Theory Of Finite Fields Research Articles

Increasing the precision of estimation of the technical flight characteristics of an aircraft

The advantages gained with the use of a nontraditional theory of finite fields based on the constructive theory of residues are demonstrated. By means of this theory, it is possible to “parallelize” processing, increase solution stability, and simplify the computations and control of the reliability of the results obtained.

A Group of Threshold Group-Signature Schemes with Privilege Subsets

Feng Deng-Guo suggested a problem so called “threshold group-signature scheme with privilege subsets”. This paper analyzes the security of such schemes at present and propose new schemes. Based on theory of finite fields, the authors firstly show there are some insufficiencies and potential hazard in the scheme proposed by Shi, et al. Secondly, using the idea of constructing group-signature scheme by individual signature scheme, a group of the ones with four variants of type of ElGamal are put fordward, which have some attractive properties, such as message recovery, shorter length of signature, etc. Finally, the security of the schemes is proved under the assumption that the respective individual signature schemes are secure.

Properties of forking in ω-free pseudo-algebraically closed fields

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data:• The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field).• The degree of imperfection of K.• The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K.They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

A Simple Slide Rule for Finite Fields

We describe how an index table for a finite field can be constructed easily by hand, provided that a primitive polynomial for that field is given. Let q be a prime number and let n be a positive integer. Addition in the finite field Fqn of qn elements is very easy if one views it as an n-dimensional vector space over Fq (the integers modulo q). Likewise, multiplication in Fqn is very easy as its multiplicative group F*n is cyclic. In order to illustrate arithmetic in Fqn, involving both operations, most undergraduate algebra texts use the representation of Fqn as the factor ring of Fq [x] modulo the principal ideal generated by the minimal polynomial of Fqn over Fq. Certainly, this illuminates the concept of factor rings, but actual computations can become tedious, as reduction modulo the ideal requires long division. A very useful tool for finite field arithmetic is an index table. Before we explain this, let us briefly recall some theory of finite fields. As F*n is cyclic, every nonzero b E Fqn can be written as a power b = ai of a generator a of that group, with a unique integer exponent 0 < i < qn _ 1. This exponent is called the index of the element b, denoted by i = ind,, (b). Such a generator a is a called a primitive element of Fqnl; it is a root (in Fqn) of a primitive polynomial f (x) E Fq [x]. Conversely, every root of a primitive polynomial is a primitive element. Suppose that we are given a primitive polynomial f (x) for Fqn over Fq. Then we can tie together the multiplicative structure of Fqn with its additive structure. We fix a root a of f, and the former is just the cyclic group generated by a. The latter is a vector space, having a basis B = I 1, a, ... ., an-l1 . So every nonzero field element can be viewed as a power of a and as a linear combination of powers of a with coefficients in Fq, The index table establishes the correspondence, listing for every exponent 0 < i < qn _ 1 (as a column) the coordinates of the vector ai with respect to B. It works like a logarithm table, as the exponentiation laws for rings imply that ind,x (b . c) = ind,x (b) + ind,x (c). Hence, all arithmetic in Fqnl becomes easy: addition is vector addition modulo q, and multiplication is carried out using the table. But where do we get the index table from? For example, the standard reference for finite fields conveniently contains nearly four pages of index tables [3, pp. 546-549]. However, we can construct our own tables very easily by hand (if n and q are not too large), provided that we know a primitive polynomial f of Fqn over Fq. This construction is the aim of this Note. The reference [3] contains many primitive polynomials, too. Before we illustrate the method, let us make the notion of an index table precise:

Model theory of finite fields and pseudo-finite fields

We give a survey of results obtained in the model theory of finite and pseudo-finite fields.

The Incongruence of Consecutive Values of Polynomials

Suppose thatf∈ Z|x|. PutDf(n) = min{k> 0|f(1), ... ,f(n) are pairwise incongruent modulok}. Special cases of this function were previously considered, using methods from elementary number theory. Results from the theory of finite fields are used to prove a theorem that for allfin a large subset of Z|x| provides a characterization ofDf(n) for allnsufficiently large. This theorem partially encompasses results due to Bremser, Schumer and Washington and to Moree and Mullen, who characterizedDf(n) for cyclic, respectively, Dickson polynomials.

The combinatorial power of the companion matrix

Using combinatorial methods, we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to Waring's formula on symmetric functions, the general solution to homogeneous linear recurrence relations, the multiplicative inverse of formal power series, the generating function of compositions (of numbers), a unified approach to Chebyshev polynomials including two recently discovered classes that satisfy analogous smallest-norm and orthogonality properties subject to different weight functions, Dickson polynomials of various kinds arising from the theory of finite fields, combinatorial expansions of Toeplitz matrices, and the recent notion of cycle dissections involving a bijective study of Waring's formula.

Cellular automata and finite fields

Abstract We rewrite some concepts in the theory of one-dimensional periodic cellular automata in the language of finite fields. The state space of an automaton with N cell and q=pf possible values for each cell (p prime) is identified with the finite field of qN elements, represented by means of a normal basis. The dynamics is given by a polynomial mapping with coefficients in the field of q elements. We illustrate the dynamical significance of several constructs of the theory of finite fields, and we suggest that cellular automata may provide a novel framework for the study of the dynamics of polynomials over finite fields.

Cellular automata and finite fields

We rewrite some concepts in the theory of one-dimensional periodic cellular automata in the language of finite fields. The state space of an automaton with N cell and q=p f possible values for each cell ( p prime) is identified with the finite field of q N elements, represented by means of a normal basis. The dynamics is given by a polynomial mapping with coefficients in the field of q elements. We illustrate the dynamical significance of several constructs of the theory of finite fields, and we suggest that cellular automata may provide a novel framework for the study of the dynamics of polynomials over finite fields.

Finding irreducible and primitive polynomials

Another important area of the theory of finite fields is designing fast algorithms for finding irreducible and primitive polynomials over finite fields. These polynomials have many applications in coding theory, cryptography, complexity theory, computer science, computational mathematics (see the books [90, 601, 705, 706, 732] and the recent papers [10, 44, 143, 224, 311, 379, 448, 450, 874]).

Composed products and module polynomials over finite fields

A general notion of composition of polynomials over a finite field was introduced by Brawley and Carlitz (1987). In their paper they developed a unique decomposition theory in two special cases of the composed product, but they left open the question of uniqueness in the general case. This paper settles this open question by proving that irreducibles do indeed decompose uniquely with respect to the general notion and, in doing so, unifies the two separate treatments of Brawley and Carlitz into a single, more general theory. The paper also offers a generalization which unifies into a single theory the notions of ‘belonging to an exponent’ and ‘belonging to a q-polynomial’, which are basic in the theory of finite fields.

Computational problems in the theory of finite fields

We give a survey of selected topics in the theory of finite fields with emphasis on computational aspects including recent advances and open problems.

On pseudo-finite near-fields which have finite dimension over the centre

In [1] J. Ax studied a class of fields with similar properties as finite fields called pseudo-finite fields. One can prove that pseudo-finite fields are precisely the infinite models of the first-order theory of finite fields. Similarly a near-field F is called pseudo-finite if F is an infinite model of the first-order theory of finite near-fields. The structure theory of these near-fields has been initiated by U. Feigner in [5].

Inconsistent nonstandard arithmetic

AbstractThis paper continues the investigation of inconsistent arithmetical structures. In §2 the basic notion of a model with identity is defined, and results needed from elsewhere are cited. In §3 several nonisomorphic inconsistent models with identity which extend the (=, &lt;) theory of the usual classical denumerable nonstandard model of arithmetic are exhibited. In §4 inconsistent nonstandard models of the classical theory of finite rings and fields modulo m, i.e. Zm, are briefly considered. In §5 two models modulo an infinite nonstandard number are considered. In the first, it is shown how to model inconsistently the arithmetic of the rationals with all names included, a strengthening of earlier results. In the second, all inconsistency is confined to the nonstandard integers, and the effects on Fermat's Last Theorem are considered. It is concluded that the prospects for a good inconsistent theory of fields may be limited.

The unreasonable effectiveness of number theory in acoustics

Some of the problems that have occupied Vern O. Knudsen during his long career in acoustics have found surprising answers through applications of number theory, a field usually considered rather abstract even within mathematics. Among these unexpected applications are the distribution of normal modes in rooms, the design of surfaces (“reflection phase gratings”) that optimally diffuse sound, and the construction of new music scales. Another application is the selection of phase angles to minimize the peak factor in waveforms for speech synthesis and sonar. Finally, precision measurements of reverberation and absorption of sound in gases, pioneered by Knudsen fifty years ago [V. O. Knudsen, J. Acoust. Soc. Am. 6, 199–204 (1935)], can be made more robust by using “pings” shaped according to the theory of finite number fields.

Decidable regularly closed fields of algebraic numbers

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

Model-complete theories of e-free Ax fields

This paper's goal is to determine which complete theories of perfect, e-free Ax fields are model-complete. A field K is e-free for a positive integer e if the Galois group g(KS∣K), where Ks is the separable closure of K, is an e-free, profinite group. A perfect field K is pseudo-algebraically closed if each nonvoid, absolutely irreducible variety defined over K has a K-rational point. A perfect, pseudo-algebraically closed field is called an Ax field. The main theorem isA complete theory of e-free Ax fields is model-complete if and only if its field of absolute numbers is e-free.The sufficiency of the latter condition is an easy consequence of a result of Moshe Jarden and Ursel Kiehne [10] and has been noted independently by A. Macintyre and K. McKenna and undoubtedly by others as well. Consequently the necessity of the latter condition is the interesting part of this paper.James Ax [3] initiated the investigation of 1-free Ax fields. He proved that these fields, which he called pseudo-finite fields, are precisely the infinite models of the theory of finite fields. He [3] also presented examples of perfect, 1-free fields which are not pseudo-algebraically closed and an example of a 1-free Ax field whose complete theory is not model-complete. Moshe Jarden [5] showed that the first examples are isolated cases in that almost all, perfect, 1-free, algebraic extensions of a denumerable, Hilbertian field are pseudo-algebraically closed. The results in this paper show that the second example is also an isolated case in that almost all complete theories of 1-free Ax fields are model-complete.

The ?ebotarev density theorem for function fields: An elementary approach

The Cebotarev density theorem is one of the major results in algebraic number theory. It provides a quantitative measure of sets of prime divisors with qualitative properties related to finite Galois extensions. The basic nature of the theorem makes it very handy for applications. Thus the t~ebotarev density theorem plays a central role in the decision procedure for the theory of finite fields of Ax [3] and in the transfer principle [7] from finite fields to the fields/((a), where K is a global field. An immediate application of the theorem, namely Bauer's theorem, is the key ingredient in Neukirch's proof [9], that two finite normal extensions of ~ with isomorphic absolute Galois group must coincide; and there are many more applications. The Cebotarev density theorem is true for number fields as well as for function fields of one variable over finite fields. The number field case has attracted the most attention. There are at least three versions of the proof, that of t~ebotarev [4], that of Artin [1, 2] and that of Deuring [5]. They have also found their way to textbooks, e.g., that of Lang [8]. Serre [11, Theorem7] sketches a unified treatment for both cases, via L-series The goal of this note is to provide an elementary proof for the t~ebotarev density theorem in the function field case.

Number Theory in Physics, Engineering and Art

Several surprising applications of number theory and the theory of finite number fields, the Galois fields, to real world problems in physics, communication engineering and artistic design are here described. They indude the design of concert hall ceilings for better ound diffusion; the confirmation of the fourth prediction by Einstein's theory of general relativity (namely the slowing of electromagnetic radiation in a gravitational field); error correcting codes and encryption; and the creation of variou artistic designs.

Theory of finite fields with an additional predicate distinguishing a subfield

Theory of finite fields with an additional predicate distinguishing a subfield