Finite Field GF Research Articles

On Matrix Representation of Extension Field GF(pL) and Its Application in Vector Linear Network Coding.

For a finite field GF(pL) with prime p and L>1, one of the standard representations is L×L matrices over GF(p) so that the arithmetic of GF(pL) can be realized by the arithmetic among these matrices over GF(p). Based on the matrix representation of GF(pL), a conventional linear network coding scheme over GF(pL) can be transformed to an L-dimensional vector LNC scheme over GF(p). Recently, a few real implementations of coding schemes over GF(2L), such as the Reed-Solomon (RS) codes in the ISA-L library and the Cauchy-RS codes in the Longhair library, are built upon the classical result to achieve matrix representation, which focuses more on the structure of every individual matrix but does not shed light on the inherent correlation among matrices which corresponds to different elements. In this paper, we first generalize this classical result from over GF(2L) to over GF(pL) and paraphrase it from the perspective of matrices with different powers to make the inherent correlation among these matrices more transparent. Moreover, motivated by this correlation, we can devise a lookup table to pre-store the matrix representation with a smaller size than the one utilized in current implementations. In addition, this correlation also implies useful theoretical results which can be adopted to further demonstrate the advantages of binary matrix representation in vector LNC. In the following part of this paper, we focus on the study of vector LNC and investigate the applications of matrix representation related to the aspects of random and deterministic vector LNC.

Almost Difference Sets from Unions of Cyclotomic Classes of Order 10

This study investigated the problem of constructing almost difference sets from single and unions of cyclotomic classes of order 10 (with and without zero) of the finite field GF(q), where q is a prime of the form q = 10n+1 for integer n≥1 and q<1000. Using exhaustive computer searches in Python, the method computed unions of two classes up to nine cyclotomic classes. Moreover, the equivalence of the generated almost difference sets having the same parameters was determined up to complementation. Results showed that a single cyclotomic class formed an almost difference set for q = 31, 71, and 151, while including zero produced an almost difference set for q=11, 31, 71, and 151. Additionally, Paley partial difference sets were constructed from the union of five cyclotomic classes. These findings addressed gaps in the literature on cyclotomy of order 10.

Encryption Method Based on Codes

This paper proposes an improvement of the McEliece asymmetric cryptosystem based on code-based cryptography by replacing the permutation matrix with a modulo operation and using a finite field GF(q) . This approach increases the complexity of the decryption process for potential attackers, providing a high level of cryptographic security without changing the length of the key. The article provides a diagram of the improved operation of the cryptosystem and describes examples of application. An analysis of the number of possible combinations of matrices has been carried out for different implementation options of code (7,4) based on different numerical systems. It has been shown that achieving cryptographic security comparable to the original McEliece cryptosystem requires the use of q≥5.

On Some New Almost Difference Sets Constructed from Cyclotomic Classes of Order 12

Almost Difference Sets have extensive applications in coding theory and cryptography. In this study, we introduce new constructions of Almost Difference Sets derived from cyclotomic classes of order 12 in the finite field GF(q), where q is a prime satisfying the form q=12n+1 for positive integers n ≥ 1 and q < 1000. We show that a single cyclotomic class of order 12 (with and without zero) can form an almost difference set. Additionally, we successfully construct almost difference sets using unions of cyclotomic classes of order 12, both for even and odd values of n. To accomplish this, an exhaustive computer search employing Python was conducted. The method involved computing unions of two cyclotomic classes up to eleven classes and assessing the presence of almost difference sets. Finally, we classify the resulting almost difference sets with the same parameters up to equivalence and complementation.

A non-degenerate chaotic bits XOR system with application in image encryption

The existing chaotic systems exhibit chaos degradation phenomena because of the limited chaotic precision in the real number field. In response to this deficiency, this paper proposes a chaotic bits XOR system (CBXS) in the integer field satisfying Devaney’s chaos. That is, CBXS has the property of sensitivity of initial value, topological transitivity, and periodic point density. Then, a series of chaotic and random test results illustrate that the sequences generated by CBXS with linear feedback shift register (LFSR) in the finite field GF(2) have good random and chaotic performance. It can overcome the chaos degradation problem because it is generated in the integer field and can avoid the limited precision in the real number field. Finally, we employ the proposed CBXS to image encryption with the optimized Arnold’s transformation. The experimental results verify that even after simple encryption, the cipher image still exhibits good encryption effect and efficiency, which illustrates the effectiveness of CBXS.

Burst error-correcting quantum stabilizer codes designed from idempotents

Certain classical codes can be viewed isomorphically as ideals of group algebras, while studying their algebraic structures help extracting the code properties. Research has shown that this was remarkably efficient in the case when the code generators are idempotents. In quantum error correction, the theory of stabilizer formalism requires classical self-orthogonal additive codes over the finite field GF(4), which, via the lens of group algebras, are essentially F_2-submodules over GF(4). Therefore, this paper provides a classification on idempotents in commutative group algebra GF(4)G, followed by a criterion that allows idempotents to generate stabilizer subgroups. Later, the construction of quantum stabilizer codes is done in the case when G is a cyclic group C_n, for n=2^m-1 and n=2^m+1. Quantum bounds on their burst error minimum distance are subsequently determined.

왼쪽 시프트 연산을 n-1번 사용하여 n비트 유한체 곱셈기 설계 및 구현

In this paper, we propose a finite fields GF(2n) multiplier algorithm by repeating the 2 multiplication algorithm n-1 times with a combination circuit composed only of AND and XOR gates. The proposed algorithm can be configured with fewer gates than the existing finite fields multiplication algorithm, and is expected to be easy to implement in hardware because it is very regular. Finally, the time to perform finite fields multiplication is also expected to be faster than other multiplication algorithms due to the smaller number of gates.

Creation of Database of Irreducible Polynomial Over A Finite Field GF(2) Based on FPGA

Creation of Database of Irreducible Polynomial Over A Finite Field GF(2) Based on FPGA

Sets of Nonbinary Sequences with a Low Level of Mutual Correlation for Systems of Digital Information Transmission

The sets of nonbinary pseudorandom sequences (NPSs) for periods N = p^S– 1 20 000 (p = 3, 5,7, and 11) generated in finite fields GF(pS) whose power is V = N + 1 and the maximum of the module of peaks of the periodic autocorrelation function (PACF) and periodic cross-correlation function (PCCF) satisfy the bounds obtained by Sidel’nikov. In addition to the minimal polynomials of elements α and α^2 , where α is a primitive element of field GF(p^S), the minimal polynomials of elements α and α^(i_d)(i_d is the decimation index) are determined on the basis of which the new sets of NPSs can be formed with the equivalent correlation properties. Sets of indices i_d 2 are determined for various combinations of parameters p and S. The cases of even and odd values of parameter S are considered, for which the values of the PACF and PCCF with the maximum module are obtained and the number and values of different levels of correlation functions are determined.

НОВЫЙ ПОДХОД К РАЗРАБОТКЕ АЛГОРИТМОВ МНОГОМЕРНОЙ КРИПТОГРАФИИ

Purpose of work is the reduction in the size of the public key of public-key algorithms of multivariate cryptography based on the computational difficulty of solving systems of many power equations with many unknowns. Research method is use of non-linear mappings defined as exponentiation operations in finite extended fields GF(qm) represented in the form of finite algebras. The latter makes it possible to perform the exponentiation operation in the field GF(qm) by calculating the values of power polynomials over the field GF(q), which define a hardly reversible nonlinear mapping of the vector space over GF(q) with a secret trapdoor. Due to the use of nonlinear mappings of this type, it is possible to specify a public key in multidimensional cryptography algorithms in the form of a nonlinear mapping implemented as a calculation of the values of a set of polynomials of the third and sixth degree. At the same time, due to the use of masking linear mappings that do not lead to an increase in the number of terms in polynomials, the size of the public key is reduced in comparison with known analogue algorithms, in which the public key is represented by a set of polynomials of the second and third degrees. The proposed approach potentially expands the areas of practical application of post-quantum algorithms for public encryption and electronic digital signature, related to multidimensional cryptography, by significantly reducing the size of the public key. Results of the study are the main provisions of a new approach to the development of algorithms of multidimensional cryptography are formulated. Hardly invertible nonlinear mappings with a secret trapdoor are proposed in the form of exponentiation operations to the second and third powers in finite extended fields GF(qm), represented in a form of a finite algebra. A rationale is given for specifying a public key in a form that includes a superposition of two non-linear mappings performed as a calculation of a set of second and third degree polynomials defined over GF(q). Techniques for implementing mappings of this type are proposed and specific options for specifying the fields GF(qm) in the form of finite algebras are considered. An estimate of the size of the public key in the algorithms developed within the framework of the new approach is made. at a given security level.. Practical relevance includes the developed main provisions of a new method for constructing multidimensional cryptography algorithms based on the computational difficulty of solving systems of many power equations with many unknowns and related to post-quantum cryptoschemes. The proposed approach expands the areas of practical application of post-quantum algorithms of this type by significantly reducing the size of the public key, which provides the prerequisites for improving performance and reducing technical resources for their implementation

An algebraic attack to the Bluetooth stream cipher E0

In this paper we study the security of the Bluetooth stream cipher E0 from the viewpoint it is a “difference stream cipher”, that is, it is defined by a system of explicit difference equations over the finite field GF(2). This approach highlights some issues of the Bluetooth encryption such as the invertibility of its state transition map, a special set of 14 bits of its 132-bit state which when guessed implies linear equations among the other bits and finally a small number of spurious keys, with 83 guessed bits, which are compatible with a keystream of about 60 bits. Exploiting these issues, we implement an algebraic attack using Gröbner bases, SAT solvers and Binary Decision Diagrams. Testing activities suggest that the version based on Gröbner bases is the best one and it is able to attack E0 in about 279 seconds on an Intel i9 CPU. To the best of our knowledge, this work improves any previous attack based on a short keystream, hence fitting with Bluetooth specifications.

Design of Low Power Versatile Bit-Serial Multiplier in Finite Field GF (2m)

Finite field arithmetic is the most important component in applications like cryptography, computer algebra and error correcting codes. Versatility is an important property the hardware industry lacks and trying to establish as much as possible. To survive in this booming technological word, the new designs should be of an adjustable one, which processes the versatile property. In this we have proposed an efficient VLSI design for versatile bit-serial multiplier in finite fields GF (2m). The versatile multiplier designed here modifications done by reducing the unwanted switching activity removed by clock gating scheme. Our design provides a solution to the power reduction. The introduced multiplier operates over a variety of binary fields up to an order of 2m. The value of m can vary up to 264 bits. Multiplier is designed using Verilog HDL.

Metode Reversible Self-Dual untuk Konstruksi Kode DNA atas Lapangan Hingga GF(4)

The DNA molecule chain consists of two complementary strands composed of a sequence of four nucleotide bases, namely adenine (A), cytosine (C), guanine (G) and thymine (T). DNA code is a set of codewords with a fixed length of the alphabet {A, C, T, G}. DNA coding is one application of coding theory over a finite field. The set {A, C, T, G} is identified as finite field GF(4) = {0, 1, w, w2} with w2 + w + 1 = 0. The reversible self-dual (RSD) code over the finite field GF(4) is a code whose dual is itself and the reverse of each codeword contained in the code. This study aims to obtain an algorithm to construct a DNA code derived from the RSD C code on the field to GF(4) which is called the Reversible Self-Dual Method. The aspects studied include the characteristics that form the basis properties of the theory in compiling the DNA code algorithm over the RSD code over GF(4). The compiled algorithm is a DNA code construction method of codeword length even that conforms to the Hamming distance constraint, reverse-complement constraint, and GC-content constraint. The input of the algorithm is a generator matrix of RSD code C with a minimum distance of d and the output is a DNA code that satisfies these three constraints.

Image encryption based on a novel memristive chaotic system, Grain-128a algorithm and dynamic pixel masking

In this paper, we first propose a memristive chaotic system and implement it by circuit simulation. The chaotic dynamics and various attractors are analysed by using phase portrait, bifurcation diagram, and Lyapunov exponents. In particular, the system has robust chaos in a wide parameter range and the initial value space, which is favourable to the security communication application. Consequently, we further explore its application in image encryption and present a new scheme. Before image processing, the external key is protected by the Grain-128a algorithm and the initial values of the memristive system are updated with the plain image. We not only perform random pixel extraction and masking with the chaotic cipher, but also use them as control parameters for Brownian motion to obtain the permutation matrix. In addition, multiplication on the finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) is added to further enhance the cryptography. Finally, the simulation results verify that the proposed image encryption scheme has better performance and higher security, which can effectively resist various attacks.

The Subfield Codes of Some [q + 1, 2, q] MDS Codes

Recently, subfield codes of geometric codes over large finite fields GF(q) with dimension 3 and 4 were studied and distance-optimal subfield codes over GF(p) were obtained, where q = pm. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of [q + 1,2,q] MDS codes over GF(q), and then studies the subfield codes over GF(p) of some of the [q+1,2,q] MDS codes over GF(q). Two families of dimensionoptimal codes over GF(p) are obtained, and several families of nearly optimal codes over GF(p) are produced. Several open problems are also proposed in this paper.

Collective Signature Schemes Using Problem of Finding Roots Modulo

Digital signature schemes in general and representative collective digital signature schemes, in particular, are often built based on the difficulty of the discrete logarithm problem on the finite field, of the discrete logarithm problem of the elliptic curve, of the problem of factor analysis, of the problem of finding the roots modulo of large primes or a combination of the difficult problems mentioned above. In this paper, we use the new difficult problem, which is to find the with root in the finite ground field GF(p) to build representative collective signature schemes, but the chosen modulo p has a special structure distinct p=Nt_0 t_1 t_2+1, where N is an even number and t0, t1, t2 are prime numbers of equal magnitude, about 80 bits. The characteristics of the proposed scheme are: i) The private key of each signer consists of 2 components (K_1, K_2), randomly selected, but the public key has only one component (Y) calculated by the formula Y= K_1^ (w_1) K_2^ (w_2); w_1 = t_0 t_1 and w_2 = t_0 t_2; and ii) The generated signature consists of a set of 3 components (e, S1, S2). We use the technique of hiding the signer's public key Y, which is the coefficient λ generated by the group manager, in the process of forming the group signature and representative collective signature to enhance the privacy of all members of the signing collective.

AMBTC-based secret image sharing by simple modular arithmetic

A distortionless secret image sharing scheme using finite field GF(PG) (PG is the largest prime less than a given number) is investigated for sharing the absolute moment block truncation coding (AMBTC) images. Two adjusting operations are devised to modify the AMBTC trios (i.e., quantization pixels and bit maps) suitable for GF(PG) sharing. Polynomials under GF(PG) are constructed for encrypting the quantization pixels and bit maps. When sharing the trios under GF(PG), the AMBTC image is perfectly recovered. Moreover, the bit map sharing under GF(PG) can be extended to GF(PS) (PS is the smallest prime larger than a given number). Another scheme using GF(PS) is constituted by combining quantization pixel sharing under GF(PG) and bit map sharing under GF(PS). But the scheme using GF(PS) is not always lossless. Experimental results show that the proposed schemes are effective. Since GF(PG) and GF(PS) are simple modular arithmetic, improved computational efficiency is obtained.

On cutting blocking sets and their codes

AbstractLetPG⁡(r,q){\operatorname{PG}(r,q)}be ther-dimensional projective space over the finite fieldGF⁡(q){\operatorname{GF}(q)}. A set𝒳{\mathcal{X}}of points ofPG⁡(r,q){\operatorname{PG}(r,q)}is a cutting blocking set if for each hyperplane Π ofPG⁡(r,q){\operatorname{PG}(r,q)}the setΠ∩𝒳{\Pi\cap\mathcal{X}}spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set ofPG⁡(3,q3){\operatorname{PG}(3,q^{3})}of size3⁢(q+1)⁢(q2+1){3(q+1)(q^{2}+1)}as a union of three pairwise disjointq-order subgeometries, and a cutting blocking set ofPG⁡(5,q){\operatorname{PG}(5,q)}of size7⁢(q+1){7(q+1)}from seven lines of a Desarguesian line spread ofPG⁡(5,q){\operatorname{PG}(5,q)}. In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimalq-ary linear code having dimension 4 and 6.

Classification of Unit Groups of Five Radical Zero Completely Primary Finite Rings Whose First and Second Galois Ring Module Generators Are of the Order pk, k = 2, 3, 4

Let R0 = GR(pkr, pk) be a Galois maximal subring of R so that R = R0 ⊕ U ⊕ V ⊕ W ⊕ Y, where U, V, W, and Y are R0/pR0 spaces considered as R0‐modules, generated by the sets {u1, ⋯, ue}, {v1, ⋯, vf}, {w1, ⋯, wg}, and {y1, ⋯, yh}, respectively. Then, R is a completely primary finite ring with a Jacobson radical Z(R) such that (Z(R))5 = (0) and (Z(R))4 ≠ (0). The residue field R/Z(R) is a finite field GF(pr) for some prime p and positive integer r. The characteristic of R is pk, where k is an integer such that 1 ≤ k ≤ 5. In this paper, we study the structures of the unit groups of a commutative completely primary finite ring R with pψui = 0, ψ = 2, 3, 4; pζvj = 0, ζ = 2, 3; pwk = 0, and pyl = 0; 1 ≤ i ≤ e, 1 ≤ j ≤ f, 1 ≤ k ≤ g, and 1 ≤ l ≤ h.

An improved upper bound on self-dual codes over finite fields GF(11), GF(19), and GF(23)

This paper gives new methods of constructing symmetric self-dual codes over a finite field GF(q) where q is a power of an odd prime. These methods are motivated by the well-known Pless symmetry codes and quadratic double circulant codes. Using these methods, we construct an amount of symmetric self-dual codes over GF(11), GF(19), and GF(23) of every length less than 42. We also find 153 new self-dual codes up to equivalence: they are [32, 16, 12], [36, 18, 13], and [40, 20, 14] codes over GF(11), [36, 18, 14] and [40, 20, 15] codes over GF(19), and [32, 16, 12], [36, 18, 14], and [40, 20, 15] codes over GF(23). They all have new parameters with respect to self-dual codes. Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades.