Elements Of Finite Field Research Articles

Binary sequence family with both small cross-correlation and large family complexity

Ahlswede, Khachatrian, Mauduit and Sárközy introduced the notion of family complexity, Gyarmati, Mauduit and Sárközy introduced the cross-correlation measure for families of binary sequences. It is a challenging problem to find families of binary sequences with both small cross-correlation measure and large family complexity. In this paper we present a family of binary sequences with both small cross-correlation measure and large family complexity by using the properties of character sums and primitive normal elements in finite fields.

On additive decompositions of primitive elements in finite fields

In this paper, we study several topics on additive decompositions of primitive elements in finite fields. Also we refine some bounds obtained by Dartyge and Sárközy as well as Shparlinski.

A zero-knowledge proof of transpose of a matrix of finite field elements

A zero-knowledge proof of transpose of a matrix of finite field elements

Pairs of r-Primitive and k-Normal Elements in Finite Fields

Pairs of r-Primitive and k-Normal Elements in Finite Fields

Maximally Recoverable Codes With Hierarchical Locality: Constructions and Field-Size Bounds

Maximally recoverable codes are a class of codes which recover from all potentially recoverable erasure patterns given the locality constraints of the code. In earlier works, these codes have been studied in the context of codes with locality. The notion of locality has been extended to hierarchical locality, which allows for locality to gradually increase in levels with the increase in the number of erasures. We consider the locality constraints imposed by codes with two-level hierarchical locality and define maximally recoverable codes with hierarchical locality. We characterize the set of all erasure patterns which can be corrected by hierarchical data-local and hierarchical local maximally recoverable codes (MRC). We show that by carefully puncturing coordinates of the code, hierarchical local MRC can be reduced to hierarchical data-local MRC. Based on picking elements of finite fields and their extensions, all of which satisfy certain linear independence properties, we provide a generic construction of parity check matrix of hierarchical local MRC for all parameters. By appropriately modifying parity check matrices of local MRCs with limited parities, we also give constructions of hierarchical local MRCs with limited parities. Finally, we also derive a lower bound on the field size of hierarchical local MRCs.

Secure Proof of the Sum of all the Elements of a Matrix of Finite Field Elements in Zero-Knowledge

Secure Proof of the Sum of all the Elements of a Matrix of Finite Field Elements in Zero-Knowledge

Secure proof of the sum of all the elements of a matrix of finite field elements in zero-knowledge

Secure proof of the sum of all the elements of a matrix of finite field elements in zero-knowledge

About r-primitive and k-normal elements in finite fields

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements: an element \(\alpha \in {\mathbb {F}}_{q^n}\) is k-normal over \({\mathbb {F}}_q\) if the greatest common divisor of the polynomials \(g_{\alpha }(x)= \alpha x^{n-1}+\alpha ^qx^{n-2}+\cdots +\alpha ^{q^{n-2}}x+\alpha ^{q^{n-1}}\) and \(x^n-1\) in \({\mathbb {F}}_{q^n}[x]\) has degree k, generalizing the concept of normal elements (normal in the usual sense is 0-normal). In this paper we discuss the existence of r-primitive k-normal elements in \({\mathbb {F}}_{q^n}\) over \({\mathbb {F}}_{q}\), where an element \(\alpha \in {\mathbb {F}}_{q^n}^*\) is r-primitive if its multiplicative order is \(\frac{q^n-1}{r}\). We provide many general results about the existence of this class of elements and we work a numerical example over finite fields of characteristic 11.

High order elements in finite fields arising from recursive towers

We illustrate a general technique to construct towers of fields producing high order elements in mathbb {F}_{q^{2^n}}, for odd q, and in mathbb {F}_{2^{2 cdot 3^n}}, for n ge 1. These towers are obtained recursively by x_{n}^2 + x_{n} = v(x_{n - 1}), for odd q, or x_{n}^3 + x_{n} = v(x_{n - 1}), for q=2, where v(x) is a polynomial of small degree over the prime field mathbb {F}_{q} and x_n belongs to the finite field extension mathbb {F}_{q^{2^n}}, for an odd q, or to mathbb {F}_{2^{2cdot 3^n}}. Several examples are provided to show the numerical efficacy of our method. Using the techniques of Burkhart et al. (Des Codes Cryptogr 51(3):301–314, 2009) we prove similar lower bounds on the orders of the groups generated by x_n, or by the discriminant delta _n of the polynomial. We also provide a general framework which can be used to produce many different examples, with the numerical performance of our best examples being slightly better than in the cases analyzed in Burkhart et al. (2009).

Redundancy AES Masking Basis for Attack Mitigation (RAMBAM)

In this work, we present RAMBAM, a novel concept of designing countermeasures against side-channel attacks and the Statistical Ineffective Fault Attack (specifically SIFA-1) on AES that employs redundant representations of finite field elements. From this concept, we derive a family of protected hardware implementations of AES. A fundamental property of RAMBAM is a security parameter d that along with other attributes of the scheme allows for making trade-offs between gate count, maximal frequency, performance, level of robustness to the first and higher-order side-channel attacks, and protection against SIFA-1. We present an analytical model that explains how the scheme reduces the leakage and how the design choices affect it. Furthermore, we demonstrate experimentally how different design choices achieve the required goals. In particular, the compact version exhibits a gate count as low as 12.075 kGE, while maintaining adequate protection. The performance-oriented version provides latency as low as one round per cycle, thus combining protection against SCA and SIFA-1 with high performance which is one of the original design goals of AES. Finally, we assess the leakage of the scheme for the first and the second (bivariate) orders using TVLA methodology on an FPGA implementation and observe resilience to at least 348M traces with 16 Sboxes.

Area efficient and high-speed Galois field multiplier for mobile edge computing devices

Mobile Edge Computing is one of the prominent technologies in 5G communication to improve throughput, scalability and reliability. In this work, area efficient and high-speed multipliers are implemented using finite field (Galois field) arithmetic which are widely utilised by edge computing devices. The performance of finite field multiplication operation is closely related to finite field elements representation. The proposed polynomial-based finite field bit parallel systolic array multiplier is able to achieve almost double the speed of existing multipliers. There is a considerable reduction in area and power for the proposed word-level normal basis finite field multiplier compared to the existing multipliers. The results clearly indicate that the proposed method improves the efficiency of finite field multipliers in terms of area, delay or power consumption. In order to process the real-time data efficiently, both the local computing and data offloading are carried out in developing a joint computation algorithm. The trade-off analysis between local computing and data offloading shows that local computing plays a more important role when the data size is small, but data offloading is preferred when the data size increases.

Existence of primitive 2-normal elements in finite fields

An element α∈Fqn is normal over Fq if B={α,αq,αq2,⋯,αqn−1} forms a basis of Fqn as a vector space over Fq. It is well known that α∈Fqn is normal over Fq if and only if gα(x)=αxn−1+αqxn−2+⋯+αqn−2x+αqn−1 and xn−1 are relatively prime over Fqn, that is, the degree of their greatest common divisor in Fqn[x] is 0. Using this equivalence, the notion of k-normal elements was introduced in Huczynska et al. (2013): an element α∈Fqn is k-normal over Fq if the greatest common divisor of the polynomials gα[x] and xn−1 in Fqn[x] has degree k; so an element which is normal in the usual sense is 0-normal.Huczynska et al. made the question about the pairs (n,k) for which there exist primitive k-normal elements in Fqn over Fq and they got a partial result for the case k=1, and later Reis and Thomson (2018) completed this case. The Primitive Normal Basis Theorem solves the case k=0. In this paper, we solve completely the case k=2 using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of k-normal elements in Fqn.

Edge even and edge odd graceful labelings of Paley Graphs

Edge even graceful labeling is a novel graceful labelling, introduced in 2017 by Elsonbaty and Daoud. A graph G with p vertices and q edges is called an edge even graceful if there is a bijection f: E(G) → {2, 4,., 2q} such that, when each vertex is assigned the sum of the labels of all edges incident to it mod 2k, where k = max (p, q), the resulting vertex labels are distinct. A labeling of G is called edge odd graceful labeling, if there exists a bijection f from the set of edges E(G) to the set {1,3,5,…,2q-1} such that the induced the map f* from the set of vertices V(G) to {0,1,2,.,.,2q-1} given by f*(u) = Σ uv∈E(G) f(uv) (mod 2q) is an injection. A graph which admits edge even (odd) graceful labeling is called an edge even (odd) graceful graph. Paley graphs are dense undirected graphs raised from the vertices as elements of an appropriate finite field by joining pairs of vertices that differ by a quadratic residue. In this paper, we study the construction of edge even (odd) graceful labeling for Paley graphs and prove that Paley graphs of prime order are edge even (odd) graceful.

Fast En/Decoding of Reed-Solomon Codes for Failure Recovery

Reed-Solomon (RS) codes are used in many storage systems for failure recovery. In popular software implementations, RS codes are defined by using a parity check matrix that is either a Cauchy matrix padded with an identity or a Vandermonde matrix. The encoding complexity can be reduced by searching for a Cauchy matrix that has a smaller number of ‘1's in its bit matrices or exploiting Reed-Muller (RM) transform in the Vandermonde matrix multiplication. This article proposes two new approaches that improve upon the previous schemes. In our first approach, different constructions of finite fields are explored to further reduce the number of ‘1's in the bit matrices of the Cauchy matrix and a new searching method is developed to find the matrices with minimum number of ‘1's. Our second approach defines RS codes using a parity check matrix in the format of a Vandermonde matrix concatenated with an identity matrix so that the multiplication with the inverse erasure columns in the encoding is eliminated and the decoding can be carried out using simplified formulas. The Vandermonde matrix in such an unconventional RS code definition needs to be constructed using finite field elements in non-consecutive order. A modification is also developed in this article to enable the application of the RM transform in this case to reduce the matrix multiplication complexity. For 4-erasure-correcting RS codes over <inline-formula><tex-math notation="LaTeX">$GF(2^8)$</tex-math></inline-formula> , the two proposed approaches increase the encoding throughput by 40 and 15 percent on average over the prior works based on Cauchy matrix and Vandermonde matrix with RM transform, respectively, for a range of codeword length. Moreover, the decoding throughput is also significantly improved.

On multiplicative order of elements in finite fields based on cyclotomic polynomials

On multiplicative order of elements in finite fields based on cyclotomic polynomials

The trace of 2-primitive elements of finite fields

Let $q$ be a prime power and $n, r$ integers such that $r\mid q^n-1$. An element of $\mathbb{F}_{q^n}$ of multiplicative order $(q^n-1)/r$ is called \emph{$r$-primitive}. For any odd prime power $q$, we show that there exists a $2$-primitive element of $\mathbb{F}_{q^n}$ with arbitrarily prescribed $\mathbb{F}_q$ trace when $n\geq 3$. Also we explicitly describe the values that the trace of such elements may have when $n=2$.

Efficient VLSI Architectures for Coupled-Layered Regenerating Codes

To enable the continued scaling of distributed storage, failure recovery schemes that have low latency and low redundancy are indispensable. Minimum storage regenerating (MSR) codes have been developed to achieve this goal. Compared to other MSR codes, the recent coupled-layered (Clay) codes have many advantages such as low sub-packetization level, small finite field for construction, and uniform repair bandwidth over data and parities. This brief proposes efficient VLSI architectures for Clay encoder and decoder. Through exploiting subfield elements of finite fields, the logic complexity is substantially lowered. Additionally, the buffer size is reduced by optimizing the coupling scheme. For an example Clay code with $(n,k)=(6,4)$ , the proposed designs achieve 12% and 24% logic complexity reduction on the encoder and decoder, respectively, under the same timing constraint.

Relaxing the Constraints on Locally Recoverable Erasure Codes by Finite Field Element Variation

To enable the continued scaling of distributed storage, locally recoverable (LRC) erasure codes are necessary to substantially reduce the number of symbols to access for failure recovery. Codes based on parity splitting and generalized integrated interleaved codes are two categories of LRC codes that achieve good tradeoffs on locality, redundancy, and complexity. The parity check and nesting matrices in these codes are modified to improve the complexity and locality. On the other hand, these structural modifications add constraints on the code parameters. This letter proposes to relax the constraints by exploiting different finite field elements, field constructions, and independent entries in the matrices. The computation complexity of finding the maximum allowed values of the parameters has been reduced by orders of magnitudes through taking advantage of isomorphic mapping and conjugacy classes, as well as developing a decomposed searching method. Significant improvements on the applicable code parameters have been achieved for both codes using the proposed schemes.

A Deterministic Construction for Jointly Designed Quasicyclic LDPC Coded-Relay Cooperation

This correspondence presents a jointly designed quasicyclic (QC) low-density parity-check (LDPC) coded-relay cooperation with joint-iterative decoding in the destination node. Firstly, a design-theoretic construction of QC-LDPC codes based on a combinatoric design approach known as optical orthogonal codes (OOC) is presented. Proposed OOC-based construction gives three classes of binary QC-LDPC codes with no length-4 cycles by utilizing some known ingredients including binary matrix dispersion of elements of finite field, incidence matrices, and circulant decomposition. Secondly, the proposed OOC-based construction gives an effective method to jointly design length-4 cycles free QC-LDPC codes for coded-relay cooperation, where sum-product algorithm- (SPA-) based joint-iterative decoding is used to decode the corrupted sequences coming from the source or relay nodes in different time frames over constituent Rayleigh fading channels. Based on the theoretical analysis and simulation results, proposed QC-LDPC coded-relay cooperations outperform their competitors under same conditions over the Rayleigh fading channel with additive white Gaussian noise.

Existence results on $k$-normal elements over finite fields

An element \alpha \in \mathbb{F}_{q^n} is normal over \mathbb{F}_q if \alpha and its conjugates \alpha, \alpha^q, \dots, \alpha^{q^{n-1}} form a basis of \mathbb{F}_{q^n} over \mathbb{F}_q . In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k -normal elements, generalizing the normal elements. In the past few years, many questions concerning the existence and number of k -normal elements with specified properties have been proposed. In this paper, we discuss some of these questions and, in particular, we provide many general results on the existence of k -normal elements with additional properties like being primitive or having large multiplicative order. We also discuss the existence and construction of k -normal elements in finite fields, providing a connection between k -normal elements and the factorization of x^n-1 over \mathbb{F}_q .