Narrow Results

In this paper, by using the Weil sums, we derive that the $c$-differential uniformity of permutation polynomial of the form $F(x)=x+\mathrm{\text{Tr}}(H(x)+H(x+1))\in {𝔽}_{2^n}[x]$ is equal to $2$ for $H(x)=x^{2^i+2^j+1}$ with $1\le i\ne j\le n-1$ and $gcd(n,i)=gcd(n,j)=gcd(n,i-j)=1$, or $H(x)={\sum }_{i,j=1}^{n-1}{x}^{2^i+2^j+1}$ with $1\le i<j\le n-1$, where $n=2k+1$. In particular, several explicit classes of APcN permutations over ${𝔽}_{2^n}$ are presented using different choices of the pair $(i,j)$ for the monomial $H(x)=x^{2^i+2^j+1}$.

Given a hypergraph ${\mathscr{H}}$, we introduce a new class of evaluation toric codes called edge codes derived from ${\mathscr{H}}$. We analyze these codes, focusing on determining their basic parameters. We provide estimations for the minimum distance, particularly in scenarios involving $d$-uniform clutters. Additionally, we demonstrate that these codes exhibit self-orthogonality. Furthermore, we compute the minimum distances of edge codes for all graphs with five vertices.

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 ${𝔽}_{2^n}$, we determine the boomerang uniformity of these permutation polynomials: (1) $f_1(x)={(x^{2^m}+x+\delta )}^{2^{2m}+1}+x$, where $n=3m$, $\delta \in {𝔽}_{2^n}$ with ${\mathrm{\text{Tr}}}_m^n(\delta )=1$; (2) $f_2(x)={(x^{2^m}+x+\delta )}^{2^{2m-1}+2^{m-1}}+x$, where $n=3m$, $\delta \in {𝔽}_{2^n}$ with ${\mathrm{\text{Tr}}}_m^n(\delta )=0$; (3) $f_3(x)={(x^{2^m}+x+\delta )}^{2^{3m-1}+2^{m-1}}+x$, where $n=3m$, $\delta \in {𝔽}_{2^n}$ with ${\mathrm{\text{Tr}}}_m^n(\delta )=0$. The results show that the boomerang uniformity of $f_1(x)$, $f_2(x)$ and $f_3(x)$ can attain $2^n$.

In this paper, we propose a new class of error-detecting codes based on quasigroups using automorphisms of the finite field ${𝔽}_{p^n}$ and the additive group $({𝔽}_p,+)$. We demonstrate that these codes are effective in detecting burst errors caused by hardware damage or noisy transmission. We also explore the codes ability to detect various types of double errors, including transposition errors, twin errors, and phonetic errors. Our analysis extends beyond bit-level errors to include block-level errors. Finally, we provide a comparative analysis and demonstrate the real-world applications of our code.

Let $k\ge 2$, $q$ be an odd prime power, and $F\in {𝔽}_q[x_1,\dots ,x_k]$ be a polynomial. An $F$-Diophantine set over a finite field ${𝔽}_q$ is a set $A\subset {𝔽}_q^{\ast }$ such that $F(a_1,a_2,\dots ,a_k)$ is a square in ${𝔽}_q$ whenever $a_1,a_2,\dots ,a_k$ are distinct elements in $A$. In this paper, we provide a strategy to construct a large $F$-Diophantine set, provided that $F$ has a nice property in terms of its monomial expansion. In particular, when $F=x_1{x}_2\dots x_k+1$, our construction gives a $k$-Diophantine tuple over ${𝔽}_q$ with size ${\gg }_{k}logq$, significantly improving the $\mathrm{\Theta }({(logq)}^{1/(k-1)})$ lower bound in a recent paper by Hammonds–Kim–Miller–Nigam–Onghai–Saikia–Sharma.

A well-designed substitution-permutation network (SPN) relies on several interspersed rounds of S-boxes and P-boxes, fulfilling the properties of Shannon’s confusion and diffusion. In any SPN, the role of the confusion constituent, S-box, is essentially decisive as one-bit change in the key leads to the change in several of the round keys. Consequentially, every round key spreads out over all the bits, changing the ciphertext in an absolutely complex manner. The unpredictable advancement of secured cryptographic systems escorts an indispensable demand to develop new S-box generators, offering optimized security features on a reduced computational cost. It has been established that the use of elliptic curves in cryptography offers elevated security with a reduced key size. In the proposed work, we deploy high resistance of elliptic curves’ point addition system against modern cryptanalysis, to evolve a substitution algorithm. Precisely, we use the complex algebraic structure of an elliptic curve group over a finite field. Our novel approach reduces the computational labor just by utilizing the nonlinear point-addition mechanism to design a substitution algorithm. The idea is supported through a step by step illustration of problem formulation. The outcomes of the proposed algorithm are judged through all the fundamental evaluation parameters and a comprehensive analysis including comparison with some recent techniques is also presented that proves the effectiveness of our model.

We present a method for constructing almost perfect nonlinear (APN) functions in odd characteristic by modifying some values of known perfect nonlinear functions. As a consequence, new APN polynomial functions such as ones over 𝔽₁₃ which are CCZ-inequivalent to known APN functions are obtained.

For cryptographic systems the method of confusion and diffusion is used as a fundamental technique to achieve security. Confusion is reflected in nonlinearity of certain Boolean functions describing the cryptographic transformation. In this paper, we present two Boolean functions which have low Walsh spectra and high nonlinearity. In the proof of the nonlinearity, a new method for evaluating some exponential sums over finite fields is provided.

A method for constructing piecewise differentially 4-uniform permutations has been recently introduced by Zha, Hu and Sun. By using this method, we provide two new infinite families of differentially 4-uniform permutations from the known APN functions. The CCZ inequivalence between these constructed functions and the known differentially 4-uniform permutations are also investigated by computation.

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length $2\nu +\delta$, size $M$, minimum subspace distance $2j$ based on $m$-dimensional totally singular subspace in the $(2\nu +\delta )$-dimensional orthogonal space ${{𝔽}_q}^{(2\nu +\delta )}$ over finite fields ${𝔽}_q$ of characteristic 2, denoted by ${(2\nu +\delta ,M,2j,m)}_q$, are presented, where $\nu$ is a positive integer, $\delta =0,1,2$, $0\le m\le \nu$, $0\le j\le m$. Then, we prove that ${(2\nu +\delta ,M,2j,m)}_q$ codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in ${ℳ}(m,0,0;2\nu +\delta )$, where ${ℳ}(m,0,0;2\nu +\delta )$ denotes the collection of all the $m$-dimensional totally singular subspaces in the $(2\nu +\delta )$-dimensional orthogonal space ${{𝔽}_q}^{(2\nu +\delta )}$ over ${𝔽}_q$ of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code ${(2\nu +\delta ,M,d)}_q$ in ${ℳ}(2\nu +\delta )$ are provided, where ${ℳ}(2\nu +\delta )$ denotes the collection of all the totally singular subspaces in the $(2\nu +\delta )$-dimensional orthogonal space ${{𝔽}_q}^{(2\nu +\delta )}$ over ${𝔽}_q$ of characteristic 2.

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In particular, on finite fields of even characteristic, GAPN functions coincide with APN functions. In this paper, we study monomial GAPN functions for odd characteristic. We give monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic. Moreover, we define a generalization of exceptional APN functions and give typical examples.

Boolean functions are fundamental bricks in the development of various applications in Cryptography and Coding theory by making benefit from the weights of related Boolean functions (Walsh spectrum). Towards this, the discrete Fourier transform (Walsh–Hadamard) plays a pivotal tool. The work in this paper is dedicated towards the algebraic and numerical degrees, together with the relationship between weights of Boolean function and their Walsh transforms. We introduce Walsh matrices and generalize them to any arbitrary Boolean function. This improves the complexity in computation of Walsh–Hadamard and Fourier transform in certain cases. We also discuss some useful results related to the degree of the algebraic normal form using Walsh–Hadamard transform.

Let 𝖥_r be a free group of rank r, 𝔽_q a finite field of order q, and let SL_n(𝔽_q) act on Hom(𝖥_r, SL_n(𝔽_q)) by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits Hom(𝖥_r, SL_n(𝔽_q))/SL_n(𝔽_q). Our first main theorem is the implementation of this algorithm in the case n = 2. As an application, we determine the E-polynomial of the character variety Hom(𝖥_r, SL₂(ℂ))//SL₂(ℂ), and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

Let $q$ be an even prime power and $m\ge 2$ an integer. By ${𝔽}_q$, we denote the finite field of order $q$ and by ${𝔽}_{q^m}$ its extension of degree $m$. In this paper, we investigate the existence of a primitive normal pair $(\alpha ,f(\alpha ))$, with $f(x)=\frac{a{x}^2+bx+c}{dx+e}\in {𝔽}_{q^m}(x)$ where the rank of the matrix $F=\left(\right.\begin{array}{ccc}\hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill 0\hfill & \hfill d\hfill & \hfill e\hfill \end{array}\left)\right.\in M_{2\times 3}({𝔽}_{q^m})$ is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for $\left(\right.\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\left)\right.$ if $q=2$ and $m$ is odd, and then we provide an explicit small list of possible and genuine exceptional pairs $(q,m)$.

The algebra of ${\mathrm{\text{GL}}}_n$-invariants of m-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants are not known even for a pair of $2\times 2$ matrices. Working over an arbitrary field we classified all ${\mathrm{\text{GL}}}_2$-orbits on m-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\mathrm{\text{GL}}}_2$-invariant polynomial functions of m-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field.

We revisit the problem of mutually unbiased measurements in the context of estimating the unknown state of a d-level quantum system, first studied (in the framework of quantum information) by Wootters and fields⁷ in 1989 and later investigated by Bandyopadhyay et al.³ in 2001 and Pittenger and Rubin⁶ in 2003. Our approach is based directly on the Weyl operators in the L²-space over a finite field when d=p^r is the power of a prime. When d is not a prime power we sacrifice a bit of optimality and construct a recovery operator for reconstructing the unknown state from the probabilities of elementary events in different measurements.

Let f be a multivariate polynomial over a finite field and its degree matrix be composed of the degree vectors appearing in f. In this paper, we provide an elementary approach to estimating the exponential sums of the polynomials with positive square degree matrices in terms of the elementary divisors of the degree matrices.

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in where n corresponds to the extension degree, and is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field 𝔽_3⁵⁷.

Several classes of permutation polynomials with given form over ${𝔽}_{p^{2m}}$ were recently proposed by Tu, Zeng, Li and Helleseth. In this paper, continuing their work, we present more permutation polynomials of the form ${(x^{p^m}-x+\delta )}^s+L(x)$ over the finite field ${𝔽}_{p^{2m}},$ where $L(x)$ is a linearized polynomial with coefficients in ${𝔽}_p$.