The Elliptic Curve Plotter is a graphical application that illustrates elliptic curves. Elliptic curves are a mathematical concept that is important in number theory, and constitutes a major area of current research. Elliptic curves find applications in elliptic curve cryptography (ECC) and integer factorization Elliptic Curve Points. Elliptic Curve Points. Log InorSign Up. This is the Elliptic Curve: 1. y 2 = x 3 + ax + b. 2. b = 2. 6. 3. a = − 1. 4. These are the two points we're adding. You can drag them around.. Elliptic Curves over Finite Fields Here you can plot the points of an elliptic curve under modular arithmetic (i.e. over \( \mathbb{F}_p\)). Enter curve parameters and press 'Draw!' to get the plot and a tabulation of the point additions on this curve

Elliptic Curve Plotter . Elliptic Curve Plotter. Version: 1.1.6 website. Installing: flatpak install flathub de.unifreiburg.ellipticcurve Removing: flatpak remove de.unifreiburg.ellipticcurve. The Elliptic Curve Plotter is a graphical application that allows the user to play and exeriment with elliptic curves and their group law. Installation via Software Center Pamac or Command line. The **Elliptic** **Curve** **Plotter** is a graphical application that illustrates **elliptic** **curves**. Users can sketch **elliptic** **curves** and experiment with their group law. Images can be saved in PNG or SVG format for later use. The program is available for desktop computers

- Elliptic Curve Plotter. Der Elliptic Curve Plotter ist eine grafische Anwendung, die elliptische Kurven illustriert. Benutzer können elliptische Kurven skizzieren und mit dem Gruppengesetz experimentieren. Bilder können zur späteren Verwendung im PNG- oder SVG-Format gespeichert werden. Das Programm ist für Desktop-Computer verfügbar. Eine eingeschränkte Version läuf
- I am trying to plot the elliptic curve secp256k1 y^2=x^3+7 in my latex-document. \begin{center} \begin{tikzpicture}[domain=-4:4, samples at ={-1.769292354238631, -1.
- Plot an arbitrary curve under modular arithmetic (i.e. over Fp F p ). Enter curve parameters and press 'Draw!'. Note: Choose p p prime (like,43,47,53,59,61,67,71,73,79,83,89,97,101,109,...), otherwise Fp F p won't be a field! Curve Plot for y2 = x3 + 2 ∗ x + 3 mod 53 y 2 = x 3 + 2 ∗ x + 3 mod 53. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34.
- The Elliptic Curve Plotter is a graphical application that allows the user to play and exeriment with elliptic curves and their group law. Website Additional informatio
- g, you can compute a list of points $(x,y)$ of the curve on your own. In view of the above curve, as Lord Shark said, real-valued points fulfill $x\geq -\sqrt[3]{7}$. Just loop over values $x$ and compute the $y$-values. Its easy with a computer algebra system like Maple
- The Elliptic Curve Plotter is a graphical application that illustrates elliptic curves. Elliptic curves are a mathematical concept that is important in number theory, and constitutes a major area of current research. Elliptic curves find applications in elliptic curve cryptography (ECC) and integer factorization. Users can sketch elliptic curves and experiment with their group law. Images can.

We can use Sage's interact feature to draw a plot of an elliptic curve modulo \(p\), with a slider that one drags to change the prime \(p\). The interact feature of Sage is very helpful for interactively changing parameters and viewing the results. Type interact? for more help and examples and visit the web page http://wiki.sagemath.org/interact A plot of elliptic curve over a finite field doesn't really make sense, it looks just like randomly scattered points. To compute square roots mod a prime, see this algorithm which should not be too difficult to implement in matlab. - President James K. Polk Feb 7 '12 at 22:37. Add a comment | 3 Answers Active Oldest Votes. 3. You have to test all points that fulfill the equation y^2 = x^3 +x.

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself Learn more advanced front-end and full-stack development at: https://www.fullstackacademy.comElliptic Curve Cryptography (ECC) is a type of public key crypto.. An elliptic curve is a curve defined by y 2 = x 3 + ax + b For example, let a = − 3 and b = 5, then when you plot the curve, it looks like this: A simple elliptic curve English: Plot of the ellipitic curve y^2 = x^3 - 3x + 1. Datum: 14. Juni 2020: Quelle: Eigenes Werk: Urheber: Googolplexian1221: Lizenz. Ich, der Urheber dieses Werkes, veröffentliche es unter der folgenden Lizenz: Diese Datei ist lizenziert unter der Creative-Commons-Lizenz Namensnennung - Weitergabe unter gleichen Bedingungen 4.0 international. Dieses Werk darf von dir verbreitet. Pure Python implementation of Elliptic Curve Cryptography - cardwizard/EllipticCurves. Pure Python implementation of Elliptic Curve Cryptography - cardwizard/EllipticCurves. Skip to content. Sign up Why GitHub? Features → Mobile → Actions → Codespaces → Packages → Security → Code review → Project management → Integrations → GitHub Sponsors → Customer stories → Team; Ente

L-function, Riemann hypothesis, Fermat's last theorem, Taniyama shimura, abc, beal, birch swinnerton dyer conjecture, modular form , p-adic of adelic space I'm trying to plot a elliptic curve in a projective Space. I got inspired by this thread. Here one can find the code. Now the point is, that the author of the thread only plotted a 2d Version of an elliptic curve. I wanted to plot this 3d equation: z*y^2=x^3-3*x*z^2+3*z^3 ( This is the projective form of the affin equation y^2 = x^3 -3x + 3 ) It already was a big problem to find plotters only. English: A plot of the type of data used by Birch and Swinnerton-Dyer to support their conjecture. The curve in question is y 2 = x 3 − 5 x (curve 800h1 of the Cremona database). This is a curve of rank 1 (and one of the curves originally looked at by Birch and Swinnerton-Dyer)

Elliptic Curves are used in public key cryptograpy to create relatively short encryption keys. They are in the form of \(y^2 = x^3 + ax + b\). This page outlines a plot for elliptic curve. The initial plot is \(y^2=x^3 - 3 x + 5\) * # Sadly enough, there is no Maple command to plot # curves in the projective plane*. # As a curiosity, the line Z=0 is a line # which contains all the points at infinity. # This line contains no affine point, and hence # cannot be seen in the affine x/y-plane. > # From now on, let us identify (x,y) with (x:y:1). # Also, it is usual to denote (0:1:0) as infinity. # The main idea is to introduce. The plot of the elliptic curve is way off, not even close. This is probably a beginner mistake so a junior programmer who takes a second to read this will probably see very quickly why I'm not getting the curve I want. python python-2.7 matplotlib elliptic-curve finite-field. Share . Improve this question. Follow edited Nov 3 '13 at 18:00. stackuser. asked Nov 3 '13 at 17:54. stackuser.

- Exhaustive Jacobi Plotter; Elliptic Curves. Adding points on an elliptic curve; Plotting an elliptic curve over a finite field; Cryptography. The Diffie-Hellman Key Exchange Protocol; Other. Continued Fraction Plotter; Computing Generalized Bernoulli Numbers; Fundamental Domains of SL_2(ZZ) Multiple Zeta Values. Computing Multiple Zeta values. Word Input; Composition Inpu
- =-4, xmax=0, y
- How to plot an elliptic curve using sage. 1. SageMathCloud: random elliptic curve. 1. plot elliptic curve over finite field using sage. 3. Second derivative of the Hankel function unsing sympy. 0. Plotting 3D Surface with Sage Math. 0. Plotting functions with SageMath, not Showing. 2. Elliptic Curve Points in sagemath . 0. Exponentiation on a point on elliptic curve unreasonably fast in.
- g Languages. Tools and Utilities. Security. Other. Flatpak Elliptic Curve Plotter. Sketches elliptic curves and allows to experiment with their group law. Education. Website Get the App; Command line instructions . The Flatpak app is included.
- If the curve is defined over real numbers, i.e., y 2 = x 3 + 25x + 17, plot the curve with -5: 4 marks: An important usage of the elliptic curves is to factorize big integers. Comparing to the difference of squares method, the advantage of EC-based factorization is that it can be parallelized easily. This question asks you to practice integer factorization with EC-based method. The smallest 3.
- Elliptic curve plotter downloads [freeware] Home | About Us | Link To Us | FAQ | Contact Serving Software Downloads in 976 Categories, Downloaded 35.495.636 Time

- elliptic curves. Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of.
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- Elliptic curve Curve25519 is used in many applications, including within Tor networks. It has the form of \(y^2 = x^3 + ax^2 + bx + c\). This page outlines a plot for elliptic curve

I'm trying to plot elliptic curves(of the form y 2 = x 3 + ax + b) in gnuplot but currently am having no success. I've tried plotting ±sqrt(x 3 + ax + b) but this leaves the gap in the middle where y=0 Any suggestions?. Also, I'm trying this in gnuplot as it was suggested to me by someone as a good program for taking data from PARI/GP, plotting it and putting it into Latex, so any suggestions. * Online-Funktionsgraph-Plotter: WZGrapher Function Grapher Developer: Walter Zorn WZGrapher is an easy-to-use and small-footprinted Function Graphing and Calculation Program written in C language, with capabilities to plot both cartesian and polar functions*. WZGrapher can also be used to graph numerical solution curves of integrals, to solve numerically and to graph ordinary differential. Elliptic curve cryptography and digital signature algorithm are more complex than RSA or ElGamal but I will try my best to hide the hairy math and the implementation details.Here is the ELI5 version in 18 lines of SageMath / Python code. I use Sage because it provides elliptic curves as first-class citizens (`FiniteField` and `EllipticCurve`) and we can take multiplication operation for granted Algebraic curve plotter. Ask Question Asked 4 years, 9 months ago. Active 1 year, 5 months ago. Viewed 810 times 14. 3 \$\begingroup\$ An algebraic curve is a certain 1D subset of the 2D-plane that can be described as set of zeros {(x,y) in R^2 : f(x,y)=0 }of a polynomial f. Here we consider the 2D-plane as the real plane R^2 such that we can easily imagine what such a curve could look.

An elliptic curve is the solution set of a non-singular cubic equation in two unknowns. In general if F is a field and f is poly with degree(f)=3, such that f(x,y) and its partial derivatives do not vanish simultaneously then E={(x,y)|f(x,y)=0} is an elliptic curve. With so called 'chord and tangent' point addition, the set E becomes an abelian group Elliptic Curve Cryptosystems Elliptic Curve Cryptography (ECC) is the newest member of the three families of established public-key algorithms of practical relevanceintroduced in Sect. 6.2.3. However, ECC has been around since the mid-1980s. ECC provides the same level of security as RSA or discrete logarithm systems with considerably shorter operands (approximately160-256 bit vs. 1024. Abstract Elliptic curves are nonsingular polynomials of degree three in two variables, as members of F[x,y]. Points on the graph of an elliptic curve can be combined using a special addition operator to turn the graph into an Abelian group. When F is a finite field, these curves are applied to problems and algorithms in cryptography and number theory To plot the curve for writing this article, and also get a sense of how things work, I wrote a Jupyter Notebook for curve plotting and calculations in Python. The plotting library is matplotlib. And if you want to play around an elliptic curve and feel how it works yourself, lucky you! I made the source code open-sourced here on GitHub, one for real numbers and one for finite field: Screenshot.

- Elliptic_curve_taniyama_10: Illustration of Wiles' theorem by some numerical examples, cf. Example 7.7. The PARI-ﬁles of these lecture notes are contained in a separate folder
- I have some elliptic curve with some points on it: I would like to give the points some names, P, etc., but cannot figure out how to do that probably simple task. The help pages ?plot,options and ?pointplot do not seem to cover it; I may be mistaken, of course. The above plot is the result of the following code: curve := y^2 = x^3 - 43*x + 166; display([ plot(+sqrt(rhs(curve)),x = -10..12.
- I am doing an experiment to prove the associativity of the addition of points on an elliptic curve. So far, I have produced a code which allows me to move points on my curve. To find their sum, I..
- In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K 2, the Cartesian product of K with itself. If the field has characteristic different from 2 and 3 then the curve can be described as a plane algebraic curve which, after a linear change of variables.
- Elliptic curves; Epidemiology: The SEIR model; Epidemiology: The SIR model; Euler line; Euler line source code; Euler's spiral (Clothoid) Even simpler function plotter ; Extended mean value theorem; Extension; F. Fermat's spiral; Fern (fractal) Fill the intersection area of three circles; Fill the intersection area of two circles; Fine tuning of labels; Five Circle Theorem; Fractal Polygons.
- plot(E) #this graphs the curve! Now let's add points! Note: The output of adding points is of the form (x:y:z). These are projective coordinates. We will only use the x and y coordinates in this course. However, it's important to note that the point at infinity will look like (0:1:0). E=EllipticCurve(QQ,[A,B]) #this defines the elliptic curve y^2=x^3+Ax+B. P=E(x1,y1) #this defines a point on.
- syms x y f(x,y) f(x,y) = x^3 - 4*x - y^2; fcontour(f,[-3 3 -4 4], 'LevelList',-6:6); colorbar title 'Contour of Some Elliptic Curves' Plot an Analytic Function and Its Approximation Using Spline Interpolant. Plot the analytic function f (x) = x exp (-x) sin (5 x)-2. syms f(x) f(x) = x*exp(-x)*sin(5*x) -2; fplot(f,[0,3]) Create a few data points from the analytic function. xs = 0:1/3:3; ys.

Deﬁnieren Sie den Begriﬀ einer elliptischen Kurve. Zeigen Sie, dass jede elliptische Zeigen Sie, dass jede elliptische Kurve eine Weierstraß-Gleichung besitzt (3.1) Cremona's databases of elliptic curves is part of Sage. The curves up to conductor 10,000 come standard with Sage, and an optional 75MB download gives all his tables up to conductor 130,000. Type sage -i database cremona ellcurve-20071019 to automatically download and install this extended table. To use the database, just create a curve by giving. sage: EllipticCurve ('5077a1') Elliptic. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchang Discrete plot of an **elliptic** **curve**. Contribute to aleph0mc/WinFormDiscreteEllipticCurvePlotSample development by creating an account on GitHub

- Error in lines 1-1 Traceback (most recent call last): File /ext/sage/sage-9.2/local/lib/python3.8/site-packages/sage/schemes/elliptic_curves/ell_point.py, line 674.
- the function on curve as elliptic.on curve instead (because it is part of the elliptic ﬁle —or, more properly, module). Try again >>> elliptic.on_curve(0,0) 1 Exercise 3.1 Modify the function on curve to work with the curve y2 = x3+8x and test with python if some points are on this curve or not. Remember to reload your function into python after the modiﬁcations are made. You can now.
- English: A plot of the type of data used by Birch and Swinnerton-Dyer to support their conjecture. The curve in question is y 2 = x 3 − 5x (curve 800h1 of the Cremona database). This is a curve of rank 1 (and one of the curves originally looked at by Birch and Swinnerton-Dyer). The horizontal axis is a bound X and the vertical axis is () = where N p is the number of points on the curve modulo p
- all elliptic curves over IF q (o r over some natural classes of curves). ¥ The Þeld IF q and the curve IE a re b oth Þxed, w e consider IE(I F qn) in the extension Þelds ¥ The curve IE is deÞned over Q (and Þxed). W e consider reductions IE(I F p) m o dulo consecutive p rimes p Rema rk: They a re describ ed in the increasing o r- der of ha rdness. 5 Group Structure of IE(I F q) and.
- Elliptic Curve Cryptography (ECC) is one of the most powerful but least understood types of cryptography in wide use today. At CloudFlare, we make extensive use of ECC to secure everything from our customers' HTTPS connections to how we pass data between our data centers.. Fundamentally, we believe it's important to be able to understand the technology behind any security system in order to.
- A Schematic Elliptic Curve Plot (credit: CloudFlare) Adding two points on the curve, A and B, is our Billiards shot. To add A and B, place the ball at point A and shoot it towards point B

You'll often see these curves depicted as planar slices of what might otherwise be a 3D plot. Elliptic Curve and Trapdoor Function # There does not appear to be a shortcut that is narrowing the gap in a Trapdoor Function based around Elliptic Curve. This means that for numbers of the same size, solving Elliptic Curve discrete logarithms is significantly harder than factoring. Since a more. ** Elliptic Curves are a type of algebraic curve with a general form described by the Diophantine equation (1) They were utilised by Andrew Wiles in his proof of Fermat's Last Theorem, and they are gaining popularity in the realm of cryptography for their security and efficiency over current cryptographic methods**. They form a large part of US National Security Agency's (NSA) Suite B of.

Plot the SECP256K1 elliptic curve and explain in simple terms the group law for elliptic curves. Question 2 Demonstrate geometric addition and scalar multiplication with arbitrary points on the curve. Question 3 Explain in simple terms, using your own words, how elliptic curves are restricted to a finite prime field Fp. Question 4 Demonstrate, with the aid of an example, geometric addition and. (c) If the curve is defined over real numbers, i.e., y2 = x3 + 25x + 17, plot the curve with -5<x<5 and -5<y<5. 4 marks An important usage of the elliptic curves is to factorize big integers Plot an elliptic curve: Plot a sum of 5 sine waves in random directions: Find the minimum of a function in a region: Show the steps taken to the minimum: Have contours at the 10% and 90% percentile values: An electrostatic potential built from a collection of point charges at positions : Charge colors, using green for negative and orange for positive: Two charges, and : Three charges, , , and.

** Elliptic curves are sometimes used in cryptography as a way to perform digital signatures**.. The purpose of this task is to implement a simplified (without modular arithmetic) version of the elliptic curve arithmetic which is required by the elliptic curve DSA protocol. In a nutshell, an elliptic curve is a bi-dimensional curve defined by the following relation between the x and y coordinates. 2 Elliptic Curve Cryptography 2.1 Introduction. If you're first getting started with ECC, there are two important things that you might want to realize before continuing: Elliptic is not elliptic in the sense of a oval circle. Curve is also quite misleading if we're operating in the field F p. The drawing that many pages show of a elliptic curve in R is not really what you need to think.

Around this main plot develop several subplots: other, more elementary approaches better suited to small ﬁelds; possible gener- alizations to point-counting on varieties more complicated than elliptic curves; further applications of our formulas for modular curves and isogenies. We steer clear only of the question of how to adapt our methods, which work most read-ily for large prime ﬁelds. English: Elliptic curve, E, 4y^2=x(x-1)(x+1) (in blue) and its polar curve, C_Q, (in red) 4y^2=2.7x^2-2x-0.9 corresponding to the point Q=(0.9,0) (in homogeneous coordinates, Q=(0.9:0:1)). The 6 black lines are the tangents to points on E that go through Q. Date: 18 September 2009: Source: Own work using mathematica, with Inkscape touch-ups. Author: RobHar: Licensing . I, the copyright holder.

You'll often see these curves depicted as planar slices of what might otherwise be a 3D plot. On the left, in transparent red, is the 3-dimensional contour plot of y2=x3-3x+z. The orange plane that intersects the 3D contour plot is shown on the right. The curve is elliptic everywhere except at the saddle point, where the curve transitions from a closed curve to an open curve. You might. Elliptic Curve Arithmetic - | Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail | To plot such a curve, we need to compute. For given values of a and b, the plot consists of positive and negative values of y for each value of x. Thus, each curve is symmetric about y = 0. Figure 10.4 shows two examples of elliptic curves. As you can see, the. I'd like to plot the secp256k1 curve, but I get Python int too large to convert to C long. Does sage have a built-in type to handle large numbers, or is there a recommended way to do this But because finite fields are, well, finite, we do not get a nice continuous curve if we try and plot points from the elliptic curve equation over them. We end up getting a scatter plot that looks like this: By using finite field addition, subtraction, multiplication, division, and exponentiation, we can actually do point addition over this curve. Although it may seem surprising that we can. Elliptic curve cryptography is used to implement public key cryptography. It was discovered by Victor Miller of IBM and Neil Koblitz of the University of Washington in the year 1985. ECC popularly used an acronym for Elliptic Curve Cryptography. It is based on the latest mathematics and delivers a relatively more secure foundation than the first generation public key cryptography systems for.

Elliptic curve cryptography, winter 2009 MICHAEL NÜSKEN, DANIEL LOEBENBERGER 3. Exercise sheet Hand in solutions until Sunday, 15 November 2009, 2359 Exercise 3.1 (Get your Weierstraß). (11 points) You are given an elliptic curveP E given by a general cubic polynomial f = i+j≤3ai,jx iyj ∈ k[x,y]. Assume you have that the point P = (0 : 1 : 0) is a ﬂex point on the curve (that is a. Here you can plot the points of an elliptic curve under modular arithmetic (i.e. over \( \mathbb{F}_p\)). Enter curve parameters and press 'Draw!' to get the plot and a tabulation of the point additions on this curve to calculate $8P$, I first calculated $2P$ by using the equation sigma = 3x^2+a/2y = (3*4^2+17)/(2*14) mod 59 = 65/28 mod 59 = 2.3214 mod 59. x3 = sigma^2-2x = -2.61 mod 59 y3. In particular, in page 2, an elliptic curve is drawn twice: the first time over the real numbers and the second one over the finite field. There is also an animation of the doubling operation over the finite field. (Hope it helps, even if is not a tool you can use to draw your curve

** A cubic equation is of the form **. Given any nine lattice points a cubic equation can be found whose plot an elliptic curve goes through all nine points as shown in the Nine-Point Cubic Demonstration. More than nine lattice points can be covered even when the lattice is tightly restricted.If a secant (or nontangent) line is drawn through two rational points on an elliptic curve it also pass $\begingroup$ It is Asymptote as given in the asnwer Appendix A: Asymptote code for affine elliptic curve plot $\endgroup$ - kelalaka Mar 17 at 21:22. 1 $\begingroup$ For the second image: I saw the Trustica videos, too, and wanted to plot it myself. I wrote them, but never got an answer. I just assume that they did it with SageMath. If you find the sourcecode please post it here! I would. We want this class to represent a point on an elliptic curve, and overload the addition and negation operators so that we can do stuff like this: p1 = Point(3,7) p2 = Point(4,4) p3 = p1 + p2 But as we've spent quite a while discussing, the addition operators depend on the features of the elliptic curve they're on (we have to draw lines and intersect it with the curve). There are a few ways. j-Invariante f ü r eine elliptische Kurve berechnen. Invarianten sind ein wichtiges Konzept auf dem Gebiet der elliptischen Kurven. Version 12 bietet Funktionen, die sowohl das direkte Arbeiten mit diesen Invarianten als auch Berechnungen und Operationen mit Halbperioden und den Werten von Halbperioden von elliptischen Funktionen erm ö glichen..

(For some values of a,b,c,d if you plot the curve it is an ellipse.) These Elliptic Curves with small keys are harder to crack than RSA with longer. They also use less resources during encryption and decryption Elliptic Curve Cryptography, as the name so aptly connotes, is an approach to encryption that makes use of the mathematics behind elliptic curves. I mentioned earlier that this can all feel a little bit abstract—this is the portion I was referring to. Let's start with what an X-axis is. And before you laugh, this is actually pretty critical to understanding ECC. Every point of the. ** Different functions can be adapted to data with the calculator: linear curve fit, polynomial curve fit, curve fit by Fourier series, curve fit by Gaussian distribution and power law curve fit**. Through the selection of further fit functions the functions are added to the chart. The style of the functions and points can be selected by means of the style information. The screenshot function. A Schematic Elliptic Curve Plot (source: CloudFlare) Adding two points on the curve, A and B, is our Billiards shot.To add A and B, place the ball at point A and shoot it towards point B.When it.

Plot two points on an elliptic curve. Now draw a straight line which goes through both points. That line will intersect the curve at some third point. That third point is the result of the addition operation. Point Doubling is similar and can be thought of as adding a point to itself. Imagine a point on the curve and draw a straight line which is a tangent to the curve at that point. The. Given an elliptic curve E a point on elliptic curve G (called the generator) and a private key k we can calculate the public key P where P = k * G. The whole idea behind elliptic curves cryptography is that point addition (multiplication) is a trapdoor function which means that given G and P points it is infeasible to find the private key k. Keep reading if you are interested to understand. The elliptic curve method (sometimes called Lenstra elliptic curve factorization, commonly abbreviated as ECM) If we plug in enough different values for x and plot the resultant y values on a graph, when we join the dots on the graph we get a curve. For the expression given above we would get a quadratic curve because the expression [math]\displaystyle{ y\,=\,3x^2\,+\,10x\,-\,9 }[/math] is. (typical plot for =) Weierstrass curves =0 =(0:1:0) Definition: a Weierstrass elliptic curve is defined by where , ∈satisfy 4 3+27 2≠0. The base point is the unique point at infinity. Can be shown: up to isomorphismevery elliptic curve is Weierstrass. Note: 1) the lines through =(0:1:0)are the vertical lines (except for the line at infinity =0). 2) The equation 2= 3+ + is. On an elliptic curve, if a line through two rational points P and Q intersects the curve again at R, then R is another rational point. This property is fundamental in number theory You beat me to solving and answering (+1, of course), so I took the liberty of adding the picture of the **elliptic** **curve** together with the tangent line at the point of interest. If you don't want your terse answer tarnished by something like that, please let me know, and I'll remove it. $\endgroup$ - Jyrki Lahtonen Nov 26 '14 at 20:1