Explicit Addition Formulae. Consider an elliptic curve E E (in Weierstrass form) Y 2 +a1XY +a3Y = X3+a2X2 +a4X+a6 Y 2 + a 1 X Y + a 3 Y = X 3 + a 2 X 2 + a 4 X + a 6. over a field K K. Let P = (x1,y1) P = ( x 1, y 1) be a point on E(K) E ( K) R= P+ Q: x. y. Point addition over the elliptic curve in 픽. The curve has points (including the point at infinity). Warning:this curve is singular. Warning:pis not a prime ∟ Algebraic Introduction to Elliptic Curves. ∟ Elliptic Curve Point Addition Example. This section provides algebraic calculation example of adding two distinct points on an elliptic curve. Now we algebraic formulas to calculate the addition operation on elliptic curves. Let's try them with some examples. The first example is adding 2 distinct points together, taken from Elliptic Curve. Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic curve cryptography as a means of producing a one-way function. The literature presents this operation as scalar multiplication, as written in Hessian form of an elliptic curve. A widespread name for this operation is also elliptic curve point multiplication, but this can convey the wrong impression of being a multiplication. The Algebra of Elliptic Curves Properties of \Addition on E Theorem The addition law on E has the following properties: (a) P +O = O +P = P for all P 2 E. (b) P +(¡P) = O for all P 2 E. (c) P +(Q+R) = (P +Q)+R for all P;Q;R 2 E. (d) P +Q = Q+P for all P;Q 2 E. In other words, the addition law + makes the points of E into a commutative group

- e a line, so it doesn't matter whether you do or ; the secant line is the same either way
- In der Mathematik sind elliptische Kurven spezielle algebraische Kurven, auf denen geometrisch eine Addition definiert ist. Diese Addition wird in der Kryptographie zur Konstruktion sicherer Verschlüsselungsmethoden verwendet. Elliptische Kurven spielen aber auch in der reinen Mathematik eine wichtige Rolle. Historisch sind sie durch die Parametrisierung elliptischer Integrale entstanden als deren Umkehrfunktionen. Eine elliptische Kurve ist eine glatte algebraische Kurve der.
- And finally, here are the two functions to compute negation and addition on the elliptic curve. The addition function is based directly on the formulas you gave (after correcting the sign of Z.y), makes use of inv_mod_p to perform the divisions modulo p, and does a final reduction modulo p for the computed x and y coordinates. def ec_inv(P): Inverse of the point P on the elliptic curve y^2.
- Elliptic Curve Calculator for elliptic curve E(F p): Y^2 =X^3+AX+B , p prime : mod p (be sure its a prime, just fermat prime test here, so avoid carmichael numbers) A: B (will be calculated so that point P is on curve) point P : x : y: point Q: x
- Elliptic Curve Key Agreement Elliptic Curve Integrated Encryption Scheme 2/29. Mathematische Grundlagen Charakteristik eines K orpers Charakteristik Es sei (K;+;) ein K orper. Die Charakteristik des K orpers ist die Ordnung des neutralen Elements der Multiplikation bez uglich der Addition. 1+1+|{z +1} n mal = 0 Gibt es kein solches n, so setzt man die Charakteristik 0. Gibt es ein solches n,
- This is the sum of the two points under elliptic curve addition
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- Elliptic curve point addition in projective coordinates Introduction. Elliptic curves are a mathematical concept that is useful for cryptography, such as in SSL/TLS and Bitcoin. Using the so-called group law, it is easy to add points together and to multiply a point by an integer, but very hard to work backwards to divide a point by a number; this asymmetry is the basis.
- Every math or cryptography student should know two fundamental facts about elliptic curves. Fundamental Fact 1. The addition rule for an elliptic curve is exactly the same as the addition rule for the circle. The addition rules are not merely similar. They are exactly the same rule applied to different curves. The addition of angles was already a familiar notion in Euclid, in history extending over centuries. Today, there are many ways to describe the addition of angles of a.
- Visualization of point addition on elliptic curve in simple Weierstrass form over real numbers and finite field. The underlying math is explained in next art..
- 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. 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, consists of solutions to: y 2 = x 3 + a x + b {\displaystyle y^{2}=x^{3}+ax+b} for some.

Addition law structure of elliptic curves David Kohel Institut de Math ematiques de Luminy Universit e de la M editerran ee 163, avenue de Luminy, Case 907 13288 Marseille Cedex 9 France Abstract The study of alternative models for elliptic curves has found recent interest from cryptographic applications, once it was recognized that such models provide more e ciently computable algorithms for. An elliptic curve addition law is said to be complete if it correctly computes the sum of any two points in the elliptic curve group. One of the main reasons for the increased popularity of Edwards curves in the ECC community is that they can allow a complete group law that is also relatively e cient (e.g., when compared to all known addition laws on Edwards curves). Such complete addition. Wolfram MathWorld gives an excellent and complete definition. But for our aims, an elliptic curve will simply be the set of points described by the equation : y 2 = x 3 + a x + b where 4 a 3 + 27 b 2 ≠ 0 (this is required to exclude singular curves). The equation above is what is called Weierstrass normal form for elliptic curves Addition on an Elliptic Curve and Modular Arithmetic involving fractions. 3. Definition of a Elliptic curve. 4. Computing the multiplicative inverse for point addition on an elliptic curve. 3. Adding two points on an elliptic curve. Where did I go wrong? 2. Elliptic curves with bad reduction on modular curves. 1. Can the base point of an elliptic curve be any point? 0. Elliptic curve scalar.

* We have skated over one issue in de ning addition on an elliptic curve, namely the fact that this operation is associative: P+ (Q+ R) = (P+ Q) + R: 3*.1 The 9th point lemma Our proof of associativity depends on the following remarkable geometric result, which asserts in e ect that any 8 points in general position on the plane determine a 9th point. Proposition 3.1 Suppose P i(i= 1 8) are 8. Faster addition and doubling on elliptic curves Daniel J. Bernstein1 and Tanja Lange2 1 Department of Mathematics, Statistics, and Computer Science (M/C 249) University of Illinois at Chicago, Chicago, IL 60607-7045, USA djb@cr.yp.to 2 Department of Mathematics and Computer Science Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands tanja@hyperelliptic.org. Elliptic Curve Addition Operations Elliptic curves have a few necessary peculiarities when it comes to addition. Two points on the curve (P, Q) will intercept the curve at a third point on the curve. When that point is reflected across the horizontal axis, it becomes the point (R) 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 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

Elliptic Curves over GF(p) Basically, an Elliptic Curve is represented as an equation of the following form. y 2 = x 3 + ax + b (Weierstrass Equation). Pre-condition: 4a 3 + 27b 2 ≠ 0 (To have 3 distinct roots). Addition of two points on an elliptic curve would be a point on the curve, too **addition** of points on an **elliptic** **curve** de nes a group structure. We only use explicit and very well{known formulas for the coordinates of the **addition** of two points. Even though the arguments in the proof are elementary, making this approach work requires several intricate arguments and elaborate computer calculations. The approach of this note was used by Laurent Th ery [Th07] to give a. Elliptic curve point addition is an operation on two point that results in a third. However, it doesn't work the way you might intuitively think. We don't just add the xs and ys. There are various rules for point addition depending on the relationship of the two points to be added. The meaning of each of these cases is covered in the book. The Point At Infinity. In the previous post where. Elliptic curves are the solutions sets of nonsingular cubic polynomials of degree three. It is possible to define an addition law for these points so that they form an abelian algebraic group. In order to add distinct points construct the line between them and determine the third point of intersection with the curve. The sum of the two points is then the reflection of the third point about the ax

**Addition** on an **Elliptic** **Curve** and Modular Arithmetic involving fractions. 3. Definition of a **Elliptic** **curve**. 4. Computing the multiplicative inverse for point **addition** on an **elliptic** **curve**. 3. Adding two points on an **elliptic** **curve**. Where did I go wrong? 2. **Elliptic** **curves** with bad reduction on modular **curves**. 1. Can the base point of an **elliptic** **curve** be any point? 0. **Elliptic** **curve** scalar. ** Note: addition of points on elliptic curves is not intuitive**. This kind of addition is defined the way it is because it has certain nice properties. It's weird, but it works. As elliptic curves form a group, there is an additive identity that is the equivalent of 0. That is, adding \$0\$ to any point will not change the result. This additive identity is the point at infinity. All lines on. Deriving Elliptic Curve Addition Formula. 0. Duplication Formula, Elliptic Curves. 0. Elliptic Curve Addition: what does it mean to have a 0 slope and what does it mean to have an infinity slope. Hot Network Questions In Tolkien's universe, what happens if a Ring of Power is worn by a member of a race other than the one the ring was made for? $1,500 Bosch Drawer Microwave causes ARC Faults. The use of elliptic curves in public-key cryptography can offer improved efﬁciency and bandwidth. Reza Rezaeian Farashahi( Dept. of Mathematical Sciences, Isfahan University of Technology, Isfahan, IranDifferential addition on Binary Elliptic Curves July 13 , 2016 7 / 38joint work with S. Gholamhossein HosseiniWAIFI 2016, Ghent , Belgium Elliptische Kurven werden bei einem bekannten Verschlüsselungsverfahren (ECC, Elliptic Curve Cryptography) verwendet. Außerdem spielen sie in einigen Gebieten der modernen Mathematik eine wichtige Rolle, unter anderem beim Beweis der berühmten Fermatschen Vermutung durch Andrew Wiles (1994). Auf elliptischen Kurven definiert man eine Addition, die zwei gegebenen Punkten P und Q (blau) der.

- Point Addition. We're given an algorithm for efficiently adding points on an elliptic curve (better than doing it geometrically every time!), which we need to implement: Using the above curve, and the points P = (493, 5564), Q = (1539, 4742), R = (4403,5202), find the point S(x,y) = P + P + Q + R by implementing the above algorithm. So let's do it! I'm going to create some simple helper.
- Associativity of point addition on an elliptic curve in fact is a non-trivial and fragile property. Messing with how we do point addition in almost any way (changing sign as proposed, using a curve with a different equation like an astroid..) breaks that property
- Addition - Given two points, we can add them to one another (or subtract) and the result would be a new point on the curve. Multiplication - Given a point, we can multiply it any number of times. Addition. Given three aligned points P, Q and R, their sum is always 0. We treat this as an inherent property of elliptic curves
- Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic curve cryptography (ECC) as a means of producing a one-way function.The literature presents this operation as scalar multiplication, as written in Hessian form of an elliptic curve.A widespread name for this operation is also elliptic curve.
- The concrete type of elliptic curves found by Weierstrass now carry his name. They are the most famous shapes of elliptic curves. Assume char≠2,3. 2 2== 33++ 2++ 3 (typical plot for =) Weierstrass curves =0 =(0:1:0) Definition: a Weierstrass elliptic curve is defined b
- In this paper we revisit the addition of elliptic curves and give an algebraic proof to the associative law by use of MATHEMATICA. The existing proofs of the associative law are rather complicated and hard to understand for beginners. An ''elementary proof to it based on algebra has not been given as far as we know. Undergraduates or non-experts can master the addition of elliptic.

** Keywords: elliptic curves, addition, doubling, explicit formulas, register allocation, scalar mul-tiplication, multi-scalar multiplication, side-channel countermeasures, uniﬁed addition formulas, complete addition formulas, eﬃcient implementation, performance evaluation 1 Introduction The core operations in elliptic-curve cryptography are single-scalar multiplication (m,P 7→ mP), double**. The elliptic curve addition formula then becomes equivalent to adding the coordinates like vectors, and then subtracting multiples of !1 and !2 until the point ends back up within the parallelogram. O is sent to the origin. Ben Wright and Junze Ye Elliptic Curves: Theory and Application. Structure of Complex Points Let E(C) be all the points with complex coordinates on the elliptic curve. Then. Isogeny class of elliptic curves over number fields; Tate-Shafarevich group; Complex multiplication for elliptic curves; The following relate to elliptic curves over local nonarchimedean fields. Local data for elliptic curves over number fields; Kodaira symbols; Tate's parametrisation of \(p\)-adic curves with multiplicative reduction; Analytic properties over \(\CC\). Weierstrass \(\wp. This paper presents further co-Z addition formulæ for various point additions on Weierstraß elliptic curves. It explains how the use of conjugate point addition and other implementation tricks allow one to develop efficient scalar multiplication algorithms making use of co-Z arithmetic. Specifically, this paper describes efficient co-Z based versions of Montgomery ladder and Joye's double.

- The addition of points on an elliptic curve E satis es the following properties: 1. (commutativity) P 1 + P 2 = P 2 + P 1 for all P 1;P 2 on E. 2. (existence of identity) P + 1= P for all points P on E. 3. (existence of inverses) Given P on E, there exists P0on E with P +P0= 1. This point P0will usually be denoted P. 4. (associativity) (P 1 + P 2) + P 3 = P 1 + (P 2 + P 3) for all P 1;P 2;P 3.
- 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.
- An elliptic curve consists of all the points that satisfy an equation of the following form: y² = x³+ax+b. where 4a³+27b² ≠ 0 (this is required to avoid singular points). Here are some example elliptic curves: Notice that all the elliptic curves above are symmetrical about the x-axis. This is true for every elliptic curve because the equation for an elliptic curve is: y² = x³+ax+b. And.
- Interactive Elliptic Curve Point Addition in the 2D Real Plane. Interactive elliptic curve calculator built in Desmos graphing tool. Mathematics of Elliptic Curve Addition and Multiplication Curve point addition on elliptic curves is defined in a very weird and interesting way. To add two curve points (x1,y1) and (x2,y2), we: D raw a line between the two points. This makes our operation.
- Keywords: Cryptography; Elliptic curve cryptography; Point addition; Point doubling. 1. Introduction. Cryptography is transformation of plain message to make them secure and immune from intruders. Elliptic Curve Cryptography (ECC) is a public key cryptography developed independently by Victor Miller and Neal Koblitz in the year 1985. In Elliptic Curve Cryptography we will be using the curve.
- 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
- g operation in classical ECC iselliptic-curve scalar multiplication: Given an integer n and an elliptic-curve pointP, compute nP. It is easy to ﬁnd the opposite of a point, so we assume n >0. Scalar multiplication is the inverse of ECDLP (given P and nP, compute n). Scalar multiplication behaves.

Example of elliptic curve addition Consider the elliptic curve defined in the previous example. 1. Let P = (3, 10) and Q = (9, 7). Then P + Q = (x 3, y 3) iS computed as follows: λ= − − = − = − =∈ 710 93 3 6 1 2 11 23. x3 = 11 2 - 3 - 9 = 6 - 3 - 9 = -6 ≡ 17 (mod 23), and y3 = 11(3 - (-6)) -10 = 11(9) -10 = 89 ≡ 20 (mod 23). Hence P + Q = (17, 20). 2. Let P = (3,10). Then 2P = P. and mechanics of cryptography, elliptic curves, and how the two manage to t together. Secondly, and perhaps more importantly, we will be relating the spicy details behind Alice and Bob's decidedly nonlinear relationship. 2 Algebra Refresher In order to speak about cryptography and elliptic curves, we must treat ourselves to a bit of an algebra refresher. We will concentrate on the algebraic.

- I want to do point subtraction on an elliptic curve on a prime field. I tried taking the points to be subtracted as (x,-y log(p)) but my answer doesn't seem to match. This is how I tried to do the subtraction: s9=point_addition(s6.a,s6.b,((s8.a)%211) ,-((s8.b)%211)); here s9, s6 and s8 are all structures with two int
- \addition on elliptic curves: P Q P + Q jjjjjj jjjjjj jjjjjj j y x OO // II Addition on y 2 5 xy = x 3 7. 2007.01.10, 09:00 (yikes!), Leiden University, part of \Mathematics: Algorithms and Proofs week at Lorentz Center: Harold Edwards speaks on \Addition on elliptic curves. Edwards What we think when we hear \addition on elliptic curves: P Q P + Q jjjjjj jjjjjj jjjjjj j y x OO // II.
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- ology, but if you can bear with me I'd like to ask this question anyway. I'll start with what I understand... You do EC multiplication with the private key to get the public key.

Point Addition is essentially an operation which takes any two given points on a curve and yields a third point which is also on the curve. The maths behind this gets a bit complicated but think of it in these terms. 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. Suitable Elliptic Curves for ECM (1) For ECM we use elliptic curves over Q (rank >0) which have a prescribed torsion subgroup. When reducing those modulo p, we know already some divisors of the group order. Theorem. Let E=Q be an elliptic curve and let m be a positive integer such that gcd(m;p)=1. If E modulo p is non-singular the reduction. Otherwise, return the curve point `x + y` where `+` is point addition on the elliptic curve `alt_bn128` specified above. This comment has been minimized. Sign in to view. axic Feb 13, 2017. Member Probably better to format it as: #### ADD Input: two curve points `(x, y)`... Output: resulting curve pointdescription... pirapira mentioned this pull request Feb 13, 2017. Byzantium changes. For point addition, we take two points on the elliptic curve and then add them together (R=P+Q). Ref: here. So if we use x³+ax+b (mod p), and we have two points P (x1,y1) and Q(x2,y2) that we. elliptic curve additions An elliptic curve addition is performed by using a few ﬁnite ﬁeld operations Implementation of elliptic curve addition op-eration requires implementation of four basic ﬁnite ﬁeld operations: addition, subtraction, multiplication, and inversion For example, addition of two distinct points requires 2 ﬁeld multiplications and 1 ﬁeld in-version Inversion is a.

Cody is a MATLAB problem-solving game that challenges you to expand your knowledge. Sharpen your programming skills while having fun 15.Elliptic Curves At the end of the last chapter we have used Picard groups to show in Proposition14.19and Remark 14.20that smooth cubic curves in P2 are not isomorphic to P1. In fact, if our ground ﬁeld is not necessarily C (so that we cannot apply topological methods as in Remark13.19), this is the ﬁrst class of smooth projective curves for which we could prove rigorously that they are. 2.2 Elliptic Curve Addition: An Algebraic Approach. Contact; Home Although the previous geometric descriptions of elliptic curves provides an excellent method of illustrating elliptic curve arithmetic, it is not a practical way to implement arithmetic computations. Algebraic formulae are constructed to efficiently compute the geometric arithmetic. 2.2.1 Adding distinct points P and Q. When P. ** Elliptic curves are curves defined by a certain type of cubic equation in two variables**. The set of rational solutions to this equation has an extremely interesting structure, including a group law. The theory of elliptic curves was essential in Andrew Wiles' proof of Fermat's last theorem. Computational problems involving the group law are also used in many cryptographic applications, and in.

Elliptic curve E Figure 1 Addition rule over an elliptic curve. 142 FUJITSU Sci. Tech. J.,36, 2,(December 2000) N. Torii et al.: Elliptic Curve Cryptosystem the point G. It is known that n is a divisor of the order of the curve E. Elliptic curves over a characteristic 2 finite field GF(2 m) which has 2 m elements have also been constructed and are being standardized for use in ECCs as. The elliptic curve method (sometimes called Lenstra elliptic curve factorization, commonly abbreviated as ECM) is a factorization method which computes a large multiple of a point on a random elliptic curve modulo the number to be factored. It is currently the best algorithm known, among those whose complexity depends mainly on the size of the factor found

E cient Elliptic Curve Exponentiation Using Mixed Coordinates Henri Cohen1, Atsuko Miyaji2, and Takatoshi Ono3 1 Laboratoire A2X, Universit e Bordeaux I 2 Multimedia Development Center, Matsushita Electric Industrial Co., Ltd. 3 Matsushita Information Systems Research Laboratory Nagoya Co., Ltd. Abstract. Elliptic curve cryptosystems, proposed by Koblitz ([12]) an Welcome to part one of exploring Programming Bitcoin's third chapter on elliptic curve cryptography in Clojure. In this section, we will be combining the subjects of the previous two chapters: Finite Fields and Elliptic Curves. Together, they make up the necessary ingredients to create the cryptographic primitives we need to build our signing and verification algorithms, which we will be.

ec_scalar_multiply([int] r, [curve_point] G): This implements curve point scalar multiplication. It gives an identical result to taking r identical copies of the curve point G and adding them together using curve point addition. For the anonymous credential operations, Findora uses a separate elliptic curve which additionally supports a pairing. Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves

Unter Elliptic Curve Cryptography (ECC) oder deutsch Elliptische-Kurven-Kryptografie versteht man asymmetrische Kryptosysteme, die Operationen auf elliptischen Kurven über endlichen Körpern verwenden. Diese Verfahren sind nur sicher, wenn diskrete Logarithmen in der Gruppe der Punkte der elliptischen Kurve nicht effizient berechnet werden können In mathematics, an elliptic curve (EC) is a smooth, projective algebraic curve of genus one, on which there is a specified point.Any elliptic curve can be written as a plane algebraic curve defined by an equation, which is non-singular; that is, its graph has no cusps or self-intersections

In classifying curves and their addition laws, it there- re makes sense to classify elliptic curves up to projective linear isomorphism. mma 3. Let E 1 and E 2 be projectively normal embeddings of an elliptic curve E defined with respect to visors D 1 and D 2 . Then there exists a projective linear isomorphism E 1 → E 2 if and only if deg(D 1 )> g(D 2 ) or D 1 ∼ D 2 . oof. An equivalence. Elliptic Curve Cryptography (ECC) Point Addition. {parts: [ {partUri:/matlab/document.xml,contentType:application/vnd.mathworks.matlab.code.document+xml,content:<?xml version=\1.0\ encoding=\UTF-8\?><w:document xmlns:w=\http://schemas.openxmlformats.org/wordprocessingml/2006/main\><w:body><w:p><w:pPr><w:pStyle. Elliptic curves E(k): elliptic curve over a eld k with char(k) 6= 2 ;3 Every elliptic curve can be written inshort Weierstrass form I Embedded in P2(k) as E : Y2Z = X3 + aXZ2 + bZ3 I The point O= (0 : 1 : 0) is called thepoint at in nity I A ne points (x : y : 1) given by y2 = x3 + ax + b I The points on E form anabelian groupunder point addition Elliptic curves appear in diverse contexts: A commutative and associative addition law that makes an elliptic curve an abelian group Over C, isomorphic to tori Over number ﬁelds F, ﬁnite generation of F-rational points (Mordell-Weil) Over number ﬁelds, ﬁnite number of rational torsion points (Nagell-Lutz and Mazur Consider the elliptic curve given by y2+ y = x3+ wx over the field GF(4) whose elements are {0,1,w,w2}. Besides O, the points on this curve have coordinates (0,0), (0,1), (w2,0) and (w2,1). Let P = (0,1) and Q = (w2,0). The line joining P and Q has the (affine) equation y = wx + 1

Elliptic curves are curves defined by a certain type of cubic equation in two variables. The set of rational solutions to this equation has an extremely interesting structure, including a group law. The theory of elliptic curves was essential in Andrew Wiles' proof of Fermat's last theorem. Computational problems involving the group law are also used in many cryptographic applications, and in algorithms for factoring large integers. (2) and (3) (1), (2), and ADDITION of POINTS on ELLIPTIC CURVES - FORMULAS) Formulas Addition of points P 1 = (x 1;y 1)and P 2 = (x 2;y 2)of an elliptic curve E : y2 = x3 + ax + b can be easily computed using the following formulas: P 1 + P 2 = P 3 = (x 3;y 3) where x 3 = 2 x 1 x 2 y 3 = (x 1 x 3) y 1 and = 8 >> < >>: (y 2 y 1) (x 2 x 1) if P 1 6= P 2; (3x2 1 + a) (2y 1) if P 1 = P 2

Elliptic curves have been used to shed light on some important problems that, at ﬁrst sight, appear to have nothing to do with elliptic curves. I mention three such problems. Fast factorization of integers There is an algorithm for factoring integers that uses elliptic curves and is in many respects better than previous algorithms. People have been factoring in-tegers for centuries, but. formal request to CFRG for new elliptic curves for usage in TLS!!! • Fastest addition law • Some have complete group law Weierstrass curves 2= 3+ + • Most general form • Prime order possible • Exceptions in group law • NIST and Brainpool curves The chosen ones. Complete addition on Edwards curves Let ≠ in and consider Edwards curve / ∶ 2+ 2=1+ 2 2 For all (!!!) 1= 1, 1, 2= Although elliptic curves had been studied earlier, indeed in great depth by Fermat, Euler's analysis clariﬁes the key points: elliptic curves (algebraic curves of genus 1) are fundamentally dif- ferent from rational curves, and not only in a negative way. They have a diﬀerent kind of symmetry, the famous group structure possessed by an elliptic curve. This paper considers two main themes. In geometric terms, the rules for addition can be stated as follows: If three points on an elliptic curve lie on a straight line, their sum is O. From this definition, we can define the rules of addition over an elliptic curve: O serves as the additive identity. Thus O = O; for any point P on the elliptic curve, P + O = P. In what follows, we assume P