Elliptic Curves and Public Key Cryptography

Size: px
Start display at page:

Download "Elliptic Curves and Public Key Cryptography"

Transcription

1 Elliptic Curves and Public Key Cryptography Jeff Achter January 7, Introduction to Elliptic Curves 1.1 Diophantine equations Many classical problems in number theory have the following form: Let R be an interesting ring, such as: Z, the integers; or Q = { a b : a, b Z} the rationals; or Z/p, the integers modulo p. Let f (X, Y) be a polynomial with coefficients in R. We would like to describe the set of R-solutions to f (X, Y) = 0, i.e., the set {(x, y) : x, y R, f (x, y) = 0}. Is this set empty? Finite? Infinite? For example, like Bachet in 1621, we might consider the equation y 2 x = 0, and look for solutions in Q. It has the easy solution (3, 5), since = 2. It also has harder solutions, since 2 can also be written as ( ) ( ) or 100 ( ) ( ) or ( ) 2 ( ) 3... How did he come up with this? It s actually easier to back up before we go forwards. About 1400 years earlier, Diophantus of Alexandria developed a number of techniques for dealing with the equations which now bear his name. For example, he looked at quadratic equations in two variables, like x 2 + 2xy + 3y 2 + x + 2y = 0 and was able to show: 1

2 1.1 Diophantine equations 1 INTRODUCTION TO ELLIPTIC CURVES Theorem 1.1. If the quadratic equation F(X, Y) = 0 has one rational solution, then it has infinitely many solutions. His proof is quite geometric. A picture has been left out here. Sketch. Let C be the graph of F(X, Y) = 0. Start with the known rational solution P. For each rational number m, let L m be the line through P with slope m. Then L m and C intersect in two points, P and a second point, Q m ; and the point Q m has rational coordinates. For example, if we start with the point ( 1, 0) satisfying x 2 + 2xy + 3y 2 + x + 2y = 0, the line through ( 1, 0) with slope m is given by So to find the points of intersection, solve y 0 = m (x ( 1)) y = mx + m x 2 + 2xy + 3y 2 + x + 2y = 0 x 2 + 2x(mx + m) + 3(mx + m) 2 + x + 2(mx + m) = 0 x 2 + 2mx 2 + 2mx + 3m 2 x 2 + 6m 2 x + 3m 2 + x + 2m = 0 (x + 1)(x + 2mx + 3m 2 x + 3m 2 + 2m) = 0 One solution is x = 1, the solution we already knew about; the other is and thus m(3m + 2) x m = 1 + 2m + 3m 2 y m = m(x + 1) m = 1 + 2m + 3m 2 y = mx + m Note that if m is rational, then both x m and y m are rational numbers. A little more work then shows that distinct values of m give rise to distinct solutions to f (x, y) = 0; and that every solution arises in this way. We may now return to Bachet s problem y 2 = x 3 2, and the known point (3, 5). We may construct the line of slope m through (3, 5) and intersect it with Bachet s curve: For instance, if we take the line of slope m = 3, we try to solve y 2 = x 3 2 y 5 = 3(x 3) 2

3 1.1 Diophantine equations 1 INTRODUCTION TO ELLIPTIC CURVES and find that y = 3x 4 (3x 4) 2 = x 3 2 x 3 9x x 18 = 0 (x 3)(x 2 6x + 6) = 0 x {3, 3 + 3, 3 3}. But we ve lost rationality! However, if we compute the tangent line to Bachet s curve at our known point, we find it has slope y 2 = x 3 2 2y dy = 3x 2 dy dx = 3x2 2y and so at (3, 5), it has slope /2 5 = ; the tangent line is given by and we obtain y 3 = 21 (x 5) 10 x x x = 0 (x 3)(x 3)(x ) = 0 and we have found a new, rational x-coordinate In general, we can explain Bachet s result: Theorem 1.2. Bachet If (x, y) is a solution to Y 2 X 3 = c, then so is as follows: ( x 4 8cx 4y 2, x6 20cx 3 + 8c 2 ) 8y 3. Compute the tangent line to the curve y 2 x 3 = c at the given point. Compute the intersection with the original curve. There will be a unique new solution. 3

4 1.2 Points on cubics 1 INTRODUCTION TO ELLIPTIC CURVES 1.2 Points on cubics Intersections Now consider a general cubic polynomial f (x, y) = 0 and the curve C it defines. Suppose P = (x 0, y 0 ) and Q = (x 1, y 1 ) are points on the curve. We would like to implement the following strategy for generating new points: Construct the line L = L PQ. Intersect L with C. And hopefully, this yields another solution. This will work, provided that #L C = 3. But there may be fewer points than we want, for a couple of different reasons. For example, consider the curve y 2 x 3 + 2: The line L : y = 3x 4 intersects C in only one Q solution. This is just because we haven t used enough numbers a statement like the one we re after can only be true if we re willing to work with K alg, instead of K. The line M : y = x intersects C in only two points, namely, (3, 5) and ( 100, ). In point of fact, we should have counted (3, 5) twice, since the line M is tangent to C (to order 2) at this point. We can see this algebraically; recall that the x-values of intersections satisfied the polynomial (x 3) 2 (x ) = 0. The vertical line x = 3 2 only intersects C at ( 3 2, 0). Multiplicity won t get us out of trouble; the algebraic equation just becomes y 2 = 0, and thus the intersection N C contains (3, 5) with multiplicity two. Fixing this last problem requires a little geometric ingenuity. If we replaced C with some other line, the sought-for claim would be: Two distinct lines meet in a single point. Which is clearly false, since parallel lines don t meet. The solution is to add a point at infinity in each direction; and two parallel lines are declared to intersect in the corresponding point at infinity. We can make sense of this geometry, and the resulting object is called the projective plane, P 2. Return to our continuing example y 2 = x 3 2; recall that the slope of the tangent line at (x, y) is 3x2 2y. As x, this tangent line becomes vertical. So we declare that C contains the point at infinity corresponding to the vertical direction. Now, in our alleged counterexample, N C consists of ( 3 2, 0) with multiplicity two and the point at infinity in the vertical direction; henceforth, this last point will be denoted O Combining points Let f (x, y) be a nonsingular cubic polynomial, C the correspondign curve, and P i = (x i, y i ) i {0, 1} be solutions to f (x, y) = 0. Then we can define a point as follows: P 2 = P 0 P 1 4

5 1.2 Points on cubics 1 INTRODUCTION TO ELLIPTIC CURVES Compute the intersection of C and the line P 0 P 1. Look for the third point. Moreover, we have: Theorem 1.3. Suppose f has coefficients in a field K, and P 0, P 1 defined over K. Then P 0 P 1 exists, and is defined over K. May have to allow a point at infinity. The rationality can be seen from explicit formulae, if desired. We compute in the special case of y 2 = x 3 2. Suppose P 0, P 1 defined over some field K, and x 0 = x 1. Then Solve y = µx + β and y 2 = x 3 2: But we know two solutions, x 0 and x 1 ; thus and L P0 P 1 : y = µx + β µ = y 1 y 0 x 1 x 0 β = y 0 µx 0 x 3 ax b y 2 = 0 x 3 ax b (µx + β) 2 = 0 x 3 µ 2 x 2 (a + 2µβ)x (b + β 2 ) = 0 x 3 µ 2 x 2 (a + 2µβ)x (b + β 2 ) = (x x 0 )(x x 1 )(x x 2 ) x 2 = µ 2 x 0 x 1 y 2 = µx 2 + β Loose ends Given a cubic polynomial ax 3 + bx 2 y + cx 2 y + dy 3 + ex 2 + f xy + gy 2 + hx + iy + j, can (eventually) change coordinates so that it looks like y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 ; and O at infinity corresponds to vertical lines. Moreover, except in characteristic two or three, can complete the square: y 1 2 (y a 1x a 3 ) 5

6 1.3 Elliptic curves 1 INTRODUCTION TO ELLIPTIC CURVES and then rescale to get y 2 = x 3 + ax + b, the short Weierstrass form of a cubic. Even though characteristic 2 is especially interesting for computer applications, in these notes I am going to ignore characteristic two, so that the formulas stay relatively manageable. So, given y 2 = f (x) = x 3 + ax + b, we have: the curve is smooth it doesn t intersect itself both derivatives never simultaneously vanish f (x) has distinct roots the discriminant = 16(4a b 2 ) is nonzero. 1.3 Elliptic curves An elliptic curve (in short Weierstrass form) is an equation y 2 = f (x),, f a smooth cubic as above. (More intrinsically, an elliptic curve E is a smooth projective irreducible curve of genus one equipped with a rational point O.) We ll use the notation E(K) for the set of points on the curve. More generally, if L/K is any extension, let E(L) be the set of points on E with coordinates in L. If P 0 P 1 = (x 2, y 2 ), define P 0 + P 1 = (x 2, y 2 ), the point obtained by reflecting P 0 P 1 across the x-axis. (Somewhat more intrinsically, this is (P 0 P 1 ) O.) Theorem 1.4. The operation + turns E(K) into an abelian group: identity element is O; P = (x, y) = (x, y). There s a clean, geometric characterization of the group law: Theorem 1.5. The set E(K), with: identity element O, and group law specified by P + Q + R = O P, Q and R collinear is an abelian group. 6

7 2 PUBLIC KEY CRYPTOGRAPHY Problems (1.1). Let E/Q be the elliptic curve y 2 = x What is ( 2, 3) + ( 1, 4)? In the next problems, we work with an elliptic curve over a field K in which 2 is invertible. (1.2). Suppose E is given in the form y 2 = f (x). (a) Describe all points P such that 2P = P + P = O. (HINT: Note that P + P = O P = P.) (b) Show that 2P = O P = O or x(p) is a root of f (x). (c) Suppose K is algebraically closed. Show that is isomorphic to Z/2 Z/2. E[2](K) = {P E(K) : 2P = O} (1.3). Suppose E is given by a equation y 2 = x 3 + ax + b, and P = (x, y) E(K) is not O, y = y(p) = 0. Find a formula for 2P: (a) What is the slope of the tangent line L to E at P? (b) Find a formula for L. (c) Find all points of intersection of L and E. You should be able to get a formula for 2P of the form (g(x, y), h(x, y)), where g and h are rational functions. If the denominator of g vanishes for some particular point P 0 = (x 0, y 0 ), we interpret this as 2P 0 = O. (1.4). (a) Find a formula for 3P. (HINT: Use problem (1.3), and the fact that 3P = P + 2P.) (b) Suppose K is algebraically closed, char(k) = 3. Given your formula, how many points P do you think satisfy 3P = O? 2 Public key cryptography 7

LECTURE 7, WEDNESDAY

LECTURE 7, WEDNESDAY LECTURE 7, WEDNESDAY 25.02.04 FRANZ LEMMERMEYER 1. Singular Weierstrass Curves Consider cubic curves in Weierstraß form (1) E : y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6, the coefficients a i

More information

Elliptic Curves. Dr. Carmen Bruni. November 4th, University of Waterloo

Elliptic Curves. Dr. Carmen Bruni. November 4th, University of Waterloo University of Waterloo November 4th, 2015 Revisit the Congruent Number Problem Congruent Number Problem Determine which positive integers N can be expressed as the area of a right angled triangle with

More information

Elliptic Curves and Mordell s Theorem

Elliptic Curves and Mordell s Theorem Elliptic Curves and Mordell s Theorem Aurash Vatan, Andrew Yao MIT PRIMES December 16, 2017 Diophantine Equations Definition (Diophantine Equations) Diophantine Equations are polynomials of two or more

More information

Grade 11/12 Math Circles Elliptic Curves Dr. Carmen Bruni November 4, 2015

Grade 11/12 Math Circles Elliptic Curves Dr. Carmen Bruni November 4, 2015 Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 11/12 Math Circles Elliptic Curves Dr. Carmen Bruni November 4, 2015 Revisit the Congruent Number

More information

LECTURE 5, FRIDAY

LECTURE 5, FRIDAY LECTURE 5, FRIDAY 20.02.04 FRANZ LEMMERMEYER Before we start with the arithmetic of elliptic curves, let us talk a little bit about multiplicities, tangents, and singular points. 1. Tangents How do we

More information

ELLIPTIC CURVES BJORN POONEN

ELLIPTIC CURVES BJORN POONEN ELLIPTIC CURVES BJORN POONEN 1. Introduction The theme of this lecture is to show how geometry can be used to understand the rational number solutions to a polynomial equation. We will illustrate this

More information

INTRODUCTION TO ELLIPTIC CURVES

INTRODUCTION TO ELLIPTIC CURVES INTRODUCTION TO ELLIPTIC CURVES MATILDE LALÍN Abstract. These notes correspond to a mini-course taught by the author during the program Two Weeks at Waterloo - A Summer School for Women in Math. Please

More information

ALGEBRAIC GEOMETRY HOMEWORK 3

ALGEBRAIC GEOMETRY HOMEWORK 3 ALGEBRAIC GEOMETRY HOMEWORK 3 (1) Consider the curve Y 2 = X 2 (X + 1). (a) Sketch the curve. (b) Determine the singular point P on C. (c) For all lines through P, determine the intersection multiplicity

More information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

2.1 Affine and Projective Coordinates

2.1 Affine and Projective Coordinates 1 Introduction Depending how you look at them, elliptic curves can be deceptively simple. Using one of the easier definitions, we are just looking at points (x,y) that satisfy a cubic equation, something

More information

Elliptic Curves, Factorization, and Cryptography

Elliptic Curves, Factorization, and Cryptography Elliptic Curves, Factorization, and Cryptography Brian Rhee MIT PRIMES May 19, 2017 RATIONAL POINTS ON CONICS The following procedure yields the set of rational points on a conic C given an initial rational

More information

CS 259C/Math 250: Elliptic Curves in Cryptography Homework 1 Solutions. 3. (a)

CS 259C/Math 250: Elliptic Curves in Cryptography Homework 1 Solutions. 3. (a) CS 259C/Math 250: Elliptic Curves in Cryptography Homework 1 Solutions 1. 2. 3. (a) 1 (b) (c) Alternatively, we could compute the orders of the points in the group: (d) The group has 32 elements (EF.order()

More information

CORRESPONDENCE BETWEEN ELLIPTIC CURVES IN EDWARDS-BERNSTEIN AND WEIERSTRASS FORMS

CORRESPONDENCE BETWEEN ELLIPTIC CURVES IN EDWARDS-BERNSTEIN AND WEIERSTRASS FORMS CORRESPONDENCE BETWEEN ELLIPTIC CURVES IN EDWARDS-BERNSTEIN AND WEIERSTRASS FORMS DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF OTTAWA SUPERVISOR: PROFESSOR MONICA NEVINS STUDENT: DANG NGUYEN

More information

a factors The exponential 0 is a special case. If b is any nonzero real number, then

a factors The exponential 0 is a special case. If b is any nonzero real number, then 0.1 Exponents The expression x a is an exponential expression with base x and exponent a. If the exponent a is a positive integer, then the expression is simply notation that counts how many times the

More information

The Group Structure of Elliptic Curves Defined over Finite Fields

The Group Structure of Elliptic Curves Defined over Finite Fields The Group Structure of Elliptic Curves Defined over Finite Fields A Senior Project submitted to The Division of Science, Mathematics, and Computing of Bard College by Andrija Peruničić Annandale-on-Hudson,

More information

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions. Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

More information

On the Torsion Subgroup of an Elliptic Curve

On the Torsion Subgroup of an Elliptic Curve S.U.R.E. Presentation October 15, 2010 Linear Equations Consider line ax + by = c with a, b, c Z Integer points exist iff gcd(a, b) c If two points are rational, line connecting them has rational slope.

More information

Integration - Past Edexcel Exam Questions

Integration - Past Edexcel Exam Questions Integration - Past Edexcel Exam Questions 1. (a) Given that y = 5x 2 + 7x + 3, find i. - ii. - (b) ( 1 + 3 ) x 1 x dx. [4] 2. Question 2b - January 2005 2. The gradient of the curve C is given by The point

More information

CONGRUENT NUMBERS AND ELLIPTIC CURVES

CONGRUENT NUMBERS AND ELLIPTIC CURVES CONGRUENT NUMBERS AND ELLIPTIC CURVES JIM BROWN Abstract. In this short paper we consider congruent numbers and how they give rise to elliptic curves. We will begin with very basic notions before moving

More information

Outline of the Seminar Topics on elliptic curves Saarbrücken,

Outline of the Seminar Topics on elliptic curves Saarbrücken, Outline of the Seminar Topics on elliptic curves Saarbrücken, 11.09.2017 Contents A Number theory and algebraic geometry 2 B Elliptic curves 2 1 Rational points on elliptic curves (Mordell s Theorem) 5

More information

Elliptic Curves and Public Key Cryptography (3rd VDS Summer School) Discussion/Problem Session I

Elliptic Curves and Public Key Cryptography (3rd VDS Summer School) Discussion/Problem Session I Elliptic Curves and Public Key Cryptography (3rd VDS Summer School) Discussion/Problem Session I You are expected to at least read through this document before Wednesday s discussion session. Hopefully,

More information

An Introduction to Elliptic Curves Dr. Reza Akhtar Algebra Short Course (SUMSRI) Miami University Summer 2002

An Introduction to Elliptic Curves Dr. Reza Akhtar Algebra Short Course (SUMSRI) Miami University Summer 2002 An Introduction to Elliptic Curves Dr. Reza Akhtar Algebra Short Course (SUMSRI) Miami University Summer 2002 Supplementary References Silverman, Joseph H. and Tate, John. Rational Points on Elliptic Curves.

More information

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

DRAFT - Math 101 Lecture Note - Dr. Said Algarni 2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that

More information

6x 2 8x + 5 ) = 12x 8

6x 2 8x + 5 ) = 12x 8 Example. If f(x) = x 3 4x + 5x + 1, then f (x) = 6x 8x + 5 Observation: f (x) is also a differentiable function... d dx ( f (x) ) = d dx ( 6x 8x + 5 ) = 1x 8 The derivative of f (x) is called the second

More information

Algebraic Geometry: Elliptic Curves and 2 Theorems

Algebraic Geometry: Elliptic Curves and 2 Theorems Algebraic Geometry: Elliptic Curves and 2 Theorems Chris Zhu Mentor: Chun Hong Lo MIT PRIMES December 7, 2018 Chris Zhu Elliptic Curves and 2 Theorems December 7, 2018 1 / 16 Rational Parametrization Plane

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013 18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and

More information

Group Law for elliptic Curves

Group Law for elliptic Curves Group Law for elliptic Curves http://www.math.ku.dk/ verrill/grouplaw/ (Based on Cassels Lectures on Elliptic curves, Chapter 7) 1 Outline: Introductory remarks Construction of the group law Case of finding

More information

Higher Portfolio Quadratics and Polynomials

Higher Portfolio Quadratics and Polynomials Higher Portfolio Quadratics and Polynomials Higher 5. Quadratics and Polynomials Section A - Revision Section This section will help you revise previous learning which is required in this topic R1 I have

More information

6x 2 8x + 5 ) = 12x 8. f (x) ) = d (12x 8) = 12

6x 2 8x + 5 ) = 12x 8. f (x) ) = d (12x 8) = 12 AMS/ECON 11A Class Notes 11/6/17 UCSC *) Higher order derivatives Example. If f = x 3 x + 5x + 1, then f = 6x 8x + 5 Observation: f is also a differentiable function... d f ) = d 6x 8x + 5 ) = 1x 8 dx

More information

Elliptic Curves: An Introduction

Elliptic Curves: An Introduction Elliptic Curves: An Introduction Adam Block December 206 Introduction The goal of the following paper will be to explain some of the history of and motivation for elliptic curves, to provide examples and

More information

Functions and Equations

Functions and Equations Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN

More information

Systems of Equations and Inequalities. College Algebra

Systems of Equations and Inequalities. College Algebra Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system

More information

Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions. Chapter 3 Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

More information

Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions Title Integration of Rational Functions by MATH 1700 MATH 1700 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 2 / 11 Rational functions A rational function is one of the form where P and Q are

More information

Partial Fractions. (Do you see how to work it out? Substitute u = ax + b, so du = a dx.) For example, 1 dx = ln x 7 + C, x x (x 3)(x + 1) = a

Partial Fractions. (Do you see how to work it out? Substitute u = ax + b, so du = a dx.) For example, 1 dx = ln x 7 + C, x x (x 3)(x + 1) = a Partial Fractions 7-9-005 Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals of the form P(x) dx, wherep(x) and Q(x) are polynomials. Q(x) The idea

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1

More information

Elliptic Curves: Theory and Application

Elliptic Curves: Theory and Application s Phillips Exeter Academy Dec. 5th, 2018 Why Elliptic Curves Matter The study of elliptic curves has always been of deep interest, with focus on the points on an elliptic curve with coe cients in certain

More information

Elliptic curve cryptography. Matthew England MSc Applied Mathematical Sciences Heriot-Watt University

Elliptic curve cryptography. Matthew England MSc Applied Mathematical Sciences Heriot-Watt University Elliptic curve cryptography Matthew England MSc Applied Mathematical Sciences Heriot-Watt University Summer 2006 Abstract This project studies the mathematics of elliptic curves, starting with their derivation

More information

Section Properties of Rational Expressions

Section Properties of Rational Expressions 88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:

More information

LECTURE 2 FRANZ LEMMERMEYER

LECTURE 2 FRANZ LEMMERMEYER LECTURE 2 FRANZ LEMMERMEYER Last time we have seen that the proof of Fermat s Last Theorem for the exponent 4 provides us with two elliptic curves (y 2 = x 3 + x and y 2 = x 3 4x) in the guise of the quartic

More information

Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions Title Integration of Rational Functions by Partial Fractions MATH 1700 December 6, 2016 MATH 1700 Partial Fractions December 6, 2016 1 / 11 Readings Readings Readings: Section 7.4 MATH 1700 Partial Fractions

More information

Calculus I Review Solutions

Calculus I Review Solutions Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.

More information

STEP Support Programme. Pure STEP 1 Questions

STEP Support Programme. Pure STEP 1 Questions STEP Support Programme Pure STEP 1 Questions 2012 S1 Q4 1 Preparation Find the equation of the tangent to the curve y = x at the point where x = 4. Recall that x means the positive square root. Solve the

More information

HOME ASSIGNMENT: ELLIPTIC CURVES OVER FINITE FIELDS

HOME ASSIGNMENT: ELLIPTIC CURVES OVER FINITE FIELDS HOME ASSIGNMENT: ELLIPTIC CURVES OVER FINITE FIELDS DANIEL LARSSON CONTENTS 1. Introduction 1 2. Finite fields 1 3. Elliptic curves over finite fields 3 4. Zeta functions and the Weil conjectures 6 1.

More information

Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website.

Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. BTW Math Packet Advanced Math Name Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. Go to the BTW website

More information

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

More information

Introduction to Arithmetic Geometry

Introduction to Arithmetic Geometry Introduction to Arithmetic Geometry 18.782 Andrew V. Sutherland September 5, 2013 What is arithmetic geometry? Arithmetic geometry applies the techniques of algebraic geometry to problems in number theory

More information

1 Functions and Graphs

1 Functions and Graphs 1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,

More information

10. Smooth Varieties. 82 Andreas Gathmann

10. Smooth Varieties. 82 Andreas Gathmann 82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

More information

Math 40510, Algebraic Geometry

Math 40510, Algebraic Geometry Math 40510, Algebraic Geometry Problem Set 1, due February 10, 2016 1. Let k = Z p, the field with p elements, where p is a prime. Find a polynomial f k[x, y] that vanishes at every point of k 2. [Hint:

More information

Chapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64

Chapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64 Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor

More information

V. Graph Sketching and Max-Min Problems

V. Graph Sketching and Max-Min Problems V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.

More information

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1

More information

Solving Equations Quick Reference

Solving Equations Quick Reference Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves

More information

Pólya Enrichment Stage Table of Contents

Pólya Enrichment Stage Table of Contents Pólya Enrichment Stage Table of Contents George Pólya (1887-1985) v Preface ix Chapter 1. Functions 1 Chapter 2. Symmetric Polynomials 15 Chapter 3. Geometry 22 Chapter 4. Inequalities 34 Chapter 5. Functional

More information

Real Analysis Prelim Questions Day 1 August 27, 2013

Real Analysis Prelim Questions Day 1 August 27, 2013 Real Analysis Prelim Questions Day 1 August 27, 2013 are 5 questions. TIME LIMIT: 3 hours Instructions: Measure and measurable refer to Lebesgue measure µ n on R n, and M(R n ) is the collection of measurable

More information

Lemma 1.3. The dimension of I d is dimv d D = ( d D+2

Lemma 1.3. The dimension of I d is dimv d D = ( d D+2 BEZOUT THEOREM One of the most fundamental results about the degrees of polynomial surfaces is the Bezout theorem, which bounds the size of the intersection of polynomial surfaces. The simplest version

More information

Congruent number elliptic curves of high rank

Congruent number elliptic curves of high rank Michaela Klopf, BSc Congruent number elliptic curves of high rank MASTER S THESIS to achieve the university degree of Diplom-Ingenieurin Master s degree programme: Mathematical Computer Science submitted

More information

Chapter 8B - Trigonometric Functions (the first part)

Chapter 8B - Trigonometric Functions (the first part) Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of

More information

MA094 Part 2 - Beginning Algebra Summary

MA094 Part 2 - Beginning Algebra Summary MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page

More information

MS 2001: Test 1 B Solutions

MS 2001: Test 1 B Solutions MS 2001: Test 1 B Solutions Name: Student Number: Answer all questions. Marks may be lost if necessary work is not clearly shown. Remarks by me in italics and would not be required in a test - J.P. Question

More information

Answers. 2. List all theoretically possible rational roots of the polynomial: P(x) = 2x + 3x + 10x + 14x ) = A( x 4 + 3x 2 4)

Answers. 2. List all theoretically possible rational roots of the polynomial: P(x) = 2x + 3x + 10x + 14x ) = A( x 4 + 3x 2 4) CHAPTER 5 QUIZ Tuesday, April 1, 008 Answers 5 4 1. P(x) = x + x + 10x + 14x 5 a. The degree of polynomial P is 5 and P must have 5 zeros (roots). b. The y-intercept of the graph of P is (0, 5). The number

More information

Points of Finite Order

Points of Finite Order Points of Finite Order Alex Tao 23 June 2008 1 Points of Order Two and Three If G is a group with respect to multiplication and g is an element of G then the order of g is the minimum positive integer

More information

Elliptic Curves Spring 2017 Lecture #5 02/22/2017

Elliptic Curves Spring 2017 Lecture #5 02/22/2017 18.783 Elliptic Curves Spring 017 Lecture #5 0//017 5 Isogenies In almost every branch of mathematics, when considering a category of mathematical objects with a particular structure, the maps between

More information

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions. Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,

More information

AN ELEMENTARY PROOF OF THE GROUP LAW FOR ELLIPTIC CURVES

AN ELEMENTARY PROOF OF THE GROUP LAW FOR ELLIPTIC CURVES AN ELEMENTARY PROOF OF THE GROUP LAW FOR ELLIPTIC CURVES Abstract. We give a proof of the group law for elliptic curves using explicit formulas. 1. Introduction In the following K will denote an algebraically

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the clear denominators theorem 5 Welcome back!

More information

Math 203A - Solution Set 1

Math 203A - Solution Set 1 Math 203A - Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in

More information

Torsion Points of Elliptic Curves Over Number Fields

Torsion Points of Elliptic Curves Over Number Fields Torsion Points of Elliptic Curves Over Number Fields Christine Croll A thesis presented to the faculty of the University of Massachusetts in partial fulfillment of the requirements for the degree of Bachelor

More information

A WHIRLWIND TOUR BEYOND QUADRATICS Steven J. Wilson, JCCC Professor of Mathematics KAMATYC, Wichita, March 4, 2017

A WHIRLWIND TOUR BEYOND QUADRATICS Steven J. Wilson, JCCC Professor of Mathematics KAMATYC, Wichita, March 4, 2017 b x1 u v a 9abc b 7a d 7a d b c 4ac 4b d 18abcd u 4 b 1 i 1 i 54a 108a x u v where a 9abc b 7a d 7a d b c 4ac 4b d 18abcd v 4 b 1 i 1 i 54a x u v 108a a //017 A WHIRLWIND TOUR BEYOND QUADRATICS Steven

More information

Symmetries and Polynomials

Symmetries and Polynomials Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we ll learn how to solve a cubic. We ll also sketch how to solve a quartic. We ll explore the connections

More information

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

More information

Section 0.2 & 0.3 Worksheet. Types of Functions

Section 0.2 & 0.3 Worksheet. Types of Functions MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2

More information

3 Polynomial and Rational Functions

3 Polynomial and Rational Functions 3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,

More information

Elliptic Curves I. The first three sections introduce and explain the properties of elliptic curves.

Elliptic Curves I. The first three sections introduce and explain the properties of elliptic curves. Elliptic Curves I 1.0 Introduction The first three sections introduce and explain the properties of elliptic curves. A background understanding of abstract algebra is required, much of which can be found

More information

Parabolas and lines

Parabolas and lines Parabolas and lines Study the diagram at the right. I have drawn the graph y = x. The vertical line x = 1 is drawn and a number of secants to the parabola are drawn, all centred at x=1. By this I mean

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Scheme-theoretic closure, and scheme-theoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:

More information

Elliptic Curves Spring 2013 Lecture #12 03/19/2013

Elliptic Curves Spring 2013 Lecture #12 03/19/2013 18.783 Elliptic Curves Spring 2013 Lecture #12 03/19/2013 We now consider our first practical application of elliptic curves: factoring integers. Before presenting the elliptic curve method (ECM) for factoring

More information

RATIONAL POINTS ON CURVES. Contents

RATIONAL POINTS ON CURVES. Contents RATIONAL POINTS ON CURVES BLANCA VIÑA PATIÑO Contents 1. Introduction 1 2. Algebraic Curves 2 3. Genus 0 3 4. Genus 1 7 4.1. Group of E(Q) 7 4.2. Mordell-Weil Theorem 8 5. Genus 2 10 6. Uniformity of Rational

More information

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions

MEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms

More information

Elliptic Curves over Q

Elliptic Curves over Q Elliptic Curves over Q Peter Birkner Technische Universiteit Eindhoven DIAMANT Summer School on Elliptic and Hyperelliptic Curve Cryptography 16 September 2008 What is an elliptic curve? (1) An elliptic

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

Elliptic curves and modularity

Elliptic curves and modularity Elliptic curves and modularity For background and (most) proofs, we refer to [1]. 1 Weierstrass models Let K be any field. For any a 1, a 2, a 3, a 4, a 6 K consider the plane projective curve C given

More information

Some Highlights along a Path to Elliptic Curves

Some Highlights along a Path to Elliptic Curves 11/8/016 Some Highlights along a Path to Elliptic Curves Part : Conic Sections and Rational Points Steven J Wilson, Fall 016 Outline of the Series 1 The World of Algebraic Curves Conic Sections and Rational

More information

Tropical Polynomials

Tropical Polynomials 1 Tropical Arithmetic Tropical Polynomials Los Angeles Math Circle, May 15, 2016 Bryant Mathews, Azusa Pacific University In tropical arithmetic, we define new addition and multiplication operations on

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #2 09/10/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #2 09/10/2013 18.78 Introduction to Arithmetic Geometry Fall 013 Lecture # 09/10/013.1 Plane conics A conic is a plane projective curve of degree. Such a curve has the form C/k : ax + by + cz + dxy + exz + fyz with

More information

Models of Elliptic Curves

Models of Elliptic Curves Models of Elliptic Curves Daniel J. Bernstein Tanja Lange University of Illinois at Chicago and Technische Universiteit Eindhoven djb@cr.yp.to tanja@hyperelliptic.org 26.03.2009 D. J. Bernstein & T. Lange

More information

12. Hilbert Polynomials and Bézout s Theorem

12. Hilbert Polynomials and Bézout s Theorem 12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of

More information

C-1. Snezana Lawrence

C-1. Snezana Lawrence C-1 Snezana Lawrence These materials have been written by Dr. Snezana Lawrence made possible by funding from Gatsby Technical Education projects (GTEP) as part of a Gatsby Teacher Fellowship ad-hoc bursary

More information

Part 2 - Beginning Algebra Summary

Part 2 - Beginning Algebra Summary Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian

More information

2-4 Zeros of Polynomial Functions

2-4 Zeros of Polynomial Functions Write a polynomial function of least degree with real coefficients in standard form that has the given zeros. 33. 2, 4, 3, 5 Using the Linear Factorization Theorem and the zeros 2, 4, 3, and 5, write f

More information

CHMC: Finite Fields 9/23/17

CHMC: Finite Fields 9/23/17 CHMC: Finite Fields 9/23/17 1 Introduction This worksheet is an introduction to the fascinating subject of finite fields. Finite fields have many important applications in coding theory and cryptography,

More information

Each is equal to CP 1 minus one point, which is the origin of the other: (C =) U 1 = CP 1 the line λ (1, 0) U 0

Each is equal to CP 1 minus one point, which is the origin of the other: (C =) U 1 = CP 1 the line λ (1, 0) U 0 Algebraic Curves/Fall 2015 Aaron Bertram 1. Introduction. What is a complex curve? (Geometry) It s a Riemann surface, that is, a compact oriented twodimensional real manifold Σ with a complex structure.

More information

Identifying the Graphs of Polynomial Functions

Identifying the Graphs of Polynomial Functions Identifying the Graphs of Polynomial Functions Many of the functions on the Math IIC are polynomial functions. Although they can be difficult to sketch and identify, there are a few tricks to make it easier.

More information

A2 HW Imaginary Numbers

A2 HW Imaginary Numbers Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest

More information

A. Incorrect! Apply the rational root test to determine if any rational roots exist.

A. Incorrect! Apply the rational root test to determine if any rational roots exist. College Algebra - Problem Drill 13: Zeros of Polynomial Functions No. 1 of 10 1. Determine which statement is true given f() = 3 + 4. A. f() is irreducible. B. f() has no real roots. C. There is a root

More information

6.2 Their Derivatives

6.2 Their Derivatives Exponential Functions and 6.2 Their Derivatives Copyright Cengage Learning. All rights reserved. Exponential Functions and Their Derivatives The function f(x) = 2 x is called an exponential function because

More information

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14 Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

More information