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| In [[mathematics]], an '''infrastructure''' is a [[Group (mathematics)|group]]-like structure appearing in [[global field]]s.
| | CMS provides the best platform to create websites that fulfill all the specifications of SEO. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. s ultimately easy to implement and virtually maintenance free. After activating, you will find their website link and get the activation code from their website. <br><br>Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. Setting Up Your Business Online Using Free Wordpress Websites. Enough automated blog posts plus a system keeps you and your clients happy. There are plenty of tables that are attached to this particular database. <br><br>A built-in widget which allows you to embed quickly video from popular websites. * Robust CRM to control and connect with your subscribers. A higher percentage of women are marrying at older ages,many are delaying childbearing until their careers are established, the divorce rate is high and many couples remarry and desire their own children. If you are looking for Hire Wordpress Developer then just get in touch with him. Where from they are coming, which types of posts are getting top traffic and many more. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. If you adored this post and you would such as to receive additional facts regarding [http://shortener.us/wordpress_backup_1441417 wordpress dropbox backup] kindly go to the web site. When you sign up with Wordpress, you gain access to several different templates and plug-in that allow you to customize your blog so that it fits in with your business website design seamlessly. You can select color of your choice, graphics of your favorite, skins, photos, pages, etc. You should stay away from plugins that are full of flaws and bugs. You can check out the statistics of page of views for your web pages using free tools that are available on the internet. |
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| == Historic development ==
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| In 1972, [[Daniel Shanks|D. Shanks]] first discovered the infrastructure of a [[Quadratic field|real quadratic number field]] and applied his [[baby-step giant-step]] algorithm to compute the [[Dirichlet's unit theorem#The regulator|regulator]] of such a field in <math>\mathcal{O}(D^{1/4+\varepsilon})</math> binary operations (for every <math>\varepsilon > 0</math>), where <math>D</math> is the [[Quadratic field#Discriminant|discriminant]] of the quadratic field; previous methods required <math>\mathcal{O}(D^{1/2+\varepsilon})</math> binary operations.<ref name="shanks-infrastructure">D. Shanks: The infrastructure of a real quadratic field and its applications. Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972), pp. 217-224. University of Colorado, Boulder, 1972. {{MR|389842}}</ref> Ten years later, [[Hendrik Lenstra|H. W. Lenstra]] published<ref name="lenstra-infrastructure">H. W. Lenstra Jr.: On the calculation of regulators and class numbers of quadratic fields. Number theory days, 1980 (Exeter, 1980), 123–150, London Math. Soc. Lecture Note Ser., 56, Cambridge University Press, Cambridge, 1982. {{MR|697260}}</ref> a mathematical framework describing the infrastructure of a real quadratic number field in terms of "circular groups". It was also described by R. Schoof<ref name="schoof-infrastructure1">R. J. Schoof: Quadratic fields and factorization. Computational methods in number theory, Part II, 235–286, Math. Centre Tracts, 155, Math. Centrum, Amsterdam, 1982. {{MR|702519}}</ref> and H. C. Williams,<ref name="williams-infrastructure1">H. C. Williams: Continued fractions and number-theoretic computations. Number theory (Winnipeg, Man., 1983). Rocky Mountain J. Math. 15 (1985), no. 2, 621–655. {{MR|823273}}</ref> and later extended by H. C. Williams, G. W. Dueck and B. K. Schmid to certain [[Cubic field|cubic number fields]] of [[Dirichlet's unit theorem|unit rank]] one<ref name="williams-dueck-schmid">H. C. Williams, G. W. Dueck, B. K. Schmid: A rapid method of evaluating the regulator and class number of a pure cubic field. Math. Comp. 41 (1983), no. 163, 235–286. {{MR|701638}}</ref><ref name="williams-dueck">G. W. Dueck, H. C. Williams: Computation of the class number and class group of a complex cubic field. Math. Comp. 45 (1985), no. 171, 223–231. {{MR|790655}}</ref> and by J. Buchmann and H. C. Williams to all number fields of unit rank one.<ref name="buchmann-williams-infrastructure">J. Buchmann, H. C. Williams: On the infrastructure of the principal ideal class of an algebraic number field of unit rank one. Math. Comp. 50 (1988), no. 182, 569–579. {{MR|929554}}</ref> In his [[Habilitation|habilitation thesis]], J. Buchmann presented a baby-step giant-step algorithm to compute the regulator of a number field of ''arbitrary'' unit rank.<ref name="buchmann-habil">J. Buchmann: Zur Komplexität der Berechnung von Einheiten und Klassenzahlen algebraischer Zahlkörper. Habilitationsschrift, Düsseldorf, 1987. [http://www.cdc.informatik.tu-darmstadt.de/~buchmann/Lecture%20Notes/habil.pdf PDF]</ref> The first description of infrastructures in number fields of arbitrary unit rank was given by R. Schoof using [[Arakelov divisor]]s in 2008.<ref>R. Schoof: Computing Arakelov class groups. (English summary) Algorithmic number theory: lattices, number fields, curves and cryptography, 447–495, Math. Sci. Res. Inst. Publ., 44, Cambridge University Press, 2008. {{MR|2467554}} [http://www.mat.uniroma2.it/~schoof/papers.html PDF]</ref>
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| The infrastructure was also described for other [[global field]]s, namely for [[algebraic function field]]s over [[finite field]]s. This was done first by A. Stein and H. G. Zimmer in the case of real [[Hyperelliptic curve|hyperelliptic]] function fields.<ref name="stein-zimmer">A. Stein, H. G. Zimmer: An algorithm for determining the regulator and the fundamental unit of hyperelliptic congruence function field. In "Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC '91," Association for Computing Machinery, (1991), 183–184.</ref> It was extended to certain cubic function fields of unit rank one by R. Scheidler and A. Stein.<ref name="stein-scheidler-cubicinfra">R. Scheidler, A. Stein: Unit computation in purely cubic function fields of unit rank 1. (English summary) Algorithmic number theory (Portland, OR, 1998), 592–606, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. {{MR|1726104}}</ref><ref name="scheidler-infrapurecubic">R. Scheidler: Ideal arithmetic and infrastructure in purely cubic function fields. (English, French summary) J. Théor. Nombres Bordeaux 13 (2001), no. 2, 609–631. {{MR|1879675}}</ref> In 1999, S. Paulus and H.-G. Rück related the infrastructure of a real quadratic function field to the divisor class group.<ref name="paulus-rueck">S. Paulus, H.-G. Rück: Real and imaginary quadratic representations of hyperelliptic function fields. (English summary) Math. Comp. 68 (1999), no. 227, 1233–1241. {{MR|1627817}}</ref> This connection can be generalized to arbitrary function fields and, combining with R. Schoof's results, to all global fields.<ref name="fontein-infrastructure">{{cite journal | first=F. | last=Fontein | title=The Infrastructure of a Global Field of Arbitrary Unit Rank | journal=Math. Comp. | volume=80 | year=2011 | number=276 | pages=2325–2357 | doi= 10.1090/S0025-5718-2011-02490-7 | arxiv=0809.1685 }}</ref> | |
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| == The one-dimensional case ==
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| === An abstract definition ===
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| A '''one-dimensional (abstract) infrastructure''' <math>(X, d)</math> consists of a [[real number]] <math>R > 0</math>, a [[finite set]] <math>X \neq \emptyset</math> together with an [[Injective function|injective]] map <math>d : X \to \mathbb{R}/R\mathbb{Z}</math>.<ref name="fontein-pohlighellman">F. Fontein: Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures. (English summary) Adv. Math. Commun. 2 (2008), no. 3, 293–307. {{MR|2429459}}</ref> The map <math>d</math> is often called the ''distance map''.
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| By interpreting <math>\mathbb{R}/R\mathbb{Z}</math> as a [[circle]] of [[circumference]] <math>R</math> and by identifying <math>X</math> with <math>d(X)</math>, one can see a one-dimensional infrastructure as a circle with a finite set of points on it.
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| === Baby steps ===
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| A '''baby step''' is an [[unary operation]] <math>bs : X \to X</math> on a one-dimensional infrastructure <math>(X, d)</math>. Visualizing the infrastructure as a circle, a baby step assigns each point of <math>d(X)</math> the next one. Formally, one can define this by assigning to <math>x \in X</math> the real number <math>f_x := \inf\{ f' > 0 \mid d(x) + f' \in d(X) \}</math>; then, one can define <math>bs(x) := d^{-1}(d(x) + f_x)</math>.
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| === Giant steps and reduction maps ===
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| Observing that <math>\mathbb{R}/R\mathbb{Z}</math> is naturally an [[Abelian group]], one can consider the sum <math>d(x) + d(y) \in \mathbb{R}/R\mathbb{Z}</math> for <math>x, y \in X</math>. In general, this is not an element of <math>d(X)</math>. But instead, one can take an element of <math>d(X)</math> which lies ''nearby''. To formalize this concept, assume that there is a map <math>red : \mathbb{R}/R\mathbb{Z} \to X</math>; then, one can define <math>gs(x, y) := red(d(x) + d(y))</math> to obtain a [[binary operation]] <math>gs : X \times X \to X</math>, called the '''giant step''' operation. Note that this operation is in general ''not'' [[Associativity|associative]].
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| The main difficulty is how to choose the map <math>red</math>. Assuming that one wants to have the condition <math>red \circ d = \mathrm{id}_X</math>, a range of possibilities remain. One possible choice<ref name="fontein-pohlighellman" /> is given as follows: for <math>v \in \mathbb{R}/R\mathbb{Z}</math>, define <math>f_v := \inf\{ f \ge 0 \mid v - f \in d(X) \}</math>; then one can define <math>red(v) := d^{-1}(v - f_v)</math>. This choice, seeming somewhat arbitrary, appears in a natural way when one tries to obtain infrastructures from global fields.<ref name="fontein-infrastructure" /> Other choices are possible as well, for example choosing an element <math>x \in d(X)</math> such that <math>|d(x) - v|</math> is minimal (here, <math>|d(x) - v|</math> is stands for <math>\inf\{ |f - v| \mid f \in d(x) \}</math>, as <math>d(x)</math> is of the form <math>v + R\mathbb{Z}</math>); one possible construction in the case of real quadratic hyperelliptic function fields is given by S. D. Galbraith, M. Harrison and D. J. Mireles Morales.<ref name="galbraith-harrison-mirelesmorales">S. D. Galbraith, M. Harrison, D. J. Mireles Morales: Efficient hyperelliptic arithmetic using balanced representation for divisors. (English summary) Algorithmic number theory, 342–356, Lecture Notes in Comput. Sci., 5011, Springer, Berlin, 2008. {{MR|2467851}}</ref>
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| === Relation to real quadratic fields ===
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| D. Shanks observed the infrastructure in real quadratic number fields when he was looking at cycles of reduced [[binary quadratic form]]s. Note that there is a close relation between reducing binary quadratic forms and [[continued fraction]] expansion; one step in the continued fraction expansion of a certain [[quadratic irrational]]ity gives an [[unary operation]] on the set of reduced forms, which cycles through all reduced forms in one equivalence class. Arranging all these reduced forms in a cycle, Shanks noticed that one can quickly jump to reduced forms further away from the beginning of the circle by [[Composition of binary quadratic forms|composing]] two such forms and reducing the result. He called this [[binary operation]] on the set of reduced forms a '''giant step''', and the operation to go to the next reduced form in the cycle a '''baby step'''.
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| === Relation to <math>\mathbb{R}/R\mathbb{Z}</math> ===
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| The set <math>\mathbb{R}/R\mathbb{Z}</math> has a natural group operation and the giant step operation is defined in terms of it. Hence, it makes sense to compare the arithmetic in the infrastructure to the arithmetic in <math>\mathbb{R}/R\mathbb{Z}</math>. It turns out that the group operation of <math>\mathbb{R}/R\mathbb{Z}</math> can be described using giant steps and baby steps, by representing elements of <math>\mathbb{R}/R\mathbb{Z}</math> by elements of <math>X</math> together with a relatively small real number; this has been first described by D. Hühnlein and S. Paulus<ref name="huehnlein-paulus">D. Hühnlein, S. Paulus: On the implementation of cryptosystems based on real quadratic number fields (extended abstract). Selected areas in cryptography (Waterloo, ON, 2000), 288–302, Lecture Notes in Comput. Sci., 2012, Springer, 2001. {{MR|1895598}}</ref> and by M. J. Jacobson, Jr., R. Scheidler and H. C. Williams<ref name="jacobson-scheidler-williams">M. J. Jacobson Jr., R. Scheidler, H. C. Williams: The efficiency and security of a real quadratic field based key exchange protocol. Public-key cryptography and computational number theory (Warsaw, 2000), 89–112, de Gruyter, Berlin, 2001 {{MR|1881630}}</ref> in the case of infrastructures obtained from real quadratic number fields. They used floating point numbers to represent the real numbers, and called these representations CRIAD-representations resp. <math>(f, p)</math>-representations. More generally, one can define a similar concept for all one-dimensional infrastructures; these are sometimes called <math>f</math>-representations.<ref name="fontein-pohlighellman" />
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| A '''set of <math>f</math>-representations''' is a subset <math>fRep</math> of <math>X \times \mathbb{R}/R\mathbb{Z}</math> such that the map <math>\Psi_{fRep} : fRep \to \mathbb{R}/R\mathbb{Z}, \; (x, f) \mapsto d(x) + f</math> is a bijection and that <math>(x, 0) \in fRep</math> for every <math>x \in X</math>. If <math>red : \mathbb{R}/R\mathbb{Z} \to X</math> is a reduction map, <math>fRep_{red} := \{ (x, f) \in X \times \mathbb{R}/R\mathbb{Z} \mid red(d(x) + f) = x \}</math> is a set of <math>f</math>-representations; conversely, if <math>fRep</math> is a set of <math>f</math>-representations, one can obtain a reduction map by setting <math>red(f) = \pi_1(\Psi_{fRep}^{-1}(f))</math>, where <math>\pi_1 : X \times \mathbb{R}/R\mathbb{Z} \to X, \; (x, f) \mapsto x</math> is the projection on $X$. Hence, sets of <math>f</math>-representations and reduction maps are in a [[one-to-one correspondence]].
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| Using the bijection <math>\Psi_{fRep} : fRep \to \mathbb{R}/R\mathbb{Z}</math>, one can pull over the group operation on <math>\mathbb{R}/R\mathbb{Z}</math> to <math>fRep</math>, hence turning <math>fRep</math> into an abelian group <math>(fRep, +)</math> by <math>x + y := \Psi_{fRep}^{-1}(\Psi_{fRep}(x) + \Psi_{fRep}(y))</math>, <math>x, y \in fRep</math>. In certain cases, this group operation can be explicitly described without using <math>\Psi_{fRep}</math> and <math>d</math>.
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| In case one uses the reduction map <math>red : \mathbb{R}/R\mathbb{Z} \to X, \; v \mapsto d^{-1}(v - \inf\{ f \ge 0 \mid v - f \in d(X) \})</math>, one obtains <math>fRep_{red} = \{ (x, f) \mid f \ge 0, \; \forall f' \in [0, f) : d(x) + f' \not\in d(X) \}</math>. Given <math>(x, f), (x', f') \in fRep_{red}</math>, one can consider <math>(x'', f'')</math> with <math>x'' = gs(x, x')</math> and <math>f'' = f + f' + (d(x) + d(x') - d(gs(x, x'))) \ge 0</math>; this is in general no element of <math>fRep_{red}</math>, but one can reduce it as follows: one computes <math>bs^{-1}(x'')</math> and <math>f'' - (d(x'') - d(bs^{-1}(x'')))</math>; in case the latter is not negative, one replaces <math>(x'', f'')</math> with <math>(bs^{-1}(x''), f'' - (d(x'') - d(bs^{-1}(x''))))</math> and continues. If the value was negative, one has that <math>(x'', f'') \in fRep_{red}</math> and that <math>\Psi_{fRep_{red}}(x, f) + \Psi_{fRep_{red}}(x', f') = \Psi_{fRep_{red}}(x'', f'')</math>, i.e. <math>(x, f) + (x', f') = (x'', f'')</math>.
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| == References ==
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| {{Reflist}}
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| {{DEFAULTSORT:Infrastructure (Number Theory)}}
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| [[Category:Algebra]]
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| [[Category:Algebraic structures]]
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