Elliptic Divisibility Sequence

	Elliptic Divisibility Sequence In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan WardMorgan Ward, Memoir on elliptic divisibility sequences, ''Amer. J. Math.'' 70 (1948), 31–74. in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including logic and cryptography. Definition A (nondegenerate) ''elliptic divisibility sequence'' (EDS) is a sequence of integers defined recursively by four initial values , , , , with ≠ 0 and with subsequent values de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Integer Sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: * Abundant numbers * Baum–Sweet sequence * Bell n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Tate Pairing In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by and extended by . applied the Tate pairing over finite fields to cryptography. See also * Weil pairing Weil may refer to: Places in Germany Weil, Bavaria Weil am Rhein, Baden-Württemberg * Weil der Stadt, Baden-Württemberg Weil im Schönbuch, Baden-Württemberg Other uses Weil (river), Hesse, Germany * Weil (surname), including people with ... References * * * * Pairing-based cryptography Elliptic curve cryptography Elliptic curves {{Crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Pairing-based Cryptography Pairing-based cryptography is the use of a pairing between elements of two cryptographic groups to a third group with a mapping e :G_1 \times G_2 \to G_T to construct or analyze cryptographic systems. Definition The following definition is commonly used in most academic papers. Let F_q be a Finite field over prime q, G_1, G_2 two additive cyclic groups of prime order q and G_T another cyclic group of order q written multiplicatively. A pairing is a map: e: G_1 \times G_2 \rightarrow G_T , which satisfies the following properties: ; Bilinearity: \forall a,b \in F_q^, P\in G_1, Q\in G_2:\ e\left(aP, bQ\right) = e\left(P, Q\right)^ ; Non-degeneracy: e \neq 1 ; Computability: There exists an efficient algorithm to compute e. Classification If the same group is used for the first two groups (i.e. G_1 = G_2), the pairing is called ''symmetric'' and is a mapping from two elements of one group to an element from a second group. Some researchers classify pairing instantiations in ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Weil Pairing Weil may refer to: Places in Germany Weil, Bavaria Weil am Rhein, Baden-Württemberg * Weil der Stadt, Baden-Württemberg Weil im Schönbuch, Baden-Württemberg Other uses Weil (river), Hesse, Germany * Weil (surname), including people with the surname Weill, Weyl * Doctor Weil (Mega Man Zero), a fictional character from the ''Mega Man'' Zero video game series * Weil-Marbach, now the Marbach Stud in Baden-Württemberg See also * Weill (other) * Weil, Gotshal & Manges Weil, Gotshal & Manges LLP is an American international law firm with approximately 1,100 attorneys, headquartered in New York City. With a gross annual revenue in excess of $1.8 billion, it is among the world's largest law firms according to ..., law firm founded in the United States * Weil's disease {{disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Elliptic Nets In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric equation is: : (x,y) = (a\cos(t),b\sin(t)) \quad \text \quad 0\leq t\leq 2\pi. Ellipses ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Katherine E Katherine, also spelled Catherine, and other variations are feminine names. They are popular in Christian countries because of their derivation from the name of one of the first Christian saints, Catherine of Alexandria. In the early Christian era it came to be associated with the Greek adjective (), meaning "pure", leading to the alternative spellings ''Katharine'' and ''Katherine''. The former spelling, with a middle ''a'', was more common in the past and is currently more popular in the United States than in Britain. ''Katherine'', with a middle ''e'', was first recorded in England in 1196 after being brought back from the Crusades. Popularity and variations English In Britain and the U.S., ''Catherine'' and its variants have been among the 100 most popular names since 1880. The most common variants are ''Katherine,'' ''Kathryn,'' and ''Katharine''. The spelling ''Catherine'' is common in both English and French. Less-common variants in English include ''Katheryn' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Hilbert's Tenth Problem Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values. For example, the Diophantine equation 3x^2-2xy-y^2z-7=0 has an integer solution: x=1,\ y=2,\ z=-2. By contrast, the Diophantine equation x^2+y^2+1=0 has no such solution. Hilbert's tenth problem has been solved, and it has a negative answer: such a general algorithm does not exist. This is the result of combined work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson which spans 21 years, with Matiyasevich completing the theorem in 1970. The theorem is now known as Matiyasevich's theorem or the MRDP theorem (an initialism for the surnames of the four principal contrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Bjorn Poonen Bjorn Mikhail Poonen (born 27 July 1968 in Boston, Massachusetts) is a mathematician, four-time Putnam Competition winner, and a Distinguished Professor in Science in the Department of Mathematics at the Massachusetts Institute of Technology. His research is primarily in arithmetic geometry, but he has occasionally published in other subjects such as probability and computer science. He has edited two books, and his research articles have been cited by approximately 1,000 distinct authors. He is the founding managing editor of the journal '' Algebra & Number Theory'', and serves also on the editorial boards of '' Involve'' and the '' A K Peters Research Notes in Mathematics'' book series.Curriculum vitae retrieved 2015-01-28. Education Poonen is a 1985 alumnus of [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Zsigmondy's Theorem In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a>b>0 are coprime integers, then for any integer n \ge 1, there is a prime number ''p'' (called a ''primitive prime divisor'') that divides a^n-b^n and does not divide a^k-b^k for any positive integer k1 and n is not equal to 6, then 2^n-1 has a prime divisor not dividing any 2^k-1 with k. Similarly, $a^n+b^n$ has at least one primitive prime divisor with the exception $2^3+1^3=9$ . Zsigmondy's theorem is often useful, especially in group theory, where it is used to prove that various groups have distinct orders except when they are known to be the same. History The theorem was discovered by Zsigmondy working in Vienna from 1894 until 1925. Generalizations Let $(a_n)_$ be a sequence of nonzero integers. The Zs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Division Polynomial In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm. Definition The set of division polynomials is a sequence of polynomials in \mathbb ,y,A,B/math> with x, y, A, B free variables that is recursively defined by: ::\psi_ = 0 ::\psi_ = 1 ::\psi_ = 2y ::\psi_ = 3x^ + 6Ax^ + 12Bx - A^ ::\psi_ = 4y(x^ + 5Ax^ + 20Bx^ - 5A^x^ - 4ABx - 8B^ - A^) ::\vdots ::\psi_ = \psi_ \psi_^ - \psi_ \psi ^_ \text m \geq 2 ::\psi_ = \left ( \frac \right ) \cdot ( \psi_\psi^_ - \psi_ \psi ^_) \text m \geq 3 The polynomial \psi_n is called the ''n''th division polynomial. Properties In practice, one sets y^2=x^3+Ax+B, and then \psi_\in\mathbb ,A,B/math> and \psi_\in 2y\mathbb ,A,B/math>. The division polynomials form a generic elliptic divisibility sequence over the ring \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Canonical Height The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example" is often used to mean "archetype". Science and technology * Canonical form, a natural unique representation of an object, or a preferred notation for some object Mathematics * * Canonical coordinates, sets of coordinates that can be used to describe a physical system at any given point in time * Canonical map, a morphism that is uniquely defined by its main property * Canonical polyhedron, a polyhedron whose edges are all tangent to a common sphere, whose center is the average of its vertices * Canonical ring, a graded ring associated to an algebraic variety * Canonical injection, in set theory * Canonical representative, in set theory a standard member of each element of a set partition Differential geometry * Canonical one-form, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Fibonacci Sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book '' Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the '' Fibonacci Quarterly''. Applications of Fibonacci numbers includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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