Galois Field GF Research Articles

Block solution of sparse linear systems over GF (q)

Probabilistic algorithms based on the Krylov / Wiedemann or the Lanczos method to solve non homogeneous N x N systems Ax = b over a Galois field GF( q ), usually require 2 N matrix---vector products and O ( n 2+ o (1) ) additional arithmetic operations. Only the block Wiedemann algorithm, as given by Kaltofen in [6], has the least number (1+ε) N + O (1) of matrix---vector products of any known algorithm. We extend its analysis to the case of singular matrices A .

Subspace subcodes of Reed-Solomon codes

We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed /spl nu/-dimensional vector subspace S of GF (2/sup m/). SSRS codes are constructed using properties of the Galois field GF(2/sup m/). They are not linear over the field GF(2/sup /spl nu//), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2/sup /spl nu//) is a subfield of GF(2/sup m/), it may not be the best /spl nu/-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems.

Systolic dual basis serial multiplier

The paper presents a systolic serial dual basis multiplier for the Galois field GF(2/sup m/) which is based on the Berlekamp-like multiplication algorithm. It needs only one control signal and is easy to simplify for the field which is generated by the irreducible polynomial x/sup m/+x+1. The multiplier's architecture is highly regular, modular and easily expandable, hence it is suitable for implementation using VLSI technologies.

Frustration and overblocking on graphs

A linear vector space V( N, 2) over Galois field GF(2) has been introduced in order to investigate frustrated systems. This approach provides a linear algorithmic method to characterise the frustrated edge configurations and conditions for overblocking.

K-System Generator of Pseudorandom Numbers on Galois Field

We analyze the structure of the periodic trajectories of the K-system generator of pseudorandom numbers on a rational sublattice which coincides with the Galois field GF[p]. The period of the trajectories increases as a function of the lattice size p and the dimension of the K-matrixd. We emphasize the connection of this approach with the one which is based on primitive matrices over Galois fields.

Shape-regular polygons in finite planes

The notion of shape in the Gaussian plane was introduced by Lester [5] and extended by Artzy [1]. In this paper we generalize this notion in the affine planesAG(2,q) over the Galois fieldGF(q), q=p r andp an odd prime. We investigate the existence of shape-regular polygons and the correspondence between shape-regularity and affine-regularity.

ACE: a VLSI chip for Galois field GF(2/sup m/) based exponentiation

Finite or Galois fields are used in numerous applications like error correcting codes, digital signal processing and cryptography. These applications often require computing exponentiations in GF(2/sup m/) which is a very computationally intensive operation. The methods proposed in the literature achieve exponentiation by iterative methods using repeated multiplications and the hardware implementations use a number of Galois field multipliers in parallel resulting in expensive hardware. In this paper, we present a new algorithm based on a pattern matching technique for computing exponentiations in GF(2/sup m/), for values of m/spl les/8. A systolic array processor architecture was developed by the authors for performing multiplication and division in GF(2/sup m/) in [13]. A similar strategy is proposed in this paper for achieving exponentiation at the rate of a new result every clock cycle. A prototype VLSI chip Called ACE implementing the proposed architecture for Galois field GF(2/sup 4/) has been designed and verified using CMOS 2 /spl mu/m technology. The chip can yield a computational rate of 40 million exponentiations per second.

Diffraction of ultrasonic waves from elastic Galois gratings

A significant attenuation of the specular echo in a wide frequency domain occurs in scattering of waves by Galois gratings. Diffraction of ultrasonic waves by immersed one-dimensional elastic gratings constructed from the finite Galois fields GF(pm) is investigated. We develop a theoretical model taking into account the elasticity of the structure, its finite thickness, the boundary conditions at the fluid–solid and solid–fluid interfaces, and the periodicity of the gratings. In addition to the typical attenuation of the specular echo predicted by discrete Fourier transform, such a model predicts minima in the spectrum of the reflection coefficient corresponding to the excitation of surface waves (generalized Rayleigh and Scholte–Stoneley waves) by mode conversion and to the excitation of Lamb-type modes. The attenuation as well as the minima are experimentally observed and these results are in good agreement with theoretically predicted ones. Galois gratings could constitute an advance in the search for new anechoic devices.

On the security of Wu and Yeh's conference key distribution system

In 1993, Wu and Yeh proposed a conference key distribution system based on cross-product operation on row vectors over the Galois Field GF( P). In this research note, we show that an opponent who knows the ID of a member in the conference can compute the conference key without knowing any secret from the system. Furthermore, we suggest a modified scheme to avoid this attack.

Scattering by immersed two-dimensional elastic galois gratings

Diffraction of ultrasonic waves by immersed elastic bigratings constructed from the Galois fields GF(23) along one direction and GF(32) along the orthogonal one is studied. Such bigratings permit a significant attenuation of the specular echo in a wide frequency domain. This attenuation is experimentally observed and agrees with the theoretical one predicted by a model which takes into account the elasticity of the structure, the boundary conditions at the fluid–solid interface, and the periodicities of the gratings. Furthermore, numerous sharp minima in the spectrum of the reflection coefficient are experimentally observed and theoretically predicted; they correspond to the excitation of surface waves (generalized Rayleigh and Scholte–Stoneley waves) and they contribute significantly to the attenuation of the specular echo.

Efficient computation of quaternary fixed polarity Reed-Muller expansions

A new fast algorithm for constructing the whole polarity coefficient matrices of fixed polarity Reed-Muller expansions over Galois Field GF(4) has been introduced. The matrices are generated in a recursive way making the algorithm computationally very effective in terms of basic field operations. The number of required additions and multiplications is advantageous when compared to the known Green's algorithms. It is also possible to generate fixed polarity quaternary Reed-Muller expansion in some chosen polarity without the necessity of going through all the steps and recursions of the full algorithm. Fast flow diagrams for implementation of the new algorithms in hardware have also been shown.

Composition algebras and PG(m, 2)

Neil A. Gordon, Trevor M. Jarvis, Johannes G. Maks, Ron Shaw The multiplication law for the non-associative algebra of Cayley numbers can be expressed (in a suitable basis) in the form exey -- (-1)f(x'Y)ex+y, x, y E V, where V = V(3,2) denotes the 3-dimensional vector space over GF(2) = $~ = {0,1}. All i=2-valued functions f which give rise to a Cayley algebra are determined, as are those which give rise to the algebra of split octonions. In the case of Cayley numbers the invariance group of f is a flag-transitive subgroup of GL(3,2) which is isomorphic to ~7 >~ ~3. In the course of some more general considerations a new proof of Hurwitz's theorem is obtained. 1. INTRODUCTION Recall that a real octonion algebra is a real composition algebra of dimension eight, that is, a real algebra ~4 with unit and equipped with a non-degenerate quadratic form Q such that Q(ala2) = Q(al)Q(a2) for all ai E Jg. Up to isomorphism there are only two such algebras: the Cayley octonion algebra of signature (8,0) and the split octonion algebra of neutral signature (4,4). Let V(3,2) denote a vector space of dimension three over the Galois field GF(2) = I= 2. It is well-known that V(3,2) may be used to label an orthonormal basis {ex} of the Cayley algebra such that exey = ~- ex§ for all x,y e V(3,2): see Freudenthal [1] and Bourbaki [2], Ch.III, Appendix, Prop.4, p.616. We adopt a projective geometry point of view by considering the seven non-zero elements of V(3,2) to be the seven points in the Fano plane PG(2,2). Three points x,y,z E PG(2,2) are collinear if and only if x + y + z = 0. Each line contains three points and, dually, through each point pass exactly three lines.

Coherent states in Finite Quantum Mechanics

The “harmonic oscillator”, introduced by Itzykson and Balian, is considered, on finite phase-space lattices with the structure of Galois fields GF [ p 2]. We study this system at the classical and quantum mechanical level. Explicit construction of the quantum eigenstates provides us with a detailed understanding of the spectrum of scales relevant to this model. We construct also the coherent states appropriate to the finite Heisenberg-Weyl and affine groups. Possible applications are discussed.

Polynomial factorization in GF(2 m)

Polynomial factorization in the finite Galois field GF(2 m ) is the basis of the design of good error correcting codes. In this paper, an efficient algorithm for polynomial factorization in GF(2 m ) is proposed. Interesting properties of prime polynomial factors are deduced.

SIGMA: a VLSI systolic array implementation of a Galois field GF(2/sup m/) based multiplication and division algorithm

Finite or Galois fields are used in numerous applications like error correcting codes, digital signal processing and cryptography. The design of efficient methods for Galois field arithmetic such as multiplication and division is critical for these applications. A new algorithm based on a pattern matching technique for computing multiplication and division in GF(2/sup m/) is presented. An efficient systolic architecture is described for implementing the algorithm which can produce a new result every clock cycle and the multiplication and division operations can be interleaved. The architecture has been implemented using 2- mu m CMOS technology. The chip yields a computational rate of 33.3 million multiplications/divisions per second.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A conference key distribution system based on cross-product

Extended from the Diffie-Hellman public key distribution system (PKDS), we propose a conference key distribution system (CKDS) based on the cross-product operations on row vectors over a Galois field GF( P), where P is a prime number. In our CKDS, the chairperson computes a conference key CK and then embeds it to some public interpolating polynomials to let only the legal intended principals recover CK, while the illegal intended principals can not. From the public parameters, an intruder or any intended principal in the network does not know how many and who are the legal intended principals in the conference. Furthermore, since the construction of the CK does not interfere with the secret keys of the intended principals, any intended principals in the network has no useful information for revealing any other principals' secret keys. Besides, our CKDS can be implemented practically.

On computing multiplicative inverses in GF(2/sup m/)

The design of a modular standard basis inversion for Galois fields GF(2/sup m/) based on Euclid's algorithm for computing the greatest common divisor of two polynomials is presented. The asymptotic complexity is linear with m both in computation time and area requirement, thus resulting in an AT-complexity of O(m/sup 2/). This is a significant improvement over the best previous proposal which achieves AT-complexity of only O(m/sup 3/). >

Efficient multiplication algorithm over the finite fields GF(qm) where q=3, 5

Galois field multiplication is central to coding theory. In many applications of finite fields, there is need for a multiplication algorithm which can be realised easily on VLSI chips. In the paper, what is called the Babylonian multiplication algorithm for using tables of squares is applied to the Galois fields GF(qm). It is shown that this multiplication method for certain Galois fields eliminates the need for the division operation of dividing by four in the original Babylonian algorithm. Also, it is found that this multiplier can be used to compute complex multiplications defined on the direct sum of two identical copies of these Galois fields.

Low complexity architecture for exponentiation in GF(2 m )

A pipeline bit serial multiplier architecture for the Galois field GF (2m) is presented. A structure for finite field exponentiation is developed based on the multiplier. The structure is regular, area efficient and suitable for VLSI implementation for large fields.

Novel approaches to the design of VLSI RNS multipliers

Two approaches are proposed for the design of a fast residue number-based multiplier over a Galois field GF(p), where p is a prime number. The first approach uses an isomorphic mapping from the additive index group, modulo (p-1), of GF(p) onto the direct sum of a set of submodular additive groups. The submoduli are selected for minimizing the hardware and increasing the speed. This is accomplished by fully exploiting the properties of a Galois field. The second one uses symmetric residue number arithmetic to perform multiplication. This uses a pseudoprimitive root as the generator for the elements of the multiplicative group of GF(p) and reduces the index storage hardware by 50% and the adder hardware by 1 bit. Multipliers designed with these approaches would be faster and use less silicon area compared to earlier designs reported in the literature.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>