Galois Field GF Research Articles

Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

Software Stream Cipher based on pGSSG Generator

Secrecy of a software stream cipher based on p-ary Generalized Self-Shrinking Generator (pGSSG) is examined in this paper. Background information for the generator’s algorithm is provided. The software architecture and key management for the cipher initialization are explained. Galois Field GF(257 32 ) and feedback polynomials are chosen for initialization of the generator. In order to examine the secrecy mathematical model of the software system is made. It is proved that the cipher is not perfect but the empirical tests result in less than 0,0125% deviation of the encrypted files’ entropy from the perfect secrecy. At last the proposed cipher is compared to four eSTREAM finalists by key length and period.

AN ELLIPTIC CURVE METRIC ON THE FUNDAMENTAL GROUP OVER $GF(2^5)$

Since the introduction of public-key cryptography by Diffe and Hellman in 1976, the potential for the use of the discrete logarithm problem in public-key cryptosystems has been recognized. Although the discrete logarithm problem as first employed by Diffe and Hellman was defined explicitly as the problem of finding logarithms with respect to a generator in the multiplicative group of the integers module a prime, this idea can be extended to arbitrary groups and in particular, to elliptic curve groups. The resulting public - key systems provide relatively small block size, high speed, and high security. In the present paper we define a metric on the fundamental group of elliptic curve over the Galois field GF(2 5 ). The fact that defining a new metric among the elliptic curves has potential application in the theory of cryptography; especially to thwart fixed table attack.

Biorthogonal quantum mechanics: super-quantum correlations and expectation values without definite probabilities

We propose mutant versions of quantum mechanics constructed on vector spaces over the finite Galois fields GF(3) and GF(9). The mutation we consider here is distinct from what we proposed in previous papers on Galois field quantum mechanics. In this new mutation, the canonical expression for expectation values is retained instead of that for probabilities. In fact, probabilities are indeterminate. Furthermore, it is shown that the mutant quantum mechanics over the finite field GF(9) exhibits super-quantum correlations (i.e. the Bell–Clauser–Horne–Shimony–Holt bound is 4). We comment on the fundamental physical importance of these results in the context of quantum gravity.

Very Fast and Economical GF(2^8) Divider Circuit Based On GF(2^2) Field Operation

This paper describes how to design a Galois field GF(2 8 )dividing circuit using double subfield transformation[5]. The circuit is much faster and more efficient than the classical GF(2 8 ) divider . Multiplier circuit is used mainly for ECC encoding and for the Encripting machine while the Divider is used for ECC decoding and the Decripting machine. The divider in the paper is designed based on the GF(2 2 ) inversing circuit and we applied double subfield transformation. We compared the new design with the classical design in both respects, speed and cost so find the new design is much better than the classical design.

Blind Identification of Nonbinary LDPC Codes Using Average LLR of Syndrome a Posteriori Probability

Prevalent adaptive modulation and coding (AMC) techniques can facilitate the flexible strategies subject to the dynamic channel quality. It would be quite intriguing for one to build a blind encoder identification technique without spectrum-efficiency sacrifice for AMC transceivers. In this paper, we make the first-ever attempt to tackle the blind nonbinary low-density parity-check (LDPC) encoder identification given a predefined encoder candidate set over the Galois field GF(q) for q-ary quadrature amplitude modulation (q-QAM) signals. Our proposed method establishes the log-likelihood ratios (LLRs) of syndrome a posteriori probabilities (APPs), which specify the potential correctness of the underlying parity-check relations, and identifies the nonbinary LDPC encoder leading to the maximum average LLR over the candidate set. Monte Carlo simulation results verify the effectiveness of our proposed new scheme.

Trellis-Based Extended Min-Sum Algorithm for Non-Binary LDPC Codes and its Hardware Structure

In this paper, we present an improvement and a new implementation of a simplified decoding algorithm for non-binary low density parity-check codes (NB-LDPC) in Galois fields GF(q). The base algorithm that we use is the Extended Min-Sum (EMS) algorithm, which has been widely studied in the recent literature, and has been shown to approach the performance of the belief propagation (BP) algorithm, with limited complexity. In our work, we propose a new way to compute modified configuration sets, using a trellis representation of incoming messages to check nodes. We call our modification of the EMS algorithm trellis-EMS (T-EMS). In the T-EMS, the algorithm operates directly on the deviation space by considering a trellis built from differential messages, which serves as a new reliability measure to sort the configurations. We show that this new trellis representation reduces the computational complexity, without any performance degradation. In addition, we show that our modifications of the algorithm allows to greatly reduce the decoding latency, by using a larger degree of hardware parallelization.

Design and Implementation of a Polynomial Basis Multiplier Architecture Over GF(2m)

In this paper, the design and circuit implementation of a polynomial basis multiplier architecture over Galois Fields GF(2m) is presented. The proposed architecture supports field multiplication of two m-term polynomials where m is a positive integer. Circuit implementations based on this parameterized architecture where m is configurable is suitable for applications in error control coding and cryptography. The proposed architecture offers low latency, polynomial basis multiplication where the irreducible polynomial P(x) = x m + p kt . x kt + … + p 1. x + 1 with m ≥ kt + 4 is dynamically reconfigurable. Results of the complexity analysis show that the proposed architecture requires less logic resources compared to existing sequential polynomial basis multipliers. In terms of timing performance, the proposed architecture has a latency of m/4, which is the lowest among the multipliers found in literature for GF(2m).

Galua Field Multipliers Core Generator

An important part of based on elliptical curves cryptographic data protection is multipliers of Galois fields. For based on elliptical curves digital signatures, not only prime but also extended Galois fields GF(pm) are used. The article provides a theoretical justification for the use of extended Galois fields GF(dm) with characteristics d > 2, and a criterion for determining the best field is presented. With the use of the proposed criterion, the best fields, which are advisable to use in data protection, are determined. Cores (VHDL descriptions of digital units) are considered as structural part of based on FPGA devices. In the article methods for cryptoprocessors cores creating were analyzed. The article describes the generator of VHDL descriptions of extended Galois field multipliers with big characteristic (up to 2998). The use of mathematical packages for calculations to improve the quality of information security is also considered. The Galois field multipliers generator creates the VHDL description of multipliers schemes, describes connections of their parts and generates VHDL descriptions of these parts as result of Quine-McCluskey Boolean functions minimization method. However, the execution time of the algorithm increases with increasing amount of input data. Accordingly, generating field multipliers with large characteristic can take frерom a few seconds to several tens of seconds. It's important to simplify the design and minimize logic gates number in a field programmable gate array (FPGA) because it will speed up the operation of multipliers. The generator creates multipliers according to the three variants. The efficiency of using multipliers for fields with different characteristics was compared in article. The expediency of using extended Galois fields GF(dm) with characteristics d > 2 in data protection tools is analyzed, a criterion for comparing data protection tools based on such Galois fields is determined, and the best fields according to the selected criterion when implemented according to a certain algorithm are determined.

Low space‐complexity digit‐serial dual basis systolic multiplier over Galois field GF(2 m ) using Hankel matrix and Karatsuba algorithm

Multiplication is one important finite field arithmetic operation in cryptographic computations. Dual basis multipliers over Galois field GF(2 m ) have been widely applied in this kind of computations because of its advantage of small chip area. Nevertheless, up to date, there are only few methods that can keep balance of low space complexity and low time complexity at the same time. In order to achieve such an efficient aim, this study presents a novel digit-serial dual basis multiplier that is different from existing ones with a modified cut-set method using Karatsuba algorithm as well as Hankel matrix. As a result, the proposed multiplier can save much space and thus be particularly suitable for some hand held devices equipped with only limited resources. The proposed digit-serial dual basis multiplier saves 55% space complexity as compared with existing similar studies with National Institute of Standards and Technology (NIST) suggested values for elliptic curve cryptosystem.

GALOIS FIELD QUANTUM MECHANICS

We construct a discrete quantum mechanics (QM) using a vector space over the Galois field GF(q). We find that the correlations in our model do not violate the Clauser–Horne–Shimony–Holt (CHSH) version of Bell's inequality, despite the fact that the predictions of this discrete QM cannot be reproduced with any hidden variable theory.

Generation of all sets of mutually unbiased bases for three-qubit systems

We propose a new method of finding the mutually unbiased bases for three qubits. The key element is the construction of the table of striation-generating curves in the discrete phase space. We derive a system of equations in the Galois field GF(8) and show that the solutions of these equations are sufficient for the construction of the general sets of complete mutually unbiased bases. A few examples are presented in order to show how our algorithm works in the following cases: striation table with three and two axes, and one and no axis in the discrete phase space.

ON SELF-INVERSE BINARY MATRICES OVER THE BINARY GALOIS FIELD

An important class of square binary matrices over t he simplest finite or Galois Field GF(2) is the cla ss of involutory or self-inverse (SI) matrices. These mat rices are of significant utility in prominent engin eering applications such as the study of the Preparata Tra nsformation or the analysis of synchronous Boolean Networks. Therefore, it is essential to devise appr opriate methods, not only for understanding the pro perties of these matrices, but also for characterizing and constructing them. We survey square binary matrices of orders 1, 2 and 3 to identify primitive SI matrices among them. Larger SI matrices are constructed as (a) the direct sum, or (b) the Kronecker product, of smalle r ones. Illustrative examples are given to demonstr ate the construction and properties of binary SI matrices. The intersection of the sets of SI and permutation binary matrices is studied. We also study higher-order SI binary matrices and describe them via recursive rel ations or Kronecker products. Our work culminates in an exposition of the two most common representations of Boolean functions via two types of Boolean SI matri ces. A better understanding of the properties and methods of constructing SI binary matrices over GF (2) is achieved. A clearer picture is attained abou t the utility of binary matrices in the representation of Boolean functions.

Nonbinary LDPC Coding and Iterative Decoding System With 2-D Equalizer for TDMR R/W Channel Using Discrete Voronoi Model

A 2-D magnetic recording (TDMR) by shingled magnetic recording (SMR) is one of the most promising technologies for realizing ultra-high areal densities. We have developed the discretized granular medium model with nonmagnetic grain boundaries and the simple writing process considering intergranular exchange fields and magnetostatic interaction fields between grains on the discrete Voronoi model for TDMR. In this paper, the bit-error rate (BER) performance of the iterative decoding system using a nonbinary low-density parity-check (LDPC) code over Galois field GF(q) with the 2-D finite-impulse-response equalizer (2D-FIRE) is obtained via computer simulation using an R/W channel model employing the writing process under TDMR specifications of 4.12 Tb/in2, and it is compared to that with the 1-D FIRE (1D-FIRE). The results show that the BER performance of the nonbinary LDPC coding and iterative decoding system with the 2D-FIRE is better than that with the 1D-FIRE.

Spin and rotations in Galois field quantum mechanics

We discuss the properties of Galois field quantum mechanics constructed on a vector space over the finite Galois field GF(q). In particular, we look at two-level systems analogous to spin, and discuss how SO(3) rotations could be embodied in such a system. We also consider two-particle ‘spin’ correlations and show that the Clauser–Horne-Shimony–Holt inequality is nonetheless not violated in this model.

Optical and THz Galois diffusers

Binary surface reliefs with sub-wavelength features making up a pseudorandom pattern based on mathematical Galois fields GF(pm) [1, 2] can scatter incoming waves into a large number of diffraction maxima within a huge solid angle. A one-dimensional (1D) Galois number sequence can be folded into a two-dimensional (2D) array by the sino-representation [2]. This concept was been verified for acoustic waves a long time ago [3, 4] and is investigated here for visible light and THz waves. Our Galois diffusers are designed as reflection reliefs and realised by electron beam lithography for the optical regime and UV photolithography for the THz regime. Our results show that optical and THz Galois surfaces are excellent diffusers for electromagnetic waves; they distribute the reflected intensity evenly over a large number of maxima nearly within the entire half solid angle in the backward direction.

Construction of New S-Boxes Over Finite Field and Their Application to Watermarking

In this work, we develop an imperceptible watermarking technique for images that employ substitution boxes constructed over Galois field GF(24). The strength of the proposed substitution box (S-box) is analyzed and its suitability is investigated for watermarking applications by applying statistical methods, which include entropy, contrast, correlation, energy, homogeneity, mean of absolute deviation (MAD), mean square error (MSE), peak-to-peak signal to noise ratio (PSNR), and structural similarity (SSIM) paradigm analysis. The application of the proposed S-box is presented for embedding copyright information in images.

A Low-Power Parallel Architecture for Finite Galois Field GF(2m) Arithmetic Operations for Elliptic Curve Cryptography

A Low-Power Parallel Architecture for Finite Galois Field GF(2<SUP>m</SUP>) Arithmetic Operations for Elliptic Curve Cryptography

S8 affine-power-affine S-boxes and their applications

The encryption process relies on the use of nonlinear mapping subsystems to create confusion in the ciphertext. The design of these nonlinear components is a challenging task and requires complex algebraic expression for their descriptions. In an effort to increase the complexity of nonlinear mappings, several implementations exhibiting interesting properties are proposed in the literature. In particular, affine-power-affine structure is designed for advanced encryption standard, which improves the complexity of its algebraic expression by increasing the number of terms. Based on the characteristics of affine-power-affine structure, we propose a new nonlinear component that uses the symmetric group permutation S8 on the Galois field GF(28) elements and provides the possibility to incorporate 40320 unique instances. A rigorous analysis is presented to evaluate the properties of these new nonlinear components by applying nonlinearity analysis, linear approximation analysis, differential approximation analysis, bit independence criterion and strict avalanche criterion. In order to determine the suitability to various encryption applications, the S-boxes are tested with generalized majority logic criterion.

Reversible secret image sharing with steganography and dynamic embedding

ABSTRACTMany traditional steganography methods do not disperse and hide secret data smoothly over all the capacity of cover images. Rather, they severely modify part of the cover image(s) to embed secret data, which results in stego images that have poor visual quality. In addition, there are still some secret image‐sharing approaches that cannot reveal the secret image losslessly without pixel expansion or extra storage or restore distortion‐free cover image(s) if they use steganography. In this paper, a novel scheme, which is based on Shamir's (t, n)‐threshold scheme (1979) and Galois Field GF(28) and uses dynamic embedding and least significant bit construction, is proposed to solve the issues mentioned above. Our experimental results showed that the dynamic embedding performance in our scheme was satisfactory and that both the secret image and the cover image can be restored losslessly without pixel expansion or extra storage. Copyright © 2012 John Wiley &amp; Sons, Ltd.