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Topic: Finite field arithmetic

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	Finite field arithmetic - Wikipedia, the free encyclopedia
		Finite fields are used in a variety of applications, including linear block codes such as BCH and RS codes in classical coding theory and in cryptography algorithms such as the Rijndael encryption algorithm.
		In a finite field with characteristic 2, addition and subtraction are identical, and are accomplished using the XOR operator.
		Multiplication in a finite field is multiplication modulo an irreducible reducing polynomial used to define the finite field.
	en.wikipedia.org /wiki/Finite_field_arithmetic (1005 words)

	Finite field - Wikipedia, the free encyclopedia
		Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory.
		The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned here in the article about fields.
		Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields.
	en.wikipedia.org /wiki/Finite_field (1268 words)

	Introduction
		Although two finite fields of the same cardinality are isomorphic, in practical applications it is often important to be guaranteed to work in a field defined by a specific polynomial.
		The prime field of a finite field F is the unique field of cardinality p, the characteristic of F. Here we mean unique not just in the mathematical sense, but in the sense that all prime fields of the same cardinality are identical in Magma, and their elements are denoted 0, 1,..., p - 1.
		For finite fields for which the complete table of Zech logarithms is stored (and which must therefore be small), printing of elements can be done in two ways: either as powers of the primitive element or as polynomials in the generating element.
	www.math.uiuc.edu /Software/magma/text306.html (1156 words)

	Finite field inverse circuit for use in an elliptic curve processor (US5982895) (Site not responding. Last check: 2007-10-21)
		A finite field inverse circuit (600) for use in an elliptic curve processor (12).
		The finite field inverse circuit (600) comprises a control circuit (610) and a data circuit (660).
		A first plurality of bits representing the finite field element to be inverted is initially loaded into a first one of the three registers.
	www.delphion.com /details?pn=US05982895__ (295 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		In the case where K is a small finite field, composition series may be found for modules of dimension up to 20,000, while (in V2.3), if K is the rational field, composition series may be found for dimensions up to 500-600.
		If R is a field, a new fast algorithm due to Allan Steel is used to construct the various matrix canonical forms: Jordan, generalized Jordan, rational, primary rational etc, while if R is an ED, recent algorithms of Havas and others are used to compute characteristic polynomials and the Hermite and Smith normal forms.
		FINITE PLANES: Although finite planes correspond to particular families of designs, separate categories are provided for both projective and affine planes in order to exploit the rich structure possessed by these objects.
	www.symbolicnet.org /ftpsoftware/magmaa2.2.txt (3296 words)

	SPFFL
		Integers, residue class rings (finite fields when the modulus is a prime number), quotient fields (small rationals).
		(finite fields when the modulus is an irreducible polynomial), quotient fields (rational functions).
		Arithmetic for integers and polynomial rings, residue class rings/fields, and quotient fields.
	members.cox.net /~kerlj/src/spffl/doc/spffl.html (991 words)

	Finite Fields
		Finite fields are the general starting point for the constructions of many combinatorial structures.
		We denote the group of all automorphisms of a field L by G(L) and the subgroup of G(L) that fixes all elements of the subfield K of L by G(L/K).
		When working with finite fields it is convenient to have both of the above representations, since the terms on the left are easy to multiply and the terms on the right are easy to add.
	www-math.cudenver.edu /~wcherowi/courses/finflds.html (3085 words)

	IEEE P1363a: Additional number-theoretic algorithms
		This summary of finite field basis conversion techniques is proposed for inclusion in IEEE P1363 Annex A. Included are some conventional basis conversion techniques, as well as some new storage-efficient basis conversion techniques.
		Finite field arithmetic is becoming increasingly important in cryptographic applications.
		It is well known that the efficiency of finite field arithmetic depends strongly on the particular way in which the field elements are represented.
	grouper.ieee.org /groups/1363/P1363a/NumThAlgs.html (715 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Furthermore, the fields of characteristic two are preferred since they provide carry-free arithmetic and at the same time a simple way to represent field elements on current processor architectures.
		Arithmetic in finite field is analogous to the arithmetic of integers.
		The fundamental arithmetic operations in finite fields are addition, multiplication, and inversion operations.
	islab.oregonstate.edu /papers/98Sunar.html (262 words)

	Circuit and method for decompressing compressed elliptic curve points (US6199086) (Site not responding. Last check: 2007-10-21)
		A storage element (250) is coupled to the finite field arithmetic logic unit (122).
		The EC control unit (123) controls the use of the operation register (124), the storage element (250) and the finite field arithmetic logic unit (122) to recursively compute the decompressed version of the compressed Y coordinate based upon the X coordinate and the compressed one-bit representation of the Y coordinate.
		a finite field arithmetic logic unit comprising a finite field square circuit, a finite field inverse circuit, a finite field multiplier circuit, and a finite field adder circuit;
	www.delphion.com /details?pn=US06199086__ (403 words)

	(No title yet)
		Over finite fields of small characteristic, the two models are equally powerful when size is considered, but Boolean circuits are exponentially more powerful than arithmetic circuits with respect to depth.
		The processor-efficiency of parallel algorithms for exponentiation in a finite field extension is studied, assuming that a normal basis over the ground field is given.
		Optimal sequential and parallel algorithms for exponentiation in a finite field extension are presented, assuming that a normal basis over the ground field is given.
	math-www.uni-paderborn.de /~aggathen/Publications/abst.html (2632 words)

	An intro to Elliptical Curve Cryptography
		Fp is the field of integers modulo p, and consists of all the integers from 0 to p-1.
		A finite field is a finite set of numbers and a set of rules for doing arithmetic with the numbers in that set.
		The operation used to do this in many finite fields is modulo: you divide the number by the size of the set, and take the remainder—as a trivial example, in a finite field of 7 elements, 2+13=1.
	www.deviceforge.com /articles/AT4234154468.html (5944 words)

	Cornell University
		A field F forms a commutative group under addition, with the additive identity element labeled as ‘0’.
		The field F without the ‘0’ element also forms a commutative group under multiplication, this time the multiplicative identity element is labeled ‘1’.
		Every finite field contains a primitive element; that is an element that when multiplied by itself produces all of the other elements in the field.
	instruct1.cit.cornell.edu /Courses/ee476/FinalProjects/s2004/abk26 (2362 words)

	Understanding elliptic-curve cryptography (Site not responding. Last check: 2007-10-21)
		In practice, the finite fields used are either integers modulo large primes, or a similar construction using 0/1 polynomials.
		The finite fields for standards-based ECC are either integers modulo a prime or binary polynomial-based.
		If the finite field is integers modulo a prime, then multiplication may use the processor's multiplication instruction, with careful code that builds up several-hundred-bit multiplication from these single-word operations.
	www.embedded.com /shared/printableArticle.jhtml;jsessionid=CC3PXIICYUE54QSNDBCCKH0CJUMEKJVN?articleID=177101463 (2039 words)

	Structure of General Factorial Designs
		When q is a prime number, finite field arithmetic is equivalent to regular integer arithmetic modulo q.
		When q=2, addition of the two elements of the finite field is equivalent to multiplication of the integers +1 and -1.
		where all the arithmetic is performed in the finite field of size q.
	www.asu.edu /it/fyi/dst/helpdocs/statistics/sas/sasdoc/sashtml/qc/chap16/sect2.htm (315 words)

	Finite Field Arithmetic (Site not responding. Last check: 2007-10-21)
		A field is a set of numbers with an addition operation and a multiplication operation.
		Contents Motivation Overview on Finite Field Arithmetic Arithmetic in GF p Arithmetic in GF m Arithmetic in GF p m Open Problems ECC WPI Why Public...
		2 binary arithmetic, we are also pursuing arithmetic architecture design for finite field (i.e.
	www.finitefieldarithmetic.info (301 words)

	[No title]
		Methods to improve performance of finite field arithmetic of characteristic two using a permuted optimal normal basis representation.
		Methods for efficient implementation of arithmetic modulo n, where n is prime or composite.
		Methods to improve performance of finite field arithmetic of a characteristic two by using subfields.
	grouper.ieee.org /groups/1363/P1363/letters/Certicom.txt (626 words)

	Efficient Systolic Architecture for Modular Multiplication over (Site not responding. Last check: 2007-10-21)
		arithmetic is fundamental to the implementation of a number of modern cryptographic systems and schemes of certain cryptographic systems.
		Most arithmetic operations, such as exponentiation, inversion, and division operations, can be carried out using just a modular multiplier or using power-sum architecture.
		The algorithm is used to design arithmetic architectures with a low hardware and time complexity.
	www.cs.utk.edu /~jerzy/para04/Abstracts/hyun_sung_kim1/hyun_sung_kim1.html (220 words)

	Introduction
		Finite fields of various kinds are supported, with optimized representations for each kind.
		In Magma, arithmetic in small non-prime finite fields is carried out using tables of Zech logarithms.
		Moreover, in passing between fields and subfields, choices regarding the embeddings have to be made, so that these embeddings are compatible (so that `diagrams commute').
	www.umich.edu /~gpcc/scs/magma/text609.htm (711 words)

	Joseph Malkevitch: Finite Arithmetics Tidbit
		The necessary concepts to show that there are indeed finite arithmetics which have the same algebraic properties as the real numbers or rational numbers are surprisingly new.
		It turns out that there is a different finite arithmetic for each prime p, and the arithmetic associated with the prime p is called Z
		These fields are traditionally known as the Galois Fields, honoring Erviste Galois who was a pioneer in their study.
	www.york.cuny.edu /~malk/tidbits/tidbit-arithmetics.html (1823 words)

	Area Efficient Multiplier based on LFSR Architecture (Site not responding. Last check: 2007-10-21)
		AOPM is a multiplier for a result with one dimensional extended fields.
		Thereby, to reduce the result with the ordinary fields element MAOPM is proposed.
		Since there are lots of applications with strict hardware requirements, we focused on area efficiency to derive a multiplier with the ordinary fields result.
	www.cs.utk.edu /~jerzy/para04/Abstracts/jae_hyung_jung/jae_hyung_jung.html (227 words)

	[No title]
		Magma V2.3 used a two-step representation where a large field was represented as an extension field of a Zech representation field, but this representation could only be used if the degree of the field had a suitable divisor and was not too large.
		The arithmetic for series over the rational field is generally 5 times faster (as a consequence of the use of fraction-free methods).
		In V2.4 the computation of the order of an elliptic curve over a finite field is performed using the plain Schoof algorithm (with the restriction that the characteristic of the field must be greater than 3).
	www.umich.edu /~gpcc/scs/magma/text72.htm (5496 words)

	Please title this page. (rs.htm)
		Finite fields are so named because all arithmetic is closed over the field (the result of any operation is still an element of the field).
		The field polynomial, which is used to determine the order of the elements in the finite field, is generally determined by specification (the example DVB specification mentioned uses eight bit symbols {m=8} and a field polynomial of 285 per specification).
		Valid field polynomials are a function of the bit width to be operated on.
	www.piclist.com /techref/method/error/rs.htm (998 words)

	WPI Cryptography and Information Security (CRIS): Spring 2003
		Elliptic curve cryptography relies heavily on the existence of efficient algorithms for finite field arithmetic.
		The arithmetic operations in OEFs are much more efficient than in characteristic two extensions or prime fields due to the use of larger characteristic base field and the selection of a binomial as the field polynomial.
		The OTF inversion algorithm was implemented on the ARM family of processors for a medium and a large sized field whose elements can be represented with 192 and 320 bits, respectively.
	www.crypto.wpi.edu /Seminars/spring2003.shtml (902 words)

	[No title]
		They pro- vide editable input and text fields, simple commands for recalculating parts of worksheets etc. A collection of examples of applications of Maple will be now be distributed in the form of worksheets.
		Automatic complex numeric arithmetic: Complex arithmetic in Maple V had to be done with the evalc function.
		Arithmetic over finite fields (Galois fields): A special representation for univariate polynomials over finite fields defined by a single alge- braic extension over Z mod p has been implemented.
	www.uic.edu /depts/adn/infwww/txt/v2632007.txt (10054 words)

	My Netscape FAQ
		More precisely, for all bivariate polynomials of total degree $n$ over a fixed finite field, the average running time is $\Oh(N)$ using fast polynomial arithmetic, and $\Oh (N^{1.5})$ using standard polynomial arithmetic where $N=n^2$ represents the input size and we ignore the logarithmic factors in our running times.
		Optimal normal bases in finite fields were introduced at the University of Waterloo by Mullin et al.\ (1989), and are used in practical hardware implementation of public-key cryptosystems.
		In cryptography, we investigate efficient arithmetic of finite fields which is crucial in practical implementation of public-key cryptosystems that are based on finite fields and elliptic curves over finite fields.
	www.math.clemson.edu /~sgao/WEB/research.html (4157 words)

	Abstract for PB-227 (Site not responding. Last check: 2007-10-21)
		We show that this is equivalent to asking whether arithmetic complexity over the prime fields is a fully general measure of complexity.
		We show that the arithmetic complexity of the above function is divided between two other canonical functions, the first to be computed modulo p and the second with modulo p^2 arithmetic.
		We thus have tied the efficacy of finite field arithmetic to specific questions about the arithmetic complexities of some fundamental functions.
	www.daimi.au.dk /PB/227/PB-227-abstract.html (184 words)

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