Finite Field Research Articles

Some classes of functions with low c-differential uniformity over finite fields

Some classes of functions with low <i>c</i>-differential uniformity over finite fields

Twisted Hessian curve over a local ring

In this paper, we study the twisted Hessian curve denoted $H^{n}_{a,d}$ over the ring $\mathbb{F}_{q} [X] / (X^{n})$, where $\mathbb{F}_{q}$ is a finite field of q elements, with q is a power of a prime number $p \geq 5$ and $n \geq 5$. In a first time, we describe these curves over this ring. In addition, we prove that when $p$ doesn't divide $\# ( H_{\pi(a), \ \pi(d)})$, then $H^{n}_{a,d}$ is a direct sum of $ H_{\pi(a), \ \pi(d)}$ and $\mathbb{F}_{q}^{n-1}$, where $ H_{\pi(a), \ \pi(d)}$ is the twisted Hessian curve over $\mathbb{F}_{q}$. Other results are deduced from, we cite the equivalence of the discrete logarithm problem on the twisted Hessian curves $H^{n}_{a,d}$ and $ H_{\pi (a), \ \pi (d)}$, which is beneficial for cryptography and cryptanalysis as well.

A Modified AES-512 Bits Algorithm for Data Encryption

Privacy is given top priority as far as communication involving transfer of confidential document or data is concerned. There is therefore a need to keep confidential data secretive from being invaded by unauthorised access, and this is done through cryptography algorithms, where the Advanced Encryption Standard algorithm has been the widely accepted symmetric block encryption algorithm for such purposes. This paper therefore focuses on developing a new AES-512 bits symmetric encryption algorithm through modification of the conventional AES-128 algorithm to be used purposefully in the classroom for document transfer.The development comes with increasing the plaintext bits of the conventional AES-128 algorithm to 512 bits plaintext which undergoes five operational transformations: STATE, SKGF, SRL, SCL and AARC through key size in the Galois field, GF(29). A numerical example is then given to explain the use of the algorithm, and finally, we provide a comparative study of this algorithm and other existing symmetric encryption models, such as the AES-128 and DES algorithms.

An Improved Algebraic Structure of Nonlinear Dynamical Systems

A substitution box is regarded as an eminent source of nonlinearity and perplexity in most of the block ciphers. In this work, we propose that the involvement of a local ring has a favorable influence on the highly desirable features of this nonlinear cryptographic system. We deploy the unit group of a local ring by considering its correspondence with the Galois field and design an algorithm that utilizes the algebraic properties of the local ring [Formula: see text], to synthesize a highly efficient S-box.

Note on Irreducible Polynomials over $\mathbb{F}_q[X]$

In this note, we provide a new criterion of polynomials's irreducibility over $\mathbb{F}_q[X]$, where $\mathbb{F}_q$ is a finite field.

On the Dimensions of Hermitian Subfield Subcodes from Higher-Degree Places

The focus of our research is the examination of Hermitian curves over finite fields, specifically concentrating on places of degree three and their role in constructing Hermitian codes. We begin by studying the structure of the Riemann–Roch space associated with these degree-three places, aiming to determine essential characteristics such as the basis. The investigation then turns to Hermitian codes, where we analyze both functional and differential codes of degree-three places, focusing on their parameters and automorphisms. In addition, we explore the study of subfield subcodes and trace codes, determining their structure by giving lower bounds for their dimensions. This presents a complex problem in coding theory. Based on numerical experiments, we formulate a conjecture for the dimension of some subfield subcodes of Hermitian codes. Our comprehensive exploration seeks to deepen the understanding of Hermitian codes and their associated subfield subcodes related to degree-three places, thus contributing to the advancement of algebraic coding theory and code-based cryptography.

A class of finite p-groups and the normalized unit groups of group algebras

Let p be a prime and F p be a finite field of p elements. Let F p G denote the group algebra of the finite p-group G over the field F p and V ( F p G ) denote the group of normalized units in F p G . Suppose that G is a finite p-group given by a central extension of the form 1 → Z p n × Z p m → G → Z p × ⋯ × Z p → 1 and G ′ ≅ Z p ,   n , m ≥ 1 and p is odd. In this paper, the structure of G is determined. And the relations of V ( F p G ) p l and G p l ,   Ω l ( V ( F p G ) ) and Ω l ( G ) are given. Furthermore, there is a direct proof for V ( F p G ) p ∩ G = G p .

High Performance M-Term Karatsuba-Like Multiplier for Finite Field Arithmetic

Abstract: Finite field multiplication plays a crucial role in cryptographic circuits due to its widespread application. However, building circuits for these multiplications poses significant challenges due to their complexity. To mitigate this, the Karatsuba algorithm is employed, dividing each number into n/2 bits to reduce space complexity. While this approach reduces space complexity, it also increases time complexity. In our research, we introduce a hybrid approach, implementing a Karatsuba-like multiplier that combines elements of both Karatsuba and SBM (school book multiplication) techniques. Here in the proposed design, we replace the Array multiplier with a Wallace tree multiplier to further enhance design performance. This combination effectively reduces both time and space complexity. Our findings, based on reported device utilization and latency, demonstrate that the proposed multiplier outperforms the standard Karatsuba multiplier in terms of speed and efficiency, particularly in the area–delay product metric.

Analysis of Isomorphism Classes of a Family of Elliptic Curves Over Finite Fields

Doche et al. constructed a family of elliptic curves (DIK elliptic curves) and proposed more efficient tripling formulas leading to a fast scalar multiplication algorithm. In this paper we present a direct method to compute the number of \(\bar{F}\)q-isomorphism classes (isomorphism over \(\bar{F}\)q ) and \(\bar{F}\)q isomorphism classes of DIK family of elliptic curves defined over a finite field \(\bar{F}\)q. We give the explicit formulae for the number of \(\bar{F}\)q-isomorphism and an estimate formulae for the number of isomorphism classes. These result can be used in the elliptic curve cryptosystems.

Sárközy’s Theorem in Various Finite Field Settings

Sárközy’s Theorem in Various Finite Field Settings

Preperiodic points of polynomial dynamical systems over finite fields

Preperiodic points of polynomial dynamical systems over finite fields

Propagation properties of a non-linear mapping based on squaring in odd characteristic

AbstractMany modern cryptographic primitives for hashing and (authenticated) encryption make use of constructions that are instantiated with an iterated cryptographic permutation that operates on a fixed-width state consisting of an array of bits. Often, such permutations are the repeated application of a relatively simple round function consisting of a linear layer and a non-linear layer. These constructions do not require that the underlying function is a permutation and they can plausibly be based on a non-invertible transformation. Recently, Grassi proposed the use of non-invertible mappings operating on arrays of digits that are elements of a finite field of odd characteristic for so-called MPC-/FHE-/ZK-friendly symmetric cryptographic primitives. In this work, we consider a mapping that we call $$\gamma $$ γ that has a simple expression and is based on squaring. We discuss, for the first time, the differential and linear propagation properties of $$\gamma $$ γ and observe that these follow the same rules up to a relabeling of the digits. This is an intriguing property that, as far as we know, only exists for $$\gamma $$ γ and the binary mapping $$\chi _{_{3}}$$ χ 3 that is used in the cryptographic permutation Xoodoo. Moreover, we study the implications of its non-invertibility on differentials with zero output difference and on biases at the output of the $$\gamma $$ γ mapping and show that they are as small as they can possibly be.

A matrix variant of the Erdős-Falconer distance problems over finite field

We study a matrix analog of the Erdős-Falconer distance problems in vector spaces over finite fields. We provide lower bounds for the cardinality of a subset of the matrix algebra over a finite field such that its distance set is the whole matrix algebra. There arises an interesting analysis of certain quadratic matrix Gauss sums.

Solving systems of algebraic equations over finite commutative rings and applications

AbstractSeveral problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A finite chain ring is a finite ring admitting exactly one maximal ideal and every ideal being generated by one element. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residue field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

A family of algebraic curves and Appell series over finite fields

A family of algebraic curves and Appell series over finite fields

ON MATRICES ARISING IN FINITE FIELD HYPERGEOMETRIC FUNCTIONS

Abstract Lehmer [‘On certain character matrices’, Pacific J. Math.6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math.10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

Vibration transmission model of buildings based on virtual spring model and energy functional variation principle

In recent years, the “metro depot + over-track buildings” development model has witnessed rapid growth in China, driven by its advantages of high land utilization and investment return rate. However, the issue of vibration and noise caused by trains entering and exiting the depot has become a major impediment to its sustainable development. To predict the vibration of over-track buildings induced by trains, this paper presented a vibration transmission model of buildings based on the virtual spring model and the energy functional variation principle. The model was then applied to an actual engineering project at a metro depot in Guangzhou, China, and its accuracy was validated by comparing the results with finite element analysis and field measurements. Additionally, the vibration transmission characteristics of the over-track building were studied. The results indicated that the vibration transmission model of buildings presented in this paper aligns well with the results obtained through finite element analysis and field measurements. This approach eliminates the need to disassemble structural members, and the introduced virtual spring model effectively simulates non-ideal connections and boundary conditions. Consequently, it offers convenient structural connection and boundary condition settings, as well as high computational efficiency. Due to the edge effects on the top floor of the buildings, vibration waves in the higher floors experience a combination of reflected and incident waves, resulting in slow attenuation and, in some cases, amplification of vibrations as they propagate from the middle floors to the top floor. The vibration of the over-track building is closely related to the natural vibration characteristics of the floor and the frequency spectrum characteristics of the vibration source. When the natural frequency of a building's floor aligns with the main frequency of the vibration source, significant vibration amplification occurs due to local resonance.

3D medical image encryption algorithm using biometric key and cubic S-box

Considering the scarcity of research on 3D medical image encryption, this paper proposes a novel 3D medical image encryption scheme based on biometric key and cubic S-box. To enhance the data security, biometric keys are utilized to overcome the limitations of traditional methods where secret keys with no practical meaning, fixed length, and finite key space, while cubic S-box is constructed to increase the nonlinearity of image cryptosystem. The proposed cryptosystem mainly consists of four phases: pseudo-random sequence generation, confusion, substitution, and diffusion. Firstly, the stepwise iterative algorithm based on coupled chaotic systems is utilized for generating pseudo-random sequences for confusion and diffusion. Secondly, the confusion algorithm based on multiple sorting can scramble pixel positions in 3D images. Thirdly, guided by the designed cubic S-box, pixel substitution is executed sequentially. Lastly, the diffusion algorithm based on ECA and finite field multiplication is capable of increasing the plaintext sensitivity of cryptosystem by concealing the statistical characteristics of plaintext. Simulation experiments performed on multiple 3D medical images demonstrate that the proposed encryption scheme exhibits favorable statistical performance, sufficiently large key space, strong system sensitivity and robustness, and can resist various typical cryptographic attacks.

Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}—similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζG(k)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G(k)^{-1}$$\\end{document} at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}. We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro-C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} groups, where C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} is a class of finite groups with prescribed composition factors. We prove that every real number a≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a \\ge 1$$\\end{document} is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} are rational functions in p-s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{-s}$$\\end{document} if G is virtually abelian. For finite groups G we calculate ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} using the rational representation theory of G.

An explicit construction for large sets of infinite dimensional q $q$‐Steiner systems

AbstractLet be a vector space over the finite field . A ‐Steiner system, or an , is a collection of ‐dimensional subspaces of such that every ‐dimensional subspace of is contained in a unique element of . A large set of ‐Steiner systems, or an , is a partition of the ‐dimensional subspaces of into systems. In the case that has infinite dimension, the existence of an for all finite with was shown abstractly by Cameron in 1995. This paper provides an explicit construction of an for all prime powers , all positive integers , and where has countably infinite dimension.