Finite Field Research Articles

Representation of Polynomials Over Finite Fields With Circulants

Representation of Polynomials Over Finite Fields With Circulants

Fast Multivariate Multipoint Evaluation over All Finite Fields

Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field \(\mathbb {F}\) that outputs the evaluations of an m -variate polynomial of degree less than d in each variable at N points in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\) . A previous work of Kedlaya and Umans (FOCS 2008 and SICOMP 2011) achieved the same time complexity when the number of variables m is at most \(d^{o(1)}\) and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar, and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than \(d^{o(1)}\) . In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski, and Williams (IPEC 2017 and Algorithmica 2019), and that of Bhargava, Ghosh, Kumar, and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and, in particular, avoids the use of the Bombieri–Vinogradov theorem. However, it requires a mild assumption that the field size is bounded by an exponential tower in d of bounded height . More specifically, our second algorithm solves the multivariate multipoint evaluation problem over a finite field \(\mathbb {F}\) in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\) , provided that the size of the finite field \(\mathbb {F}\) is at most \((\exp (\exp (\exp (\cdots (\exp (d)))))\) , where the height of this tower of exponentials is fixed.

External codes for multiple unicast networks via interference alignment.

We introduce a formal framework to study the multiple unicast problem for a coded network in which the network code is linear over a finite field and fixed. We show that the problem corresponds to an interference alignment problem over a finite field. In this context, we establish an outer bound for the achievable rate region and provide examples of networks where the bound is sharp. We finally give evidence of the crucial role played by the field characteristic in the problem.

Enhancing image data security with chain and non-chain Galois ring structures

The local ring-based cryptosystem is built upon the core mathematical operations of the algebraic structure of local rings, which provides a significant advantage in ensuring security against advanced cryptanalysis. However, using the entire set of units within this structure would be computationally impractical in practical applications. Thus, we present a novel approach for designing 16×16 and 32×32 substitution boxes over chain and non-chain Galois ring respectively, by utilizing subgroups of local rings in a computationally feasible manner. Our proposed scheme significantly reduces memory usage by integrating both chain and non-chain rings. Specifically, the 16×16 and 32×32 S-boxes require only 16×28and 32×28 memory cells, respectively, whereas S-boxes of these dimensions over fields have been shown to be highly inefficient due to their extensive memory requirements (16×216and 32×232,respectively). The proposed method offers a more efficient solution for constructing S-boxes over large Galois fields and integrates chain and non-chain local Galois rings, as well as finite fields, for efficient transmission designs in smart devices. The algebraic structures used in this approach are used to establish the most vital aspect of a block cipher, the substitution box. We use each of these algebraic structures, each with a different bit size, to construct three distinct S-boxes. We discuss the performance and sensitivity of the proposed S-boxes to demonstrate their effectiveness in data protection. In this technique, the substitution boxes established by the non-chain ring, maximal cyclic subgroup of the Galois ring, and Galois field are applied for the processes of substitution, exclusive-or function, and diffusion in image encryption, respectively. Furthermore, we have performed numerous standard analyses on the encrypted image, including statistical analysis, differential analysis, and NIST tests. The results of these tests demonstrate the effectiveness of the proposed approach for various cryptographic purposes. Overall, this work provides a valuable contribution to the field of cryptography, particularly in constructing efficient S-boxes over large Galois fields.

Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Let D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and 1<n1≤⋯≤nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < n_1 \\le \\cdots \\le n_k$$\\end{document}, there exists an integer-valued polynomial on D, that is, an element of Int(D)={f∈K[X]∣f(D)⊆D}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D) = \\{ f \\in K[X] \\mid f(D) \\subseteq D \\}$$\\end{document}, which has precisely k essentially different factorizations into irreducible elements of Int(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D)$$\\end{document} whose lengths are exactly n1,…,nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1, \\ldots , n_k$$\\end{document}. Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.

The associative algebra of derivations of a group algebra

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group [Formula: see text] over a finite field [Formula: see text] are determined such that the set of derivations of the group algebra form an associative [Formula: see text]-algebra. The derivations of [Formula: see text] form a nontrivial associative [Formula: see text]-algebra if and only if [Formula: see text] has characteristic 2 and [Formula: see text] is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting [Formula: see text]-algebra is determined.

Residue sums of Dickson polynomials over finite fields

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

A Selberg Trace Formula for GL3(Fp)∖GL3(Fq)/K

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analog of the upper half-space and an action of GL3(Fq) on it. To compute the orbital sums, we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula, we decompose the induced representation ρ=IndΓG1 for G=GL3(Fq) and Γ=GL3(Fp).

An efficient high throughput BCH module for multi-bits error correction mechanism on hardware platform

The bose-chaudhuri-hocquenghem (BCH) codes are a cyclic error correction codes (ECC) class. The BCH is constructed by using a polynomial over the Galois field. The BCH codes can detect and correct the multi-bits with an easy decoding mechanism. The BCH codes are used in most of the storage device's cryptography, disk drives, and satellite applications. This manuscript presents an efficient high-throughput BCH module with an encoding and decoding mechanism for multi-bit corrections. The BCH code of (15, k) is used to construct the encoder and decoder architectures. The BCH encoder decoder (ED) module with single error correction (SEC), double error correction (DEC), and triple-error correction (TEC) are discussed in detail. The BCH encoder module uses a linear feedback shift register (LFSR). The BCH decoder with SEC and DEC is constructed using the syndrome generator module (SGM) and chien search module (CSM). The BCH decoder with TEC is designed using SGM, inversion-based berlekamp-massey-algorithm (BMA), and CSMs. The BCH-ED module with SEC, DEC, and TEC utilizes &amp;lt;1 % chip area on Artix-7 FPGA. The BCH-ED with SEC, DEC, and TEC achieves a throughput of 7.13 Gbps, 1.2 Gbps, and 0.803 Gbps, respectively. Lastly, the BCH module is compared with existing BCH approaches with better improvement in chip area, frequency, and throughput parameters.

On Quotients of Stochastic Networks Over Finite Fields

On Quotients of Stochastic Networks Over Finite Fields

Appell-Lauricella hypergeometric functions over finite fields and algebraic varieties

Appell-Lauricella hypergeometric functions over finite fields and algebraic varieties

Finite algebras in the design of multivariate cryptography algorithms

A new approach to the design of multivariate public-key cryptalgorithms is introduced. It envisages using non-linear mappings defined as squaring and cubic operations in finite fields represented as finite algebras. The developed approach allows significant reduction of the size of public key and thereby make post-quantum algorithms of multivariate cryptography much more practical. In the developed algorithms, the secret key includes a set of values of structural constants that determine the modifications of the finite fields used and the coefficients in the set of sixth degree polynomials that make up the public key

A new approach for the numerical calculation of diffraction patterns using starshades

Context. We studied the imaging of exoplanetary systems using starshades, which are externally occulted coronagraphs in space. Aims. We provide a new method for precisely evaluating the stray light due to the star and a rapid calculation of the point spread functions in the presence of vignetting effects from the external occulter. Our study used shaped occulter configurations published in the literature, in particular, the SISTER NI2 and NW2 systems. Methods. The wavefront at the telescope aperture was computed using the classic Fresnel filtering method. The Fourier transform of the occulter was obtained with the highest possible precision using an approach initially developed for radio antennas, known as the polygonal shape factor. Results. We show that the Fresnel diffraction for a finite spatial field operates at very low frequencies only, and that it is sufficient to calculate the Fourier transforms there. Diffraction patterns computed numerically fully agree with theoretical predictions. The central parts of diffractions of petal and apodized occulters are identical over a large central area that increases in size with the number of petals. These diffraction patterns are used to compute the point spread functions. We computed the stray light for a non-point source star; this shows that starshades are not sensitive to star leakage, with a star diameter limit for a given configuration. We also computed signal-to-noise ratios for a perfect experiment limited by photon noise.

A blockchain-based hybrid encryption technique with anti-quantum signature for securing electronic health records

Electronic health records (EHRs) are important for the efficient management of healthcare data. However, Healthcare data travels across an open route, i.e., the Internet, making EHR security a difficult process to do. This puts healthcare data vulnerable to cyber assaults. A possible method for protecting EHRs is blockchain technology. In this work, we develop an EHR architecture based on blockchain, which ensures all stakeholder's safety and privacy. We analyze various security architectures used for EHRs and the standard encryption system is integrated with quantum computing (QC). To safeguard the conventional traditional encrypting system against quantum assaults, we provide a hybrid signature technique that combines the Elliptic Curve Digital Signature Algorithm (ECDSA) and Dilithium within the anti-quantum lattice-based blind signature. Based on the difficulty of lattice problems over finite fields, Dilithium is a lattice-based signature method that is substantially safe against selected message assaults. The developed technique creates high entropy secret keys using the lattice basis delegation mechanism. The combination of ECDSA and Dilithium provides an efficient and secure signature system that is resilient to quantum attacks. The proposed scheme ensures that only authorized users with a defined role can use the database to access the data. We evaluate the efficiency of our scheme by comparing its performance to other state-of-the-art solutions in terms of transaction throughput, resource utilization, and communication cost. Results demonstrate that the developed technique outperforms the existing techniques in terms of efficiency and security.

Classification the elements of the twisted Hessian curves in the ring $\mathbb{F}_{q}[e], e^{3}=e^2$

Let $\mathbb{F}_{q}$ denote the finite field of $q$ elements, where $q$ is a prime power. In this paper, we study the twisted Hessian curves denoted $H_{a,d}(\mathbb{F}_{q}[e])$ over the ring $\mathbb{F}_{q}[e]$, where $e^{3}=e^2$ and $(a,d)\in (\mathbb{F}_{q}[e])^{2}$. More precisely, we study some arithmetical properties of this ring and using the Twisted Hessian equation, we define the twisted Hessian curves $H_{a,d}(\mathbb{F}_{q}[e])$. This work study the twisted Hessian curve helped us to define two twisted Hessian over the finite field $\mathbb{F}_{q}$. We end this paper by giving the classification of the elements in $H_{a,d}(\mathbb{F}_{q}[e])$.

Human Head Transcranial Magnetic Stimulation Using Finite Element Method

Transcranial magnetic stimulation (TMS) is a wearable neuromodulation technique. It is approved for several therapies for various neurological disorders, including major depressive disorder, traumatic brain injury, Parkinson’s disease, and post-traumatic stress disorder. This method became an alternative neuromodulation technique for such brain-related disorders. However, it has shown significant improvement in this alternative approach. Studies based on this technique have shown limited efficacy. They might be associated with current levels, poor coil locality, optimal coil size, and neuromodulator settings. It has been shown in this research that coil heating is related to higher levels of current. Thus, it is required to analyze the impact of the current levels on the induced magnetic distribution to define the optimal current range for the TMS coils. It is not feasible to investigate this research with experimental tests and analytic methods. Alternatively, using an advanced computational model of the coils and accounting for different human head anatomical layers, coil current capacity can be optimized based on finite element magnetic field distribution. This paper aims to investigate the impact of the coil current levels on the induced magnetic field distribution. The current capacity of the coils can be optimized based on the required magnetic field. In this way, the overheating may be reduced and may result in increased efficacy. As a proof-of-concept, a prototype coil and multi-layered geometrical human head models were generated using geometric shapes. The fundamental human head tissue layers were generated based on their average thickness. The model was simulated based on a finite element magnetic simulation using appropriate boundary conditions and neuromodulator settings. The various coil current levels were applied to analyze the outcome. The models were simulated, and the results were recorded based on these current levels. Results showed that there is a direct relation between applied current levels and induced magnetic flux density in the region of interest.

Numerical study of plasmas start-up by electron cyclotron waves in NCST spherical tokamak and CN-H1 stellarator

According to the physics of tokamak start-up, this study constructs a zero-dimensional (0D) model applicable to electron cyclotron (EC) wave assisted start-up in NCST spherical torus (spherical tokamak) and CN-H1 stellarators. Using the constructed 0D model, the results obtained in this study under the same conditions are compared and validated against reference results for pure hydrogen plasma start-up in tokamak. The results are in good agreement, especially regarding electron temperature, ion temperature and plasma current. In the presence of finite Ohmic electric field in the spherical tokamak, a study on the EC wave assisted start-up of the NCST plasma at frequency of 28 GHz is conducted. The impact of the vertical magnetic field B v on EC wave assisted start-up, the relationship between EC wave injection power P inj, Ohmic electric field E, and initial hydrogen atom density n H0 are explored separately. It is found that under conditions of Ohmic electric field lower than ITER (~ 0.3 V m−1), EC wave can expand the operational space to achieve better plasma parameters. Simulating the process of 28 GHz EC wave start-up in the CN-H1 stellarator plasma, the plasma current in the zero-dimensional model is replaced with the current in the poloidal coil of the stellarator. Plasma start-up can be successfully achieved at injection powers in the hundreds of kilowatts range, resulting in electron densities on the order of 1017–1018 m–3.

Local Isomeric Polar Nanoclusters Enabled Superior Capacitive Energy Storage Under Moderate Fields in NaNbO3-Based Lead-Free Ceramics.

The high-field energy-storage performance of dielectric capacitors has been significantly improved in recent years, yet the high voltage risks of device failure and large cost of insulation technology increase the demand for high-performance dielectric capacitors at finite electric fields. Herein, a unique superparaelectric state filled with polar nanoclusters with various local symmetries for lead-free relaxor ferroelectric capacitors is subtly designed through a simple chemical modification method, successfully realizing a collaborative improvement of polarization hysteresis, maximum polarization, and polarization saturation at moderate electric fields of 20-30kVmm-1. Therefore, a giant recoverable energy density of ≈5.0Jcm-3 and a high efficiency of ≈82.1% are simultaneously achieved at 30kVmm-1 in (0.9-x)NaNbO3-0.1BaTiO3-xBiFeO3 lead-free ceramics, showing a breakthrough progress in moderate-field comprehensive energy-storage performances. Moreover, superior charge-discharge performances of high-power density ≈182MWcm-3, high discharge energy density ≈4.3Jcm-3 and ultra-short discharge time <70ns as well as excellent temperature stability demonstrate great application potentials for dielectric energy-storage capacitors in pulsed power devices. This work provides an effective and paradigmatic strategy for developing novel lead-free dielectrics with high energy-storage performance under finite electric fields.

Cryptanalysis of a Cayley Hash Function Based on Affine Maps in one Variable over a Finite Field

Cryptanalysis of a Cayley Hash Function Based on Affine Maps in one Variable over a Finite Field

Multi‐lingual encryption technique using Unicode and Riemann zeta function and elliptic curve cryptography for secured routing in wireless sensor networks

AbstractSecure routing and communication with confidentiality based on encryption of texts in multiple natural languages are challenging issues in wireless sensor networks which are widely used in recent applications. The existing works on Elliptic Curve Cryptography based secured routing algorithms are focused only on the encryption and decryption of single language text encrypted over a Prime finite field. In this article, a new algorithm called Multi‐Language ECC encrypted Secure Routing algorithm with trust management is proposed, in order to ensure confidentiality and integrity which focuses on the encryption of plain text using Riemann's zeta function and Elliptic Curve Cryptography for improving the key strength which is applied for encryption over a range of multi languages namely Tamil, English, Hindi French and German which are supported by Unicode and routing the text security. From the experiments conducted using the proposed multi‐lingual encryption algorithm with network routing, we prove that the suggested method provides greater security than the current secure routing algorithms due to the use of Zeta function and Gamma function with ECC key and trust management. but also boasts reduced complexity compared to other existing multi‐lingual encryption algorithms.