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Let a prime $ p\equiv 1(\text{mod}3) $ and $ z $ be non-cubic in $ \mathbb{F}_p $. Gauss proved that the number of solutions of equation $ x_1^3+x_2^3+zx_3^3 = 0 $ in $ \mathbb{F}_p $ was $ p^2+\frac{1}{2}(p-1)(9d-c) $, where $ c $ was uniquely determined and $ d $, except for the sign, was defined by$ 4p = c^2+27d^2,\ \ c\equiv 1(\text{mod}3). $In 1978, Chowla, Cowles, and Cowles determined the sign of $ d $ for the case of 2 being non-cubic in $ \mathbb{F}_p $. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field $ \mathbb{F}_q $ with $ q = p^k $, $ p\equiv 1(\text{mod}3) $, and determined the sign of $ d $ for the case of 3 being non-cubic.

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.

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.

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.

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].

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.

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.

Abstract We provide the structure of the unit group of $${\mathbb {F}}_{p^k}S_n$$ F p k S n , where $$p>n$$ p > n is a prime and $$S_n$$ S n denotes the symmetric group on n letters. We also provide the complete characterization of the unit group of the group algebra $${\mathbb {F}}_{p^k}A_6$$ F p k A 6 for $$p\ge 7$$ p ≥ 7 , where $$A_6$$ A 6 is the alternating group on 6 letters.

A linear rank inequality is a linear inequality that holds by dimensions of vector spaces over any finite field. A characteristic-dependent linear rank inequality is also a linear inequality that involves dimensions of vector spaces but this holds over finite fields of determined characteristics, and does not in general hold over other characteristics. In this paper, using as guide binary matrices whose ranks depend on the finite field where they are defined, we show a theorem which explicitly produces characteristic-dependent linear rank inequalities; this theorem generalizes results previously obtained in the literature.

We investigate the lattice $ L( V)$ of subspaces of an $m$-dimensional vector space $ V$ over a finite field ${\rm GF}(q)$ with a prime power $q=p^n$ together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice $ L( V)$ satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when $ L( V)$ is orthomodular. For $m>1$ and $p\nmid m$ we show that $ L( V)$ contains a $(2^m+2)$-element (non-Boolean) orthomodular lattice as a subposet. Finally, for $q$ being a prime and $m=2$ we characterize orthomodularity of $ L( V)$ by a simple condition.

Let $F_q$ be a field with $q$ elements, where $q$ is a power of a prime number $p\geq 5$. For any integer $m\geq 2$ and $a\in F_q^*$ such that the polynomial $x^m-a$ is irreducible in $F_q[x]$, we combine two different methods to explicitly construct elements of high order in the field $F_q[x]/\langle x^m-a\rangle$. Namely, we find elements with multiplicative order of at least $5^{\sqrt[3]{m/2}}$, which is better than previously obtained bound for such family of extension fields.

We give the characterization of the unit group of $\mathbb{F}_qSL(2, \mathbb{Z}_5)$, where $\mathbb{F}_q$ is a finite field with $q = p^k$ elements for prime $p > 5,$ and $SL(2, \mathbb{Z}_5)$ denotes the special linear group of $2 \times2$ matrices having determinant $1$ over the cyclic group $\mathbb{Z}_5$.

Let $q$ be an odd prime power. We discuss possible definitions over $\FF_{q^2}$ (using the Hermitian form) of circles, unit segments and half-lines. If we use our unit segments to define the convex hulls of a set $S\subset \FF_{q^2}^n$ for $q\notin \{3,5,9\}$ we just get the $\FF_q$-affine span of $S$.

Encryption schemes like AES require finite field modular multiplication. The encryption speed is highly dependent on the performance of the finite field multiplier. Several high-speed systolic bit parallel multipliers with low area complexity have been proposed in the literature. In this paper, a modular multiplication algorithm is used to propose a bit parallel polynomial basis systolic multiplier which has achieved 89% less and 17% less area-delay product than the best existing multipliers. It has been observed that the area complexity of the proposed systolic multiplier for irreducible polynomials matches with the best-reported multiplier with 17% less time complexity. The results are further verified with the help of the FPGA implementation of the proposed multiplier for m = 8,163. Being generic, the proposed multiplier can be optimized further for trinomials and pentanomials.

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 p2 and be one of the rings Zp,Zp2,GF(p2),Zp(+)Zp or Zp×Zp. We determine the geodetic number g(G(R)) associated with each such ring.

We characterize the unit group of semisimple group algebras $\mathbb{F}_qG$ of some non-metabelian groups, where $F_q$ is a field with $q=p^k$ elements for $p$ prime and a positive integer $k$. In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_3\times C_3)\rtimes C_3)\rtimes C_2$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix B ∈ 𝔽n×n is defined as ℓ1ℓ2· · · ℓn, where ℓj ∈ {A, S, N} according to whether all, some but not all, or none of the principal minors of order j of B are nonzero. Building upon the second author’s recent classification of the epr-sequences of symmetric matrices over the field 𝔽 = 𝔽2, we initiate a study of the case 𝔽= 𝔽3. Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.

Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ].

There are many recent results on reverse-engineering (potentially hidden) structure in cryptographic S-boxes. The problem of recovering structure in the other main building block of symmetric cryptographic primitives, namely, the linear layer, has not been paid that much attention so far. To fill this gap, in this work, we develop a systematic approach to decomposing structure in the linear layer of a substitutionpermutation network (SPN), covering the case in which the specification of the linear layer is obfuscated by applying secret linear transformations to the S-boxes. We first present algorithms to decide whether an ms x ms matrix with entries in a prime field Fp can be represented as an m x m matrix over the extension field Fps . We then study the case of recovering structure in MDS matrices by investigating whether a given MDS matrix follows a Cauchy construction. As an application, for the first time, we show that the 8 x 8 MDS matrix over F28 used in the hash function Streebog is a Cauchy matrix.

In this paper, by employing the AGW criterion and determining the number of solutions to some equations over finite fields, we further investigate nine classes of permutation polynomials over $ \mathbb{F}_{p^n} $ with the form $ (x^{p^m}-x+\delta)^{s_1}+(x^{p^m}-x+\delta)^{s_2}+x $ and propose five classes of complete permutation polynomials over $ \mathbb{F}_{p^{2m}} $ with the form $ ax^{p^m}+bx+h(x^{p^m}-x) $.

Let $ F $ be a finite field of characteristic $ p $ having $ q = p^n $ elements and $ G $ be an abelian group. In this paper, we determine the structure of the group of units of the group algebra $ FG $, where $ G $ is an abelian group of order $ 17\leq |G|\leq 20 $.

The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.

Similarities between biological and digital communication systems have been investigated since biology also uses a discrete alphabet to represent and transmit information. The genetic information of an organism is encoded in DNA molecules by units called bases. However, there is no a definitive model and the question as what error-correcting code underlies DNA sequences remains an open problem. Recent works show that DNA sequences can be identified as codewords in a class of cyclic error-correcting codes known as BCH codes. We propose improvements regarding the code construction process that resulted in a novel algorithm for searching BCH codes whose codeword differ from a given DNA sequence (mapped to finite field $\mathbb {F}_{4}$ ) in up to only one symbol. The most important improvement is to replace brute force decoding with syndrome decoding. In this sense, based on a statistical analysis, we verify whether in a collection of sequences with the same taxonomic rank there is a code that identifies most of these sequences, called dominant code. Furthermore, we check whether the dominant code can provides a biological information to DNA classification being an alignment-free method. Finally, we show that the probability of a DNA sequences with odd-length $n$ be identified by a BCH code tends to analytical probability of the same code identifying a random vector.

In 2006, Hubert, Mauduit and Sárközy extended the notion of binary sequences to n-dimensional binary lattices and introduced the measures of pseudorandomness of binary lattices. In 2011, Gyarmati, Mauduit and Sárközy extended the notions of family complexity, collision and avalanche effect from binary sequences to binary lattices. In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in finite fields and study the pseudorandom measure of order k, family complexity, collision and avalanche effect. Results indicate that such binary lattices are “good,” and their families possess a nice structure in terms of family complexity, collision and avalanche effect.

An approach enabling partial generalization of the Green–Sasao hierarchy for polynomial forms of Boolean functions to the case of an arbitrary finite field was introduced. The exact value of the Shannon function was obtained for the class of pseudo-Kroneker and free-Kronecker forms of n-ary functions over an arbitrary finite field Fq. The value found is equal to qn−1. The previously known result for Boolean functions was generalized.

Let F(2m) will be an arbitrary finite field of characteristic 2. It’s square extension will be considered as an algebra with basiс elements 1 and e over the field F(2m). Here 1 is considered as the unit element of the algebra, and e satisfies the relation: e2 = e + α. An element α maybe arbitrary from the field F(2m), but it is not satisfying to the condition α = x + x2 for some x element from F(2m). Let us n0 (α) denote the order of basis element e. Then the main result of the paper can be formulated as: The irreducible polynomial 1 + t + αt2 divides the polynomial 1 + tn if and only the order if n0 (α) divides a natural n. The similar results for arbitrary elements of field F(2m) follow from main theorem. The proof of main result based on the properties of the recurrence relations between the polynomials Pn (α) and Qn (α), definite for all n = 0, 1, 2, … by the relations en = Pn (α) + Qn (α) e. The formulas for the generating series of these polynomials contain the most important such properties. The formulas were obtained.

Let $\mathbb{F}_q$ be the finite field of order $q$ and $f(x)$ be an irreducible polynomial of degree $n$ over $\mathbb{F} _q$. For a positive divisor $n_1$ of $n$, define the $n_1$-traces of $f(x)$ to be $\mathrm{Tr}(\alpha;n_1)=\alpha+\alpha^q+\cdots+\alpha^{q^{n_1-1}}$ where $\alpha$'s are the roots of $f(x)$. Let $N_q^*(n;n_1)$ denote the number of monic irreducible polynomials of degree $n$ over $\mathbb{F} _q$ with nozero $n_1$-traces. Ruskey, Miers and Sawada have found the formula for $N_q^*(n;n)$. Based on the properties of linearized polynomials, we obtain the formula for $N_q^*(n;n_1)$ in the general case, including a new proof to the result by Ruskey, Miers and Sawada.

Let $p$ be a odd prime number and let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements. In this paper, by using the Gauss sum and Jacobi sum, we give an explicit formula for the number $N(x_1^4+x_2^4=c)$ of solutions of the following two-variable diagonal quartic equations over $\mathbb{F}_q$: $x_1^4+x_2^4=c$ with $c\in\mathbb{F}_q^*$. From this result, one can deduce that $N(x_1^4+x_2^4=c)=q+O(q^{\frac{1}{2}}).$

Let $p$ be an odd prime and let $\mathbb{F}_q$ be a finite field of characteristic $p$ with order $q=p^s$. For $f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, ..., x_n]$, we denote by $N(f(x_1, \cdots, x_n)=0)$ the number of $\mathbb{F}_q$-rational points on the affine hypersurface $f(x_1, \cdots, x_n)=0$. In 1981, Myerson gave a formula for $N(x_1^4+\cdots+x_n^4=0)$. Recently, Zhao and Zhao obtained an explicit formula for $N(x_1^4+x_2^4=c)$ with $c\in\mathbb{F}_q^*:=\mathbb{F}_q\setminus \{0\}$. In this paper, by using the Gauss sum and Jacobi sum, we arrive at explicit formulas for $N(x_1^4+x_2^4+x_3^4=c)$ and $N(x_1^4+x_2^4+x_3^4+x_4^4=c)$ with $c\in\mathbb{F}_q^*$.

Codebooks with low coherence have extensive applications in many fileds such as code division multiple access (CDMA) communication systems, MIMO communications, compressed sensing and so on. In this paper, based on additive characters over finite fields, we propose a construction of optimal codebook with respect to the Levenshtein bound and verify that it is a new construction. By shortening the length of the optimal codebooks, we present a construction of codebooks asymptotically meeting the Levenshtein bound. To the best of our knowledge, the parameters of the asymptotically optimal codebooks are new.

Substitution box (S-Box) has a prominent significance being the fundamental nonlinear component of block cipher which fulfils confusion, one of the properties proposed by Claude Shannon in 1949. In this paper, we proposed an S-Box by using the action of modular group PSL $\left ({2,\mathbb {Z} }\right)$ on projective line PL $\left ({F_{257} }\right)$ over Galois field GF $\left ({2^{8} }\right)$ . In the first step we obtained elements of GF $\left ({2^{8} }\right)$ by using powers of $\alpha $ , where $\alpha $ is the primitive root of irreducible polynomial $p\left ({x }\right)$ of order 8 over field $\mathbb {Z}_{2}$ , then applied the generators of PSL $\left ({2,\mathbb {Z} }\right)$ and followed steps to get rid of infinity from output. In the final step of proposed scheme, one of the permutations of $S_{16}$ is applied which enhanced the possible number of S-Boxes obtained by any single specific irreducible polynomial $p(x)$ over field $\mathbb {Z}_{2}$ of order 8. We analyzed performance of the proposed $8\times 8$ S-Box under cryptographic properties such as strict avalanche criterion, bit independence criterion, nonlinearity, differential approximation probability, linear approximation probability; and compared obtained results with a number of renowned S-Boxes. Lastly, we performed statistical analysis (which comprises of contrast analysis, homogeneity analysis, energy analysis, correlation analysis, entropy analysis and mean of absolute deviation analysis) on our proposed S-Box and obtained results have been compared with adequate number of S-Boxes.

Owing to mathematical theory and computational power evolution, modern cryptosystems demand ingenious trapdoor functions as their foundation to extend the gap between an enthusiastic interceptor and sensitive information. This paper introduces an adaptive block encryption scheme. This system is based on product, exponent, and modulo operation on a finite field. At the heart of this algorithm lies an innovative and robust trapdoor function that operates in the Galois Field and is responsible for the superior speed and security offered by it. Prime number theorem plays a fundamental role in this system, to keep unwelcome adversaries at bay. This is a self-adjusting cryptosystem that autonomously optimizes the system parameters thereby reducing effort on the user’s side while enhancing the level of security. This paper provides an extensive analysis of a few notable attributes of this cryptosystem such as its exponential rise in security with an increase in the length of plaintext while simultaneously ensuring that the operations are carried out in feasible runtime. Additionally, an experimental analysis is also performed to study the trends and relations between the cryptosystem parameters, including a few edge cases.

We consider vector mappings over the set of 0 and 1 given by the set of Boolean functions. Boolean functions included in the map are given in ANF. Having fixed the rule according to which the binary vectors are associated with the elements of a finite field of characteristic two, we obtain a one-to-one correspondence between the mappings of the vector space into itself and the functions over the finite field. The finite field is considered as a ring of polynomials with the operations of addition and multiplication modulo the selected irreducible polynomial. It is known that any function over a finite field can be written as a polynomial. Moreover, for various irreducible polynomials, the polynomials over a finite field, corresponding to a given vector map, may in general be different and have different properties. We consider the problem of finding an irreducible polynomial such that the degree of a polynomial over a finite field is minimal, provided that the correspondence between binary vectors and elements of a finite field is given by using a polynomial basis of a finite field. We present an algorithm to solve this problem. Firstly we rewrite Boolean functions in trace form and calculate expressions for finite field polynomial coefficients. Then we calculate elements of dual basis as an expression of polynomial basis. Using them we obtain system pf Boolean equations. Finally we solve system of Boolean equations by SAT-solver and filter reducible polynomials. We estimate of the complexity of obtaining the specified system of Boolean equations is bitwise operations.

Inspired by the Montgomery and generalized polynomial basis (GPB) squaring operation, we introduce and study the notion of asymptotic square root over binary extension fields GF(2m). This new notion is a natural generalization of Montgomery-like square root, a special form of square root that was recently introduced by Li et al. Given an arbitrary element A ∈ GF(2m) and a fixed element w, the asymptotic square root is defined as A1/2 · w. We show that, by choosing a proper parameter w regarding different kind of irreducible polynomials that define GF(2m), such square root operation can achieve better space and time complexity. Meanwhile, we have proved that the complexity of asymptotic is linear with the generating polynomials. Specifically, for the field GF(2m) is defined by two type of specific irreducible pentanomials, i.e., Type C.1 and Type C.2 pentanomials, we derived explicit formulae for space and time complexities associated with the asymptotic square root operator. On top of that, a practical application of the asymptotic square root in exponentiation computation is also presented.

Secret sharing schemes are used to distribute a secret value among a group of users so that only authorized set of them can reconstruct the original secret correctly. The modular secret sharing scheme (MSSS) we are studying is based on the Chinese Remainder Theorem. In this scheme the secrets s(x), S(x), s1(x),…, sk(x) are defined as follows s(x) = S(x) = mod m(x), si(x) = S(x) mod mi(x), i = 1, 2, …, k. All the secrets and moduli are chosen from polynomial ring Fp[x], and the reconstruction of secret s(x) is carried out by applying the above-mentioned Chinese Remainder Theorem. The verification of any secret sharing scheme is understood as the protocol of verification by the participants of their partial secrets and (or) the protocol for verifying the legitimacy of the actions of the dealer. In this paper, we introduce a perfect verification protocol of MSSS. It means that none information leaks under distribution and verification. Two verification protocols are introduced in this paper. The first one is simpler and it depends on assumption about dealer honesty. If there is no such assumption verification is more complex. Both protocols are based on one work by J. Benalo and generalize the protocol proposed earlier by M. Vaskovsky and G. Matveev in two ways. First, the general, not only the threshold access structure is verified, and secondly, the dealer is not necessarily honest. Earlier, N. Shenets found the perfection condition of MSSS. Thus, if these conditions аre met, both the MSSS and its verification protocol are perfect.

In the present paper new scheme of secret verification are constructed. Verification with trusted party participation is conducted with help of an external device, which takes an arbitrary polynomial S(x), input element x0 ∈ Fpn and returns a value ξS(x0), where ξ is an Fpn – valued uniformly distributed random variable. It is shown that using of such device allows any user to verify his secret. Polynomial verification scheme is based on verification of divisibility g(x)|f(x) in the ring Z[x]. Only a value of polynomial S(x) in unknown point x = l is disclosed at the proposed verification method. Benaloh’s verification of the modular scheme allows any shareholder to ensure in consistency of all partial secrets, i. e. any legal group of shareholders can restore the secret S(x) correctly. None information about the secret S(x), excepting a prior information, is disclosed. The proposed protocols can be used safely for schemes over arbitrary finite fields without additional restrictions on a size of a filed.

One of the directions of the investigation of functions over finite fields is the study of their representations, including polynomial ones. In the area of polynomial representations of functions the problem of estimating the complexity of such representations can be highlighted. The complexity of the polynomial, representing the function, is the number of its nonzero terms. Each function can be represented by several different polynomials from the same class. The complexity of a function in the class of polynomials is the least possible complexity of a polynomial from this class, representing the function. The complexity of the given set of functions in the class of polynomials is the maximal complexity of a function from the set in this class of polynomials. In the case of functions over a finite field of order 2 (Boolean functions), exact values of the complexity of such representations are known for many classes of polynomial forms. But for functions over finite fields of order greater than two, even in fairly simple classes of polynomials, only mismatched upper and lower bounds of complexity have been found. This paper is devoted to the study of the representation of seven-valued functions by polarized polynomials. The polynomials of this class have the form of a sum of a finite number of products of a certain type. For the case of Boolean and three-valued functions, effective lower bounds for the complexity in the class of polarized polynomials are known, as well as a weaker power estimate for functions over a finite field of prime order. In previous papers, the authors obtained effective lower bounds for the complexity of functions over finite fields of order 4 and 5 in the class of polarized polynomials. In this paper an effective lower bound for the complexity of seven-valued functions in the class of polarized polynomials has been obtained.

Linear complexity is an important criterion to characterize the unpredictability of pseudo-random sequences, and large linear complexity corresponds to high cryptographic strength. Pseudo-random Sequences with a large linear complexity property are of importance in many domains. In this paper, based on the theory of inverse Gray mapping, two classes of new generalized cyclotomic quaternary sequences with period pq are constructed, where pq is a product of two large distinct primes. In addition, we give the linear complexity over the residue class ring Z4 via the Hamming weights of their Fourier spectral sequence. The results show that these two kinds of sequences have large linear complexity.

Let Nq denote the number of solutions of the generalized Markoff-Hurwitz-type equations over the finite field , where n≥2, mi, k, kj and t≥n are positive integers, ai, , for 1≤i≤n and 1≤j≤t. Recently, some researches considered the above equation with k = k1 = … = kt = 1 and obtained some generalizations of Carlitz's results. In this paper, we determine Nq explicitly in some other cases. This extends the previous conclusions.

Multiplication in finite fields is used in many applications, especially in cryptography. It is a basic and the most computationally intensive operation from among all such operations. Several systolic multipliers are proposed in the literature that offer low hardware complexity or high speed. In this paper, a bit‐parallel polynomial basis systolic multiplier for generic irreducible polynomials is proposed based on a modified interleaved multiplication method. The hardware complexity and delay of the proposed multiplier are estimated, and a comparison with the corresponding multipliers available in the literature is presented. Of the corresponding multipliers, the proposed multiplier achieves a reduction in the hardware complexity of up to 20% when compared to the best multiplier for m = 163. The synthesis results of application‐specific integrated circuit and field‐programmable gate array implementations of the proposed multiplier are also presented. From the synthesis results, it is inferred that the proposed multiplier achieves low power consumption and low area complexitywhen compared to the best of the corresponding multipliers.

In this paper, we develop the method presented previouslyby Hong, Qin and Zhao to obtain several results on the permutationalbehavior of the reversed Dickson polynomial $D_{n,k}(1,x)$of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$.Particularly, we present the explicit evaluation of thefirst moment $\sum_{a\in {\mathbb F}_{q}}D_{n,k}(1,a)$.Our results extend the results of Hong, Qin and Zhaoto the general $k\ge 0$ case.

Substitution-box (s-box) is a basic component of block cipher which performs a substitution. Two powerful cryptanalysis techniques applied to block ciphers are linear cryptanalysis and differential cryptanalysis. The resistance against differential cryptanalysis can be achieved by eliminating high-probability differential trails. We should choose an s-box where the maximum difference propagation probability is as small as possible to eliminating high-probability differential trails. Nyberg proposed a method to construct the s-box by using the inverse mapping on a finite field then implements affine transformations on . In this study, we generate 47.104 s-box according to Nyberg. The experimental results showed that s-boxes have the maximum difference propagation probability with the same frequency.

Let Fq be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a first time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we define the elliptic curve Ea,b(Fq[e]) and we will show that Eπ0(a),π0(b)(Fq) and Eπ1(a),π1(b)(Fq) are two elliptic curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and Eπ0(a),π0(b)(Fq)×Eπ1(a),π1(b)(Fq).

Recently, the interest to polynomial representations of functions over finite fields and over finite rings is being increased. Complexity of those representations is widely studied. This paper introduces new upper bounds on complexity of discrete functions over particular finite fields in class of polarized polynomials. The results are state in the terms of matrix forms. A matrix form is representation of functions vector of values as a product of nonsingular matrix and a vector of coefficients. The complexity of matrix form of a special kind is equal to complexity of polarized polynomial for same function. A complexity of a matrix form is a number of nonzero coefficients in its vector. Every function can be represented by variety of matrix forms of the same class. A complexity of a function in a class of matrix forms is the minimal complexity of forms in the class representing this function. This paper introduces new upper bounds on complexity of functions in class of polarized polynomials over fields of orders $2^k$ and $p^k$, $p$ is prime and $p \geqslant 3$.

The representations, including polynomial, of functions over final fields have been actively investigated. The complexity of such representations is the main stream of research. Polynomial representations of Boolean functions have been studied well enough. The exact values of the complexity have been found for a lot of polynomial classes. Recently, the interest to polynomial representations of functions over finite fields and over finite rings is being increased. There are a lot of difficulties in studying of the complexity of these representations. Only not equal upper and lower bounds has been obtained, even for sagnificantly simple classes of polynomials. This paper is about polarized polynomials over finite field of order 4. Such a polynomial is a finite sum of products. Every polynomial represents an n-variable function over finite field. A complexity of a polynomial is a number of nonzero summands in it. Every function can be represented by several polynomials, which are belongs to the same class. A complexity of a function in a class of polynomials is the minimal complexity of polynomials in the class, which represent this function. Previously, the constructive lower bounds in the class of polarized polynomials have been known only for the case of Boolean and three-valued functions. Also, the weaker, non-constructive lower bound has been known for the case of functions over arbitrary prime finite field. In this paper the constructive lower bound has been obtained for functions over finite field of order 4 in the class of polarized polynomials. The lower bound is equivalent to previously known lower bound for Boolean and three-valued functions.

Polynomial representations of Boolean functions have been studied well enough. Recently, the interest to polynomial representations of functions over finite fields and over finite rings is being increased. There are a lot of difficulties in studying of the complexity of these representations. Only not equal upper and lower bounds has been obtained, even for sagnificantly simple classes of polynomials. This paper is about polarized polynomials over finite fields and their generalizations: differentially and generically polarized polynomials. Such a polynomial is a finite sum of products. The difference between classes of polynomials is in constraints, applied to the products. Every polynomial represents an $n$-variable function over finite field. A complexity of a polynomial is a number of nonzero summands in it. Every function can be represented by several polynomials, which are belongs to the same class. A complexity of a function in a class of polynomials is the minimal complexity of polynomials in the class, which represent this function. Previously, the upper bounds were known for arbitrary $n$-variable function over primary finite field of order, greater than 2, for the classes of polarized and differentially polarized polynomials, as well as for the class of generically polarized polynomials. A representation of an $n$-ary function over finite field of order $q$ by a polarized polynomial or its generalization can be considered as a Kroneker form. This means, that the function, considered as a vector in appropriate linear space, is computed by a linear transformation of a vector of coefficients of the polynomial, where the matrix of this linear transformation is a Kroneker product of $n$ nonsingular matrices with rank $q$. This approach helped us to improve the upper bound for polarized and differentially polarized polynomials for the case of any finite field of odd order, and to improve the upper bound for generically polarized polynomials for the case of any finite field of order, greater than 2.