Field extensions

For an -extension of an algebraic number field satisfying certain appropriate conditions the author obtains a formula analogous to the Riemann-Hurwitz formula. This formula connects the Iwasawa invariants of the fields and , where is some -extension of the field . It is not assumed that and are fields of CM-type.

Recovery of a manifold with boundary and its continuity as a function of its metric tensor

A new coiled tubing (CT) manufacturing technology was developed and introduced to commercial field operations in 2015. The new technology uses new steels and new heat treatment processes for complete microstructure transformation to produce significantly improved and uniform microstructure and properties along the entire coiled tubing string, including bias welds. Numerous straight and tapered wall thickness CT strings have been commercially deployed and used in various field operations and environments. Information has been collected for CT field performance evaluation of the new technology 125-ksi and 110-ksi minimum yield strength CT grades. Field results show new technology CT performance and utilization far beyond any conventional CT capability and experience. Additionally, testing of CT strings after extensive field use shows significant remaining string life and confirms observations and conclusions from previous extensive laboratory and field testing.The new CT manufacturing technology also has been extended to produce a new, 95-ksi yield strength CT grade. This new CT grade has significantly better general performance than comparable conventional CT grades. It also has substantially better performance in sour, aqueous H2S-containing environments. Non-sour and sour test results for the new CT grade and comparisons with other CT grades are presented in the paper. Sour fatigue tests were conducted on full-body CT samples after 4-day sour exposure. Sulfide stress cracking and sour fatigue testing in H2S environments show that the new 95-ksi CT grade has much better sour performance than the conventional 90-ksi grade that is currently the highest CT grade regularly used in sour environments.

Affine-invariant codes are extended cyclic codes of length p^m invariant under the affine-group acting on {\F}_{p^m}. This class of codes includes codes of great interest such as extended narrow-sense BCH codes. In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1. We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools. In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field.

This paper proposes an exponentiation method with Frobenius mappings. The main target is an exponentiation in an extension field. This idea can be applied for scalar multiplication of a rational point of an elliptic curve defined over an extension field. The proposed method is closely related to so-called interleaving exponentiation. Unlike interleaving exponentiation methods, it can carry out several exponentiations of the same base at once. This happens in some pairing-based applications. The efficiency of using Frobenius mappings for exponentiation in an extension field was well demonstrated by Avanzi and Mihailescu. Their exponentiation method efficiently decreases the number of multiplications by inversely using many Frobenius mappings. Compared to their method, although the number of multiplications needed for the proposed method increases about 20%, the number of Frobenius mappings becomes small. The proposed method is efficient for cases in which Frobenius mapping cannot be carried out quickly.

This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field Fp using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.

Results and questions on matchings in abelian groups and vector subspaces of fields

A method for polynomial multiplication over finite fields using field extensions and polynomial interpolation is introduced. The proposed method uses polynomial interpolation as Toom-Cook method together with field extensions. Furthermore, the proposed method can be used when Toom-Cook method cannot be applied directly. Explicit formulae improving the previous results in many cases are obtained.

We define a diagram associated with a purely inseparable field extension of finite exponent. We show that, under this definition, for any given field extension the shape of its diagram is unique. Thus our diagram improves the diagram defined by Sweedler in [2, p. 402]. In §2 we define an “allowable” shape for a diagram. Given any “allowable” shape for a diagram representing a finite field extension, we construct a field extension whose diagram has that shape.

The meeting Pro- p Extensions of Global Fields and pro- p Groups was organised by Nigel Boston (Madison), John Coates (Cambridge) and Fritz Grunewald (Düsseldorf). As the name of the meeting conveys, a primary aim was to bring together group theorists working in the field of pro- p groups and number theorists interested in pro- p extensions of global fields. The workshop consisted of over 25 talks, supplemented by informal discussions. Topics included: Galois groups of extensions with restricted ramification, self-similar and automata groups, non-commutative Iwasawa theory, groups acting on rooted trees. This meeting was well attended with over 50 participants; more than 30 of these came from countries other than Germany. The range of topics and the diverse backgrounds of the participants led to a stimulating exchange of recent results, challenging problems and general ideas. The organisers and participants thank the Mathematisches Forschungsinstitut Oberwolfach for providing the setting for this successful workshop. The following extended abstracts appear in the chronological order of the talks; they were collected and edited by Benjamin Klopsch (Düsseldorf).

In modern days security issues play a vital and integral part of information technology and its related applications such as ATM and smart cards. Cryptography has evolved as a important element of security process. Elliptic curve cryptography is used to develop a variety of scheme for security purpose. Point addition and point doubling arithmetic used in elliptic curve is applicable for prime field. When same arithmetic performs over extension field it increases the complexity of the application but enhances the security in the field of communication technology. This paper communicates Weierstrass equation as the root of elliptic curve. Also described the group operation on elliptic curve that is point addition and point doubling used for encryption application based on prime field Fp. The proposed work is based on elliptic curve group operations solved in extension field F p 2. This new approach is verified using System for Algebra and Geometry and Experimentation (SAGE) open source software.

In this chapter we shall analyze arbitrary extension fields of a field Φ. A study of finite dimensional extension fields and a partial study of algebraic extensions has been made in Chapter I. In this chapter our primary concern will be with infinite dimensional extensions and we shall begin again with the algebraic ones. We define algebraically closed fields and prove the existence of an algebraic closure of any field. We shall extend the classical Galois theory to apply to infinite dimensional normal and separable extensions. After this we shall consider arbitrary extension fields and we shall show that these can be built up in two stages: first a purely transcendental one and then on top of this an algebraic extension. The invariant of this mode of generating a field is the transcendency degree which is the cardinal number of a transcendency basis. We shall obtain conditions for the existence of a transcendency basis such that the extension is separable algebraic over the purely transcendental extension determined by the basis. We shall also give a definition of separability of an extension field that generalizes the notion of algebraic separability. The notion of a derivation plays an important role in these considerations.KeywordsPrime IdealStructure TheoryGalois GroupHigh DerivationAlgebraic ClosureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A recognition model which defines a measure of shape similarity on the direct output of multiscale and multiorientation Gabor filters does not manifest qualitative aspects of human object recognition of contour-deleted images in that: (a) it recognizes recoverable and nonrecoverable contour-deleted images equally well whereas humans recognize recoverable images much better, and (b) it distinguishes complementary feature-deleted images whereas humans do not. Adding some of the known connectivity pattern of the primary visual cortex to the model in the form of extension fields (connections between collinear and curvilinear units) among filters increased the overall recognition performance of the model and: (a) boosted the recognition rate of the recoverable images far more than the nonrecoverable ones, and (b) increased the similarity of complementary feature-deleted images, but not part-deleted ones, more closely corresponding to human psychophysical results. Interestingly, the performance was approximately equivalent for narrow (/spl plusmn/15/spl deg/) and broad (/spl plusmn/90/spl deg/) extension fields. The described method is most promising for the processing of noisy input images.

The RSA public key cryptosystem is based on a single modular equation in one variable. A natural generalization of this approach is to consider systems of several modular equations in several variables. In this paper we consider Patarin’s Hidden Field Equations (HFE) scheme, which is believed to be one of the strongest schemes of this type. We represent the published system of multivariate polynomials by a single univariate polynomial of a special form over an extension field, and use it to reduce the cryptanalytic problem to a system of ∈m 2 quadratic equations in m variables over the extension field. Finally, we develop a new relinearization method for solving such systems for any constant ∈ > 0 in expected polynomial time. The new type of attack is quite general, and in a companion paper we use it to attack other multivariate algebraic schemes, such as the Dragon encryption and signature schemes. However, we would like to emphasize that the polynomial time complexities may be infeasibly large for some choices of the parameters, and thus some variants of these schemes may remain practically unbroken in spite of the new attack.KeywordsQuadratic EquationSignature SchemeUnivariate PolynomialMultivariate PolynomialMultivariate Signature SchemeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Network Time Protocol version 4 (NTPv4) defines the optional usage of extension fields. An extension field, as defined in RFC 5905, is an optional field that resides at the end of the NTP header and that can be used to add optional capabilities or additional information that is not conveyed in the standard NTP header. This document updates RFC 5905 by clarifying some points regarding NTP extension fields and their usage with Message Authentication Codes (MACs).