Summation polynomials and the discrete logarithm problem on elliptic curves

"Series expansion" redirects here. For other notions of the term, see series.As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivativesat a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named after English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin.DefinitionThe Taylor series of a real or complex function f(x) that is infinitely differentiable in a neighborhood of a real or complex number a, is the power serieswhich in a more compact form can be written aswhere n! is the factorial of n and f (n)(a) denotes the n th derivative of f evaluated at the point a; the zeroth derivative of f is defined to be f itself and (x−a)0 and 0! are both defined to be 1.Often f(x) is equal to its Taylor series evaluated at x for all x sufficiently close to a. This is the main reason why Taylor series are important.In the particular case where a= 0, the series is also called a Maclaurin series.[edit] ExamplesThe Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for (1 −x)−1 is the geometric seriesso the Taylor series for x−1 at a = 1 isBy integrating the above Maclaurin series we find the Maclaurin series for −log(1 −x), where log denotes the natural logarithm:and the corresponding Taylor series for log(x) at a = 1 isThe Taylor series for the exponential function e x at a = 0 isThe above expansion holds because the derivative of e x is also e x and e0 equals 1. This leaves the terms (x− 0)n in the numerator and n! in the denominator for each term in the infinite sum.[edit] ConvergenceThe sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.The Taylor polynomials for log(1+x) only provide accurate approximations in the range −1 < x≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.Taylor series need not in general be convergent, but often they are. The limit of a convergent Taylor series of a function f need not in generalbe equal to the function value f(x), but often it is. If f(x) is equal to its Taylor series in a neighborhood of a, it is said to be analytic in this neighborhood. If f(x) is equal to its Taylor series everywhere it is called entire. The exponential function e x and the trigonometric functions sine and cosine are examples of entire functions. Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan. For these functions the Taylor series do not converge if x is far from a.Taylor series can be used to calculate the value of an entire function in every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for entire functions include:1.The partial sums (the Taylor polynomials) of the series can be usedas approximations of the entire function. These approximations are good if sufficiently many terms are included.2.The series representation simplifies many mathematical proofs.Pictured on the right is an accurate approximation of sin(x) around the point a = 0. The pink curve is a polynomial of degree seven:The error in this approximation is no more than |x|9/9!. In particular, for |x| < 1, the error is less than 0.000003.In contrast, also shown is a picture of the natural logarithm function log(1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. This is similar to Runge's phenomenon.Taylor's theorem gives a variety of general bounds on the size of the error in R n(x) incurred in approximating a function by its n th-degree Taylor polynomial.[edit] HistoryThe Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility: the result was Zeno's paradox. Later, Aristotle proposed a philosophicalresolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.[1] Liu Hui independently employed a similar method a few centuries later.[2]In the 14th century, the earliest examples of the use of Taylor series and closely-related methods were given by Madhava of Sangamagrama.[3] Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor[4], after whom the series are now named.The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.[edit] PropertiesThe function e−1/x²is not analytic at x= 0: the Taylor series is identically 0, although the function is not.If this series converges for every x in the interval (a−r, a + r) and the sum is equal to f(x), then the function f(x) is said to be analytic in the interval(a−r, a+ r). If this is true for any r then the function is said to be an entire function. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.The importance of such a power series representation is at least fourfold. First, differentiation and integration of power series can be performed term by term and is hence particularly easy. Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available. Third, the (truncated) series can be used to compute function values approximately (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm).Fourth, algebraic operations can often be done much more readily on the power series representation; for instance the simplest proof of Euler's formula uses the Taylor series expansions for sine, cosine, and exponential functions. This result is of fundamental importance in such fields as harmonic analysis.Another reason why the Taylor series is the natural power series for studying a function f is given by the probabilistic interpretation of Taylor series. Given the value of f and its derivatives at a point a, the Taylor series is in some sense the most likely function that fits the given data.Note that there are examples of infinitely differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). For instance, the function defined pointwise by f(x) = e−1/x² if x≠ 0 and f(0) = 0 is an example of a non-analytic smooth function. All its derivatives at x = 0 are zero, so the Taylor series of f(x) at 0 is zero everywhere, even though the function is nonzero for every x≠ 0. This particular pathology does not afflict Taylor series in complex analysis. There, the area of convergence of a Taylor series is always a disk in the complex plane (possibly with radius 0), and where the Taylor series converges, it converges to the function value. Notice that e−1/z²does not approach 0 as z approaches 0 along the imaginary axis, hence this function is not continuous as a function on the complex plane.Since every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined onthe real line, the radius of convergence of a Taylor series can be zero.[5] There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.[6]Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f(x) = e−1/x² can be written as a Laurent series.[edit] List of Taylor series of some common functionsSee also List of mathematical seriesThe cosine function in the complex plane.An 8th degree approximation of the cosine function in the complex plane.The two above curves put together.Several important Maclaurin series expansions follow.[7] All these expansions are valid for complex arguments .Exponential function:Natural logarithm:Finite geometric series:Infinite geometric series:Variants of the infinite geometric series:Square root:Binomial series (includes the square root for α = 1/2 and the infinite geometric series for α = −1):with generalized binomial coefficientsTrigonometric functions:where the B s are Bernoulli numbers.Hyperbolic functions:Lambert's W function:The numbers B k appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. The E k in the expansion of sec(x) are Euler numbers.[edit] Calculation of Taylor seriesSeveral methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. Particularly convenient is the use of computer algebra systems to calculate Taylor series.[edit] First exampleCompute the 7th degree Maclaurin polynomial for the function.First, rewrite the function as.We have for the natural logarithm (by using the big O notation)and for the cosine functionThe latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big O notation:Since the cosine is an even function, the coefficients for all the odd powers x, x3, x5, x7, . . . have to be zero.[edit] Second exampleSuppose we want the Taylor series at 0 of the function.We have for the exponential functionand, as in the first example,Assume the power series isThen multiplication with the denominator and substitution of the series of the cosine yieldsCollecting the terms up to fourth order yieldsComparing coefficients with the above series of the exponential function yields the desired Taylor series[edit] Taylor series as definitionsClassically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. For example the exponential function is the function which is everywhere equal to its own derivative, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series.Taylor series are used to define functions in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may defineanalytical functions of matrices and operators, such as the matrix exponential or matrix logarithm.In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.[edit] Taylor series in several variablesThe Taylor series may also be generalized to functions of more than one variable withFor example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:where the subscripts denote the respective partial derivatives.A second-order Taylor series expansion of a scalar-valued function of more than one variable can be compactly written aswhere is the gradient and is the Hessian matrix. Applyingthe multi-index notation the Taylor series for several variables becomesin full analogy to the single variable case.[edit] See also∙Taylor's theorem∙Linear approximation∙Power series∙Laurent series∙Holomorphic functions are analytic— a proof that a holomorphic function can be expressed as a Taylor power series ∙Newton's divided difference interpolation∙Difference engine∙Mean value theorem[edit] Notes1.^ Kline, M. (1990) Mathematical Thought from Ancient to Modern Times.Oxford University Press. pp. 35-37.2.^Boyer, C. and Merzbach, U. (1991) A History of Mathematics. John Wileyand Sons. pp. 202-203.3.^"Neither Newton nor Leibniz - The Pre-History of Calculus and CelestialMechanics in Medieval Kerala". MAT 314. Canisius College. Retrieved on 2006-07-09.4.^ Taylor, Brook, Methodus Incrementorum Directa et Inversa [Direct andReverse Methods of Incrementation] (London, 1715), pages 21-23(Proposition VII, Theorem 3, Corollary 2). Translated into English inD. J. Struik, A Source Book in Mathematics 1200-1800 (Cambridge,Massachusetts: Harvard University Press, 1969), pages 329-332.5.^ Exercise 12 on page 418 in Walter Rudin, Real and Complex Analysis.McGraw-Hill, New Dehli 1980, ISBN 0-07-099557-56.^ Exercise 13, same book7.^ Most of these can be found in (Abramowitz & Stegun 1970).。

椭圆曲线离散对数问题是基于椭圆曲线上的离散对数难题，通常用于椭圆曲线密码学中的一种加密算法，比如椭圆曲线数字签名算法（ECDSA）和椭圆曲线Diffie-Hellman密钥交换（ECDH）等。

下面我将详细解释椭圆曲线离散对数问题的背景和基本概念：
1. 椭圆曲线基础
椭圆曲线是一个由满足特定方程的点组成的集合。

通常，椭圆曲线的方程可以表示为：
y2=x3+ax+b
其中，a和b是定义曲线的参数。

在有限域上，我们通常选择一个素数域（有限域）作为底域，比如GF(p)，其中p是一个素数。

曲线上的点和加法运算定义了一个群
结构。

2. 离散对数问题
离散对数问题是在一个有限群中找到给定元素的幂等于另一个给定元素的指数。

对于椭圆曲线离散对数问题，我们考虑椭圆曲线上的点和它们的倍数。

具体而言，对于椭圆曲线上的点P和整数k，问题是找到整数n，使得nP=Q，
其中Q是已知的椭圆曲线上的点。

这里，n就是离散对数。

3. 安全性
椭圆曲线离散对数问题是一种困难的数学问题，目前尚未找到高效的经典算法来解决。

这使得椭圆曲线密码学方案变得相对安全。

然而，随着量子计算技术的发展，部分经典加密算法（如RSA、DSA）和椭圆曲线密码学也面临潜在的风险。

因此，未来的密码学研究重点之一是发展抵抗量子计算攻击的密码学算法，以确保信息安全。

总体来说，椭圆曲线离散对数问题在密码学中扮演着重要的角色，它为很多加密算法提供了安全性基础。

台湾国⽴交通⼤学数学视频数学视频Calculus I 台湾国⽴交通⼤学 Michael Fuchs⽼師 36集（点击进⼊我的淘宝店）Calculus II 台湾国⽴交通⼤学 Michael Fuchs⽼師 29集（点击进⼊我的淘宝店）Chapter1　Functions and Model1-5　Exponential Functions1-6　Inverse Functions and LogarithmsChapter2　Limits and Derivatives2-2　The Limit of a Function2-4　The Precise Definition of a Limit2-3　Calculating Limits Using the Limit Laws2-6　Limits at Infinity; Horizontal Asymptotes2-5　Continuity2-8　Derivatives2-9　The Derivative as a FunctionChapter3　Differentiation Rules3-1　Derivatives of Polynomials and Exponential Functions3-2　The Product and Quotient Rules3-4　Derivatives of Trigonometric Functions3-5　The Chain Rule3-6 Implicit Differentiation3-8 Derivatives of Logarithmic Functions3-10　Related Rates3-7 Higher Derivatives3-11　Linear Approximations and DifferentialsChapter4　Applications of Differation4-1　Maximum and Minimum Values4-2　The Mean Value Theorem4-3 How Derivatives Affect the Shape of a Graph4-4 Indeterminate Forms a nd L’Hospital’s Rule4-7　Optimization Problems4-5　Summary of Curve Sketching4-10　AntiderivativesChapter5　Integrals5-1　Areas and Distances5-2　The Definite Integral5-3　The Fundamental Theorem of Calculus5-4　Indefinite Integrals and the Total Change Theorem5-5　The Substitution Rule5-6　The Logarithm Defined as an IntegralChapter6　Applications of Integration6-1　Areas between Curves6-2　Volumes6-3　Volumes be Cylindrical ShellsChapter7　Techniques of Integration7-1　Integration by Parts7-2　Trigonometric Integrals7-3　Trigonometric Substitution7-4　Integration of Rational Functions by Partial Fractions7-8　Improper Integrals7-7　Approximate IntegrationChapter8　Further Applications of Integration8-1　Arc Length8-2　Area of a Surface of RevolutionChapter10　Parametric Equations and Polar Coordinates10-1　Curves Defined by Parametric Equations10-2　Calculus with Parametric Curves10-3　Polar Coordinates10-4　Areas and Lengths in Polar Coordinates微积分(⼀) 台湾国⽴交通⼤学莊重⽼師　24集（点击进⼊我的淘宝店）微积分(⼆)台湾国⽴交通⼤学莊重⽼師 24集（点击进⼊我的淘宝店）課程章節第⼀章Functions and Model第⼆章Limits and derivatives第三章Differentiation Rules第四章The Properties of Gases第五章Integrals第六章Applications of Integration第七章Techniques of Integration第⼋章Further Applications of Integration第⼗章Parametric Equations and Polar Coordinates第⼗⼀章Infinite Sequences and Series第⼗⼆章Vectors and the Geometry of Space第⼗三章Vector Functions第⼗四章Partial Derivatives第⼗五章Multiple Integrals⾼等微积分(⼀)台湾国⽴交通⼤学⽩啟光⽼師　29集（点击进⼊我的淘宝店）⾼等微积分(⼆) 台湾国⽴交通⼤学　⽩啟光⽼師　27集（点击进⼊我的淘宝店）第⼀章The Real and Complex Number SystemsFields Axioms, Order Axioms　Completeness Axioms第⼆章Basic TopologyCardinality of SetsMetric SpacesCompact SetsConnected Sets第三章Numerical Sequences and SeriesConvergent SequencesCauchy SequencesUpper and Lower LimitsSeries of Nonnegative TermsThe Root and Ratio TestAbsolute Convergence, Rearrangements第四章ContinuityLimits of Functions and Continuous FunctionsContinuity and CompactnessContinuity and Connectednessdiscontinuities, Infinite Limits and Limits at Infinity第五章Differentiation　The Derivative of a Real Function, Mean Value TheoremL’Hopital’s RuleTaylor’s TheoremDifferentiation of Vector-valued Functions第六章The Riemann-Stieltjes Integral　Definition and Existence of the IntegralProperties of the IntegralIntegration and DifferentiationIntegration and Differentiation第六章The Riemann-Stieltjes IntegralIntegration and Differentiation第七章Sequence and Series of FunctionsSequence and Series of Functions --- the Main ProblemUniform Convergence and ContinuityUniform Convergence and IntegrationUniform Convergence and DifferentiationEquicontinuous Family of FunctionsThe Stone-Weierstrass Theorem第⼋章Some Special FunctionsPower seriesSome Special FunctionsFourier SeriesThe Gamma Function第九章Functions of several variablesFunction of Several VariablesFunction of Several Variables：DifferentiationFunction of Several Variables：DifferentiationThe Inverse Function TheoremThe Implicit Function TheoremThe Rank TheoremDeterminantsDifferentiation of Integrals偏微分⽅程(⼀) 台湾国⽴交通⼤学林琦焜⽼师 3.8GB （点击进⼊我的淘宝店）偏微分⽅程(⼆) 台湾国⽴交通⼤学林琦焜⽼师 3.4GB （点击进⼊我的淘宝店）内容纲要第⼀章 The Single First-Order Equation1-1 Introduction Partial differential equations occur throughout mathematics. In this part we will give some examples1-2 Examples1-3 Analytic Solution and Approximation methods in a simple example 1-st order linear example1-4 Quasilinear Equation The concept of characteristic1-5 The Cauchy Problem for the Quasilinear-linear Equations1-6 Examples Solved problems1-7 The general first-order equation for a function of two variables characteristic curves, envelope1-8 The Cauchy Problem characteristic curves, envelope1-9 Solutions generated as envelopes第⼆章Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables2-1 Characteristics for Linear and Quasilinear Second-Order Equations Characteristic2-2 Propagation of Singularity Characteristic curve and singularity2-3 The Linear Second-Order Equation classification of 2nd order equation2-4 The One-Dimensional Wave Equation dAlembert formula, dimond law, Fourier series2-5 System of First-Order Equations Canonical form, Characteristic polynominal2-6 A Quasi-linear System and Simple Waves Concept of simple wave第三章 Characteristic Manifolds and Cauchy Problem3-1 Natation of Laurent Schwartz Multi-index notation3-2 The Cauchy Problem Characteristic matrix, characteristic form3-3 Real Analytic Functions and the Cauchy-Kowalevski Theorem Local existence of solutions of the non-characteristic 3-4 The Lagrange-Green Identity Gauss divergence theorem3-5 The Uniqueness Theorem of Ho ren Uniqueness of analytic partial differential equations3-6 Distribution Solutions Introdution of Laurent Schwartzs theory of distribution (generalized function)第四章 The Laplace Equation4-1 Greens Identity, Fundamental Solutions, and Poissons Equation Dirichlet problem, Neumann problem, spherical symmetry, mean value theorem, Poisson formula4-2 The Maximal Principle harmonic and subharmonic functions4-3 The Dirichlet Problem, Greens Function, and Poisson Formula Symmetric point, Poisson kernel4-4 Perrons method Existence proof of the Dirichlet problem4-5 Solution of the Dirichlet Problem by Hilbert-Space Methods Functional analysis, Riesz representation theorem, Dirichlet integra第五章 Hyperbolic Equations in Higher Dimensions5-1 The Wave Equation in n-Dimensional Space(1) The method of sphereical means(2) Hadmards method of descent(3) Duhamels principle and the general Cauchy problem(4) mixed problem5-2 Higher-Order Hyperbolic Equations with Constant Coefficients(1) Standard form of the initial-value problem(2) solution by Fourier transform,(3) solution of a mixed problem by Fourier transform5-3 Symmetric Hyperbolic System(1) The basic energy inequality(2)Finite difference method(3) Schauder method第六章 Higher-Order Elliptic Equations with Constant Coefficients6-1 The Fundamental Solution for Odd n Travelling wave6-2 The Dirichlet Problem Lax-Milgram theorem, Garding inequality6-3 Sobolev Space Weak solution and Hibert space第七章 Parabolic Equations7-1 The Heat Equation Self-Similarity, Heat kernel, maximum principle7-2 The Initial-Value Problem for General Second-Order Parabolic Equations(1) Finite difference and maximum principle(2) Existence of Initial Value Problem第⼋章 H. Lewys Example of a Linear Equation without Solutions8-1 Brief introduction of Functional Analysis Hilbert and Banach spaces, projection theorem, Leray-Schauder theorem8-2 Semigroups of linear operator Generation, representation and spectral properties8-3 Perturbations and Approximations The Trotter theorem8-4 The abstract Cauchy Problem Basic theory8-5 Application to linear partial differential equations Parabolic equation, Wave equation and Schrodinger equation8-6 Applications to nonlinear partial differential equations KdV equation, nonlinear heat equation, nonmlinear Schrodinger equation变分学导论应⽤数学系林琦焜⽼师台湾国⽴交通⼤学 2GB （点击进⼊我的淘宝店）内容纲要第⼀章变分学之历史名题1.1 Bernoulli 最速下降曲线1.2 最⼩表⾯积的迴转体1.3 Plateau问题(最⼩曲⾯)1.4 等周长问题1.5 古典⼒学之问题第⼆章 Euler- Lagrange⽅程2.1 变分之原理2.2 折射定律与最速下降曲线2.3 ⼴义座标2.4 Dirichlet 原理与最⼩曲⾯2.5 Lagrange乘⼦与等周问题2.6 Euler-Lagrage ⽅程之不变量2.7 Sturm-Liouville问题2.8 极值(积分)问题第三章 Hamilton系统3.1 Legendre变换3.2 Hamilton⽅程3.3 座标变换与守恒律3.4 Noether定理3.5 Poisson括号第四章数学物理⽅程4.1 波动⽅程4.2 Laplace与Poisson⽅程4.3 Schrodinger ⽅程4.4 Klein-Gordon ⽅程4.5 KdV ⽅程4.6 流体⼒学⽅程　课程书⽬变分学导论 (Lecture note by Chi-Kun Lin).向量分析台湾国⽴交通⼤学林琦焜 3.3GB （点击进⼊我的淘宝店）向量分析主要是要谈”梯度、散度与旋度”这三个重要观念，⽽对应的则是⽅向导数、散度定理、与Stokes定理因此重⼼就在於如何釐清线积分、曲⾯积分以及他们所代表的物理意义。

标题:【原创】椭圆曲线加密分析：FLEXLM ECC问答[7月16日更新到第7部分]作者:readyu时间: 2012-06-25,19:00:18链接: /showthread.php?t=152615说明：我曾经写过ECCTool用于学习和研究椭圆曲线加密在软件上的应用。

椭圆曲线密码学工具 ECCTooL v1.04原帖年代久远不再更新。

v1.05版本我放在本帖3#。

FLexLM ECC是椭圆曲线签名的一个典型应用，不可不察。

我整理了一下FLEXLM ECC方面的研究心得，准备把它完全写出来。

后面会有算法描述的例子代码。

FLEXLM ECC问答 (1)readyu 2012.61. FLEXLM是个虾米东西？FLEXlm是应用广泛的License管理工具。

宣称数千家EDA工具采用它管理授权。

FLEXlm对厂商来说，它的优点是证书管理功能强大，支持平台众多。

对最终用户来说，它的缺点是不够友好，比如Windows下，经常由于某些原因，flexlm服务启动错误。

对于破解者来说，它有着为数众多的教本。

维基百科上的条目为：/wiki/FLEXlmFLEXLM本来属于GLOBEtrotter。

Macrovision曾经收购GLOBEtrotter。

2006年的时候，FLEXLM原来的开发组跑路，另起灶头，产品叫RLM。

/index.php2008年Macrovision把FLEXLM卖了。

现在叫FlexNet Publisher。

/products/flexnet-publisher.htm2. FLexLM 的ECC（椭圆曲线加密）怎么来的？早期的flexlm采用的常规加密，不安全，能被做出lic。

这样的教程是很多的。

flexlm的安全性完全得不到保障，所以，自 v8(大约2001-2002)版本引入了公钥加密算法：椭圆曲线加密。

从此，它一直是flexlm的金钟罩。

包括v9(2003-) , v10(2004-) , v11(2007-), 到最新的版本 v11.10 (2012) 。