Representations Of Finite Fields Research Articles

The Primitive Matrix Representation and Symmetric Matrix Representation of Finite Fields

The Primitive Matrix Representation and Symmetric Matrix Representation of Finite Fields

A Novel Realization of Threshold Schemes over Binary Field Extensions

This paper presents a novel technique to realize Karnin et al.'s (k,n)-threshold schemes over binary field extensions as a software. Our realization uses the matrix representation of finite fields and matrix-vector multiplications, and enables rapid operations in software implementation. The theoretical evaluation and computer simulation reveal that our realization of Karnin et al.'s scheme achieves much faster processing time than the ordinary symbol oriented realization of the scheme. Further, we show that our realization has comparable performance to the existing exclusive-OR-based fast schemes of Fujii et al. and Kurihara et al.

Representations of finite fields and the Naini-Seberry partitioning problem

Safavi-Naini and Seberry suggested a subliminal channel based on partitioning a set /spl Gamma/ of the generator matrices of error-correcting codes. We give a systematic procedure to partition /spl Gamma/ into E/sub i/'s with |E/sub i/|=2/sup m/ for any m, which is more efficient than the method in Yang, C-N et al., (1997).

On "A new representation of elements of finite fields GF(2/sup m/) yielding small complexity arithmetic circuits"

For original article see G. Drolet, ibid., vol. 47, no. 9, p. 938-946, (Sept 1998). We characterize the smallest n with GF(2)[X]/(X/sup n/ + 1) containing an isomorphic copy of GF(2/sup m/). This characterization shows that the representation of finite fields described in a previous issue of the IEEE Transactions on Computers is not "optimal" as claimed. The representation considered there can often be improved significantly.

Applications of the representation of finite fields by matrices

We consider the matrix well-known representation of K[X]/(P), when P is monic irreducible polynomial, with coefficients in K. This representation enables us to give a fast algorithm to solve the equation x d=a in a finite field.

Lattices of Compatibly Embedded Finite Fields

The design of a computational facility for finite fields that allows complete freedom in the manner in which fields are constructed, is complicated by the fact that a field of fixed isomorphism typeKmay be constructed in many different ways. It is desirable that the user be able to perform simultaneous computations in different versions ofKin such a way that isomorphisms identifying elements in the different versions are applied automatically whenever necessary. This paper presents a coherent scheme for solving this problem based on an efficient method for compatibly embedding one field within another. This scheme forms a central component of theMAGMAmodule for finite fields. The paper also outlines the different representations of finite fields employed in the package and comments briefly on some of the major algorithms.

Factoring polynomials using binary representations of finite fields

The factorization of polynomials over finite fields is considered. A new deterministic algorithm is proposed that solves the equal-degree factorization problem by combining Berlekamp's (1970) trace algorithm iterative method with the concept of binary representations of finite fields. The interesting aspects of the new algorithm are its simple structure, the easy proof of its correctness, and its efficiency when an efficient realization of the mapping from the finite field to a binary representation is known. Some results about binary representations of finite fields are derived to show that the new factoring algorithm is also nonasymptotically efficient for every finite field. The only practical drawback may be the precomputation of some constants needed in the binary representation, but several suggestions are given to improve this when more about the finite field is known.

Binary Representations of Finite Fields and Their Application to Complexity Theory

Binary representations of finite fields are defined as an injective mapping from a finite field tol-tuples with components in {0, 1} where 0 and 1 are elements of the field itself. This permits one to study the algebraic complexity of a particular binary representation, i.e., the minimum number of additions and multiplications in the field needed to compute the binary representation. The two-way complexity of a binary representation is defined as the sum of the algebraic complexities of the binary representation and of its inverse mapping. Two particular binary representations are studied: the standard representation and the logarithmic representation. A method of surrogate computation is developed and used to deduce relationships between the algebraic complexities of certain functions. The standard representation of a finite field is shown to be among the two-way easiest representations of this field. In particular, the standard representation of a finite field with characteristicpis two-way easy wheneverp− 1 has only small prime factors. For any finite field having a two-way easy binary representation, the algebraic complexity in this field is shown to be essentially equivalent to Boolean circuit complexity. For any finite field, the Boolean circuit complexity of Zech's (or Jacobi's) logarithm is shown to be closely related to the Boolean circuit complexity of the discrete logarithm problem that is used in public-key cryptography.

Computing Complex Convolutions Using Surrogate Fields

In this contribution we show how representations of finite fields as Gaussian integers modulo a Gaussian prime can be used for computing complex convolutions. The main idea is to switch from the representation of finite fields as Gaussian integers to the representation of a finite field as integers modulo a prime p and to perform the convolution modulo p. The transformation from the representation of the finite field as Gaussian integers modulo a Gaussian prime to the representation as integers modulo p is done via a simple formula which stems from the extended Euclidean algorithm. The transformation back can easily be done using the complex modulo function.

Matrix Representation of Finite Fields

Abstract : Finite fields (also called Galois Fields) have been studied since their introduction by Evariste Galois in 1832 and the publication of his work in 1846. In the last few decades, finite fields have become important to information theory, coding theory, and cryptography. This report presents a simple method for representing a finite field in terms of powers of a single matrix over the integers modulo the characteristic of the field. The addition and multiplication in the field are immediately obtained as the results of ordinary matrix addition and multiplication. This representation called the canonical cyclic representation, makes it easy to understand the field structure and to carry out computations in the field.

Trace- and norm-compatible extensions of finite fields

J. H. Conway suggested to use norm-compatible polynomials when constructing extension fields in order to get finite field embeddings, which are computationally easy to handle. Analogous to norm-compatibility for the multiplicative group of ¯Fq we exhibit the notion of trace-compatibility for its additive group. By this we get computationally simple embeddings in the case of normal basis representations of finite fields. We give an algorithm for computing tracecompatible polynomials and count the number of distinct compatible polynomials of a fixed degree of both kinds.

On Subcodes of Generalized Second Order Reed–Muller Codes

We study sets of quadratic forms in m variables over $GF ( q )$ such that the difference of any two quadratic forms in the set has rank m. It is easily seen that the cardinality of such a set is at most $q^m $, and we shall give four constructions of such sets which have cardinality exactly $q^m $ (where q is an odd prime power). The first construction works only in the case when m is odd, and uses a BCH code. The second is an ad hoc construction in the case when $m = 2$ The third construction is due to H. A. Wilbrink, and generalizes the second (though not in an obvious way). The fourth uses symmetric representations of finite fields and is due to G. Serousi and A. Lempel. These sets correspond to subcodes of the generalized second order Reed-Muller code, and we give the weight distribution of these codes.

On Symmetric Representations of Finite Fields

This paper presents a complete characterization of symmetric representations of finite fields, and representations that are closed under transposition. It is also shown that every finite field has a symmetric representation, and conditions are given under which closure under transposition is equivalent to symmetry.

Computation with finite fields

A technique for systematically generating representations of finite fields is presented. Relations which must be physically realized in order to implement a parallel arithmetic unit to add, multiply, and divide elements of finite fields of 2 n elements are obtained. Finally, techniques for using a maximal length linear recurring sequence to modulate a radar transmitter and the means of extracting range information from the returned sequence are derived. * Operated with support from the U.S. Army, Navy and Air Force.