime to $ q-1 $, we use the Smith normal form of exponent matrices to derive exact formulas for the numbers of rational points on the affine varieties over $ \mathbb F_q $ defined by $ \left\{ \begin{aligned} &a_1x_1^{d_{11}}...x_n^{d_{1n}}+... +a_sx_1^{d_{s1}}...x_n^{d_{sn}} = b_1,\\ &a_{s+1}x_1^{d_{s+1,1}}...x_n^{d_{s+1,n}}+... +a_{s+t}x_1^{d_{s+t,1}}...x_n^{d_{s+t,n}} = b_2 \end{aligned} \right. $ and $ \left\{ \begin{aligned} &c_1x_1^{e_{11}}...x_m^{e_{1m}}+... +c_rx_1^{e_{r1}}...x_m^{e_{rm}} = l_1,\\ &c_{r+1}x_1^{e_{r+1,1}}...x_m^{e_{r+1,m}}+... +c_{r+k}x_1^{e_{r+k,1}}...x_m^{e_{r+k,m}} = l_2,\\ &c_{r+k+1}x_1^{e_{r+k+1,1}}...x_m^{e_{r+k+1,m}}+... +c_{r+k+w}x_1^{e_{r+k+w,1}}...x_m^{e_{r+k+w,m}} = l_3, \end{aligned} \right. $ respectively, where $ d_{ij}, e_{i'j'}\in \mathbb Z^+, a_i, c_{i'}\in \mathbb F_q^*, i = 1, ..., s+t,$ $j = 1, ..., n, i' = 1, ..., r+k+w, j' = 1, ..., m, $ and $ b_1, b_2, l_1, l_2, l_3\in \mathbb F_q $. These formulas extend the theorems obtained by Q. Sun in 1997. Our results also give a partial answer to an open question posed by 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].

Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than n over a finite field Fq with q elements can be multiplied in time O(n logq log(n logq)), uniformly in q. Under the same hypothesis, we show how to multiply two n-bit integers in time O(n logn); this algorithm is somewhat simpler than the unconditional algorithm from the companion paper [22]. Our results hold in the Turing machine model with a finite number of tapes. [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]

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]

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]

This paper investigates conditions under which representability of each element from the field as the sum , where , , and are fixed natural numbers >1, implies a similar representability of each square matrix over the field . We propose a general approach to solving this problem. As an application we describe fields and commutative rings where 2 is a unit, over which each square matrix is the sum of two 4-potent matrices. [ABSTRACT FROM AUTHOR]

An improved lower bound is found for the bilinear complexity of the multiplication of arbitrary matrices over finite fields. [ABSTRACT FROM AUTHOR]

Let Fq be a finite field, where q is a power of a prime p such that p ≥ 5. Let α be a root of a monic polynomial of a minimal degree over Fq. In this paper, we will study elliptic curves over (Fq[α];+; *), where + is the usual addition and * represent a non-standard product law over Fq[α]. Using elliptic curves over this ring can result in a cryptographic method that is fast, simple and secure. [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]

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]

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]

Let $\mathbb F$ be a finite field of odd order and $a,b\in\mathbb F\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$ , where χ is the extended quadratic character on $\mathbb F$. Let $Q_{a,b}$ be the quasigroup over $\mathbb F$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \geqslant 0$ , and $(x,y) \mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\} = \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \operatorname{Aut}(\mathbb F)$. We also characterize $\operatorname{Aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results, we also characterize the minimal subquasigroups of $Q_{a,b}$. [ABSTRACT FROM AUTHOR]

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over F p of degree at most 4 p 3 / 4 + 1 whose points form blocking sets. [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]

One of the classical problems in the subject of group algebras is that of deducing Wedderburn decomposition of a finite semisimple group algebra. In this short note, we discuss how to check whether a matrix ring over a finite field is a Wedderburn component of the Wedderburn decompo- sition of a group algebra or not. Finally, we formulate an open problem in this direction. [ABSTRACT FROM AUTHOR]

We consider all the non-metabelian groups G of order 144 that have exponent either 36 or 72 and deduce the unit group U(픽qG) of semisimple group algebra 픽qG. Here, q denotes the power of a prime, i.e., q = pr for p prime and a positive integer r. Up to isomorphism, there are 6 groups of order 144 that have exponent either 36 or 72. Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order 144 that are a direct product of two nontrivial groups. In all, this paper covers the unit groups of semisimple group algebras of 17 non-metabelian groups. [ABSTRACT FROM AUTHOR]

Let Fq be the finite field, q = pk, with p being a prime and k being a positive integer. Let F* q be the multiplicative group of Fq, that is F* q = Fq \{0}. In this paper, by using the Jacobi sums and an analog of Hasse-Davenport theorem, an explicit formula for the number of solutions of cubic diagonal equation x³ 1 + x³ 2 + ... + x³ n = c over Fq is given, where cεF* q and p º 1(mod 3). This extends earlier results. [ABSTRACT FROM AUTHOR]

Let $ q $ be an even prime power and let $ \mathbb{F}_{q} $ be the finite field of $ q $ elements. Let $ f $ be a nonzero polynomial over $ \mathbb{F}_{q^2} $ of the form $ f = a_{1}x_{1}^{m_{1}}+\dots+a_{s}x_{s}^{m_{s}}+y_{1}y_{2}+\dots+y_{n-1}y_{n}+y_{n-2t-1}^{2}+\dots +y_{n-3}^2+y_{n-1}^{2}$ $+b_{t}y_{n-2t}^{2}+ $ $ \dots +b_{1}y_{n-2}^{2}+b_{0}y_{n}^{2} $, where $ a_{i}, b_j\in \mathbb{F}_{q^{2}}^{*}, $ $ m_i\ne 1, $ $ (m_{i}, m_{k}) = 1, $ $ i\ne k, $ $ m_{i}|(q+1), $ $ m_{i}\in \mathbb{Z}^{+}, $ $ 2|n $, $ n > 2 $, $ 0\leq t\leq \frac{n}{2}-2 $, $ {\mathrm{Tr}}_{\mathbb{F}_{q^2}/\mathbb{F}_{2}}(b_{j}) = 1 $ for $ i, k = 1, \dots, s $ and $ j = 0, 1, \dots, t $. For each $ b \in \mathbb{F}_{q^2} $, let $ N_{q^2}(f = b) $ denote the number of $ \mathbb{F}_{q^2} $-rational points on the affine hypersurface $ f = b $. In this paper, we obtain the formula of $ N_{q^2}(f = b) $ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.

Let $ a, n $ be positive integers and let $ p $ be a prime number. Let $ \mathbb F_q $ be the finite field with $ q = p^a $ elements. Let $ \{a_i\}_{i = 1}^\infty $ be an arbitrary given infinite sequence of elements in $ \mathbb F_q $ and $ a_1\neq 0 $. For each positive integer $ i $, let $ \{d_{i+j, i}\}_{j = 0}^\infty $ be an arbitrary given sequence of positive integers with $ d_{ii} $ coprime to $ q-1 $. For each integer $ n\ge 1 $, let $ N_n $, $ \bar N_n $ and $ \widetilde{N}_n $ denote the number of $ \mathbb F_q $-rational points of the hypersurfaces defined by the following three equations: $ a_1x_1+\cdots+a_nx_n = b, $ $ x_1^2+\cdots+x_n^2 = b $ and $ a_1 x_1^{d_{11}}+a_2 x_1^{d_{21}}x_2^{d_{22}}+ \cdots+a_n x_1^{d_{n1}}x_2^{d_{n2}} \cdots x_n^{d_{nn}} = b, $ respectively. In this paper, we show that the generating function $ \sum_{n = 1}^{\infty}N_nt^n $ is a rational function in $ t $. Moreover, we show that if $ p $ is an odd prime, then the generating functions $ \sum_{n = 1}^{\infty}\bar N_nt^n $ and $ \sum_{n = 1}^{\infty}\widetilde{N}_nt^n $ are both rational functions in $ t $. Moreover, we present the explicit rational expressions of $ \sum_{n = 1}^{\infty}N_nt^n $, $ \sum_{n = 1}^{\infty}\bar N_nt^n $ and $ \sum_{n = 1}^{\infty}\widetilde{N}_nt^n $, respectively.

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 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]

The nonlinearity of Boolean functions is well known and the behaviour of the nonlinearity of random Boolean functions has been studied, showing that they concentrate around one point. We show that it is the same for the autocorrelation of Boolean function. As an application, we show that the autocorrelation can distinguish a non random binary sequence from a random one. [ABSTRACT FROM AUTHOR]

For a finite field F q , where q = 2 s , s ∈ N , we present some results regarding the constructions of irreducible polynomials of degree 4n over F q by using the method of composition of polynomials from a given irreducible polynomial p(x) having degree n ≥ 1 over F q . [ABSTRACT FROM AUTHOR]

Due to its excellent diffusion avalanche performance, MDS linear transformation based on rotational-XOR shift operation is frequently employed in the construction of block cipher diffusion layer. In existing studies, the search for the best linear transformation on the small finite field (F2n)~4 is almost complete, however when the number of XOR terms is between 9 and 11, there is no best linear transformation on the huge finite field (F2n)~8. For the actual use of the 64-bit environment, it is crucial to investigate the most recent optimum linear transformation of the huge finite field (F2~8)~8. This work chooses a huge finite field (F2n)~8 as its research object based on the findings of earlier studies. The branch number test problem is converted into a Boolean Satisfiability problem (SAT) by examining the structural properties of the rotational-XOR diffusion layer. The issue is solved using the Satisfiability Modulo Theories (SMT) solver Boolector, and the final linear transformation group of the huge finite field (F2n)~8 is investigated. The experimental findings demonstrate the existence of a suboptimal linear transformation in the huge finite field (F2~8)~8 with up to 8 branches. Along with the test, a particular rotational-XOR form is provided, and a different approach is suggested for determining the true requirement for the ideal diffusion layer. Future research will focus on the theoretical properties and broadened applications of the suboptimal linear transformation of the huge finite field (F2n)~8. [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]

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]

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]

Let r ≥ 3 be a positive integer and F q the finite field with q elements. In this paper, we consider the r-regular complete permutation property of maps with the form f = τ ∘ σ M ∘ τ - 1 where τ is a PP over an extension field F q d and σ M is an invertible linear map over F q d . When τ is additive, we give a general construction of r-regular CPPs for any positive integer r. When τ is not additive, we give many examples of regular CPPs over the extension fields for r = 3 , 4 , 5 , 6 , 7 and for arbitrary odd positive integer r. These examples are the generalization of the first class of r-regular CPPs constructed by Xu et al. (Des Codes Cryptogr 90:545–575, 2022). [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]

Let q be an even prime power and let F q be the finite field of q elements. Let f be a nonzero polynomial over F q 2 of the form f = a 1 x 1 m 1 + ⋯ + a s x s m s + y 1 y 2 + ⋯ + y n − 1 y n + y n − 2 t − 1 2 + ⋯ + y n − 3 2 + y n − 1 2 + b t y n − 2 t 2 + ⋯ + b 1 y n − 2 2 + b 0 y n 2 , where a i , b j ∈ F q 2 ∗ , m i ≠ 1 , (m i , m k) = 1 , i ≠ k , m i | (q + 1) , m i ∈ Z + , 2 | n , n > 2 , 0 ≤ t ≤ n 2 − 2 , T r F q 2 / F 2 (b j) = 1 for i , k = 1 , ... , s and j = 0 , 1 , ... , t. For each b ∈ F q 2 , let N q 2 (f = b) denote the number of F q 2 -rational points on the affine hypersurface f = b. In this paper, we obtain the formula of N q 2 (f = b) by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q. In this paper, we investigate the permutation property of the trinomial x (p - 1) q + 1 + α x pq + x q + p - 1 over F q 2 , where α ∈ F q ∗ and q = p k with k being a positive integer. More precisely, we prove that for p = 7 , α ∈ F q ∗ and k > 1 , the trinomial x 6 q + 1 + α x 7 q + x q + 6 is a permutation polynomial over F q 2 if and only if α = - 1 and k = 2 . We also prove that for p > 3 , α ∈ F q ∗ and k = 1 , the trinomial x (p - 1) q + 1 + α x pq + x q + p - 1 is a permutation polynomial over F q 2 if and only if α = - 3 . [ABSTRACT FROM AUTHOR]

Several classes of complete permutation polynomials (CPPs) in the form of (xpm − x+ δ )s+ axpm + bx over Fp2m are described in this paper. For fractional exponent s, the permutation properties of the given polynomials are proved by the AGW criterion and for determining the bijectivity of the corresponding piecewise functions. For exponent ssatisfying s= i(pm − 1) + pj, the main proofs here depend on the AGW criterion and the quantity of solutions for certain linearised equations. Several results generalise some earlier results on permutation polynomials (PPs).

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.

Let a , n be positive integers and let p be a prime number. Let F q be the finite field with q = p a elements. Let { a i } i = 1 ∞ be an arbitrary given infinite sequence of elements in F q and a 1 ≠ 0. For each positive integer i , let { d i + j , i } j = 0 ∞ be an arbitrary given sequence of positive integers with d i i coprime to q − 1. For each integer n ≥ 1 , let N n , N ¯ n and N ~ n denote the number of F q -rational points of the hypersurfaces defined by the following three equations: a 1 x 1 + ⋯ + a n x n = b , x 1 2 + ⋯ + x n 2 = b and a 1 x 1 d 11 + a 2 x 1 d 21 x 2 d 22 + ⋯ + a n x 1 d n 1 x 2 d n 2 ⋯ x n d n n = b , respectively. In this paper, we show that the generating function ∑ n = 1 ∞ N n t n is a rational function in t. Moreover, we show that if p is an odd prime, then the generating functions ∑ n = 1 ∞ N ¯ n t n and ∑ n = 1 ∞ N ~ n t n are both rational functions in t. Moreover, we present the explicit rational expressions of ∑ n = 1 ∞ N n t n , ∑ n = 1 ∞ N ¯ n t n and ∑ n = 1 ∞ N ~ n t n , respectively. [ABSTRACT FROM AUTHOR]