The Ubiquity of Elliptic Curves

Joseph Silverman (Brown University)Public Lecture – Dublin

Tuesday, 4 September 2007, 7:30 PM

Elliptic CurvesGeometry, Algebra, Analysis and

Beyond…

• An elliptic curve is an object with a dual nature:

• On the one hand, it is a curve, a geometric object.

• On the other hand, we can “add” points on the curve as if they were numbers, so it is an algebraic object.

• The addition law on an elliptic curve can be described:

• Geometrically using intersections of curves• Algebraically using polynomial equations• Analytically using functions with complex

variables

• Elliptic curves appear in many diverse areas of mathematics, ranging from number theory to complex analysis, and from cryptography to mathematical physics.

What is an Elliptic Curve?

- 3 -

The Equation of an Elliptic CurveAn Elliptic Curve is a curve given by an equation

E : y2 = f(x) for a cubic or quartic polynomial f(x)

We also require that the polynomial f(x) has no double roots. This ensures that the curve is nonsingular.

- 4 -

After a change of variables, the equation takes the simpler form

E : y2 = x3 + A x + B

Finally, for reasons to be explained shortly, we toss in an extra point O “at infinity,” so E is really the set

E = { (x,y) : y2 = x3 + A x + B } { O }

A Typical Elliptic Curve E

E : Y2 = X3 – 5X + 8

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Surprising Fact: We can use geometry to take two points P and Q on the elliptic curve and define their “sum”

P+Q.

Surprising Fact: We can use geometry to take two points P and Q on the elliptic curve and define their “sum”

P+Q.

The Addition Law on anElliptic Curve

Adding Points P + Q on E

P

Q

P+Q

R

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Doubling a Point P on E

P

2*P

RTangent Line to E at P

- 8 -

Vertical Lines and an Extra Point at Infinity

Vertical lines have no third intersection point

Q

Add an extra point O “at infinity.”The point O lies on every vertical line.

O

P

Q = –P

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Properties of “Addition” on E

Theorem: The addition law on E has the following properties:

a) P + O = O + P = P for all P E.

b) P + (–P) = O for all P E.

c) (P + Q) + R = P + (Q + R) for all P,Q,R E.

d) P + Q = Q + P for all P,Q E.

In mathematical terminology, the addition law + makes the points of E into a commutative group.

All of the group properties are easy to check except for the associative law (c). The associative law can be verified by a lengthy computation using explicit formulas, or by using more advanced algebraic or analytic methods.

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An Example

Using the tangent line construction, we find that

2P = P + P = (– 7/4, – 27/8).

Using the secant line construction, we find that

3P = P + P + P = (553/121, – 11950/1331)

Similarly, 4P = (45313/11664, 8655103/1259712).

As you can see, the coordinates become complicated.

E : Y2 = X3 – 5X + 8

The point P = (1,2) is on the curve E.

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An Addition Formula for ESuppose that we want to add the points

P1 = (x1,y1) and P2 = (x2,y2)

on the elliptic curve

E : y2 = x3 + Ax + B.

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. if 2

3 and if Let 21

1

21

2112

12 PPy

AxPP

xx

yy

).2,( Then 1213

212

21 yxxxxPP

Quite a mess!!!!! But…

Crucial Observation: If A and B are rational numbers and if the coordinates of P1 and P2 are rational numbers,

then the coordinates P1+ P2 and 2P1 are rational numbers.

The Group of Points on E with Rational Coordinates

The elementary observation on the previous slide leads to an important result:

Theorem (Poincaré, 1900): Suppose that an elliptic curve E is given by an equation of the form

y2 = x3 + A x + B with A,B rational numbers.

Let E(Q) be the set of points of E with rational coordinates,

E(Q) = { (x,y) E : x,y are rational numbers } { O }.

Then sums of points in E(Q) remain in E(Q).

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In mathematical terminology, E(Q) is a subgroup of E.

The Group of Points on E with Other Sort of Coordinates

In an earlier slide, we drew a picture of E in the plane. This was the set of points of E with real coordinates:

E(R) = { (x,y) E : x,y are real numbers } { O }.

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Similarly, we can look at the points of E whose coordinates are complex numbers:

E(C) = { (x,y) E : x,y are complex numbers } { O }.

And later we’ll look at the set of points E(Fp) whose coordinates are in a “finite field” Fp.

Key Fact: In any of these sets, we can add points and stay within the set.

What Does E(R) Look Like?

We saw one example of E(R). It is also possible for E(R) to have two connected components.

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E : Y2 = X3 – 9X

Elliptic Curves and Complex Numbers

Or…How the Elliptic Curve Acquired Its Unfortunate Moniker

The Arc Length of an Ellipse

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The arc length of a half circle

-a a

x2+y2=a2

a

a xa

dxa22

is given by the familiar integral

dx

xa

xabaa

a

22

2222 /1

is more complicatedThe arc length of a half ellipse

x2/a2 + y2/b2 = 1

-a

b

a

An Elliptic Curve!

The Arc Length of an Ellipse

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Let k2 = 1 – b2/a2 and change variables x ax. Then the arc length of an ellipse is

1

1 2

22

1

1dx

x

xka

dxy

xka

1

1

221LengthArc

with y2 = (1 – x2) (1 – k2x2) = quartic in x.

An elliptic integral is an integral , where R(x,y) is a rational function of the coordinates (x,y) on an “elliptic curve”

E : y2 = f(x) = cubic or quartic in x.

dxyxR ),(

1

1 222

22

)1)(1(

1dx

xkx

xka

Elliptic Integrals and Elliptic Functions

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Doubly periodic functions are called elliptic functions.

Its inverse function w = sin(z) is periodic with period 2.

The circular integral is equal to sin-1(w).

w

x

dx0 21

The elliptic integral has an inverse

w = (z) with two independent complex periods 1 and 2.

w

BAxx

dx 3

(z + 1) = (z + 2) = (z) for all z C.

Elliptic Functions and Elliptic Curves

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This equation looks familiar

BzAzz )()()( 32

The -function and its derivative satisfy an algebraic relation

The double periodicity of (z) means that it is a function on the quotient space C/L, where L is the lattice

L = { n11 + n22 : n1,n2 Z }.

1

2

1+ 2 L

(z) and ’(z) are functions on a fundamental parallelogram

The Complex Points on an Elliptic Curve

E(C) =

- 21 -

The -function gives a complex analytic isomorphism

Thus the points of E with coordinates in the complex numbers C form a torus, that is, the surface of a donut.

E(C) )(),( zz

L

C

Parallelogram with opposite sides identified = a torus

Elliptic Curves andNumber Theory

Rational Points on Elliptic Curves

E(Q) : The Group of Rational PointsA fundamental and ancient problem in number theory is that of solving polynomial equations using integers or rational numbers.

The description of E(Q) is a landmark in the modern study of Diophantine equations.

Theorem (Mordell, 1922): Let E be an elliptic curve given by an equation

E : y2 = x3 + A x + B with A,B Q.

There is a finite set of points P1,P2,…,Pr so that every point P in E(Q) can be obtained as a sum

P = n1P1 + n2P2 + … + nrPr with n1,…,nr Z.

In math terms, E(Q) is a finitely generated group.- 23 -

E(Q) : The Group of Rational Points

A point P has finite order if some multiple of P is O. The elements of finite order in E(Q) are quite well understood.

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Theorem (Mazur, 1977): The group E(Q) contains at most 16 points of finite order.

Conjecture: The number of points needed to generate E(Q) may be arbitrarily large.

The minimal number of points needed to generate the group E(Q) is much more mysterious!

Current World Record: (Elkies 2006) There is an elliptic curve with

Number of generators for E(Q) 28.

E(Z) : The Set of Integer PointsIf P1 and P2 are points on E having integer coordinates, then P1 + P2 will have rational coordinates, but there is no reason for it to have integer coordinates.

Indeed, the formulas for P1 + P2 are so complicated, it seems unlikely that P1 + P2 will have integer coordinates.

Complementing Mordell’s finite generation theorem for rational points is a famous finiteness result for integer points.

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Theorem (Siegel, 1928): An elliptic curve

E : y2 = x3 + A x + B with A,B Z

has only finitely many points P = (x,y) with integer coordinates x,y Z.

Elliptic Curves and Finite Fields

Finite Fields

You may have run across clock arithmetic, where after counting 0, 1, 2, 3,…,11, you go back to 0.

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Another way to view clock arithmetic is that whenever you add or multiply numbers together, you should divide by 12 and just keep the remainder.

We want to do the same thing, but instead of using 12, we’ll use a prime number p, for example 3 or 7 or 37.

The Finite Field Fp is the set of numbers

0, 1, 2, …, p–1

with the rule that when we add or multiply two of them, we are required to divide by p and just keep the remainder.

An Example of a Finite Field

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For example, in the finite field F7,

245 645

since 9 divided by 7 leaves remainder 2.

since 20 divided by 7 leaves remainder 6.

Similarly, in the field F41, we have

11 x 15 = 1 and 23 x 25 = 1 and 19 x 13 = 1 and …

This illustrates why we use a prime p, instead of a number like 12. In a finite field Fp, every nonzero number has a reciprocal. So Fp is a lot like the rational numbers Q and the real numbers R:

In Fp, not only can we can add, subtract, and multiply, we can also divide by nonzero numbers

Elliptic Curves over a Finite Field

The formulas giving the addition law on E are fine if the points have coordinates in any field, even if the geometric pictures don’t make sense.

For example, we can take points with coordinates in Fp.

Example:The curve E : Y2 = X3 – 5X + 8 modulo 37

contains the pointsP = (6,3) and Q = (9,10).

Using the addition formulas, we can compute in E(F37):

2P = (35,11) 3P = (34,25) 4P = (8,6) 5P = (16,19) …

P + Q = (11,10) 3P + 4Q = (31,28) …

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E(Fp) : The Group of Points Modulo pNumber theorists also like to solve polynomial equations modulo p.

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Theorem (Hasse, 1922): An elliptic curve equation

E : y2 x3 + A x + B (modulo p)

has p + 1 +

solutions (x,y) mod p, where the error satisfies

.2 p

This is much easier than finding solutions in Q, since there are only finitely many solutions in the finite field Fp!

One expects E(Fp) to have approximately p+1 points.

A famous theorem of Hasse (later vastly generalized by Weil and Deligne) quantifies this expectation.

Elliptic Curves andCryptography

The (Elliptic Curve) Discrete Log ProblemSuppose that you are given two points P and Q in E(Fp).

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• If the prime p is large, it is very very difficult to find m.

• Neal Koblitz and Victor Miller (1985) independently invented Elliptic Curve Cryptography in 1985 when they suggested building a cryptosystem around the ECDLP.

• The extreme difficulty of the ECDLP yields highly efficient cryptosystems that are in widespread use protecting everything from your bank account to your government’s secrets.

The Elliptic Curve Discrete Logarithm Problem (ECDLP) is to find an integer m satisfying

Q = P + P + … + P = mP.

m summands

Elliptic Curve Diffie-Hellman Key Exchange

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Public Knowledge: A group E(Fp) and a point P of order n.

BOB ALICE

Choose secret 0 < b < n Choose secret 0 < a < n

Compute QBob = bP Compute QAlice = aP

Compute bQAlice Compute aQBob

Bob and Alice have the shared value bQAlice = abP = aQBob

Presumably(?) recovering abP from aP and bP requiressolving the elliptic curve discrete logarithm problem.

Send QBob to Alice

to Bob Send QAlice

Elliptic Curves andClassical Physics

The Pit and the Pendulum

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The Pit and the Pendulum

- 36 -

This leads to a simple harmonic motion for the pendulum.

In freshman physics, one assumes that is small and derives the formula

22

2

d

dk

t

But this formula is only a rough approximation. The actual differential equation for the pendulum is

)sin(d

d 22

2

k

t

How to Solve the Pendulum Equation

toequaliskt2

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.1 withd

21

d2

)cos(

d 42

4xy

y

x

x

x

Conclusion: tan( /2) = Elliptic Function of t

An Elliptic Curve!!!An Elliptic Integral!!!

and do a bunch of algebra.

As a favor, I’ll spare you the details and just tell you the answer!!

To solve the pendulum equation, we make the substitution

2

tanx

Elliptic Curves andModern Physics

Elliptic Curves and String Theory

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In string theory, the notion of a point-like particle is replaced by a curve-like string.

As a string moves through space-time, it traces out a surface.

For example, a single string that moves around and returns to its starting position will trace a torus.

So the path traced by a string looks like an elliptic curve!

In quantum theory, physicists like to compute averages over all possible paths, so when using strings, they need to compute integrals over the space of all elliptic curves.

Elliptic Curves andNumber Theory

Fermat’s Last Theorem

Fermat’s Last Theorem and Fermat Curves

- 41 -

Fermat’s Last Theorem says that if n > 2, then the equation

an + bn = cn

has no solutions in nonzero integers a,b,c.

It is enough to prove the case that n = 4 (already done by Fermat himself) and the case that n = p is an odd prime.

If we let x = a/c and y = b/c, then solutions to Fermat’s equation give rational points on the Fermat curve

xp + yp = 1.

But Fermat’s curve is not an elliptic curve. So how can elliptic curves be used to study Fermat’s problem?

Elliptic Curves and Fermat’s Last Theorem

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Frey suggested that Ea,b,c would be such a strange curve, it shouldn’t exist at all. More precisely, Frey doubted that Ea,b,c could be modular.

Ribet verified Frey’s intuition by proving that Ea,b,c is indeed not modular.

Wiles completed the proof of Fermat’s Last Theorem by showing that (most) elliptic curves, in particular elliptic curves like Ea,b,c, are modular.

Gerhard Frey (and others) suggested using an hypothetical solution (a,b,c) of Fermat’s equation to “manufacture” an elliptic curve

Ea,b,c : y2 = x (x – ap) (x + bp).

Elliptic Curves and Fermat’s Last Theorem

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To Summarize:

Suppose that ap + bp = cp with abc 0.

Ribet proved that Ea,b,c is not modular

Wiles proved that Ea,b,c is modular.

Conclusion: The equation ap + bp = cp has no solutions.

Ea,b,c : y2 = x (x – ap) (x + bp)

But what does it mean for an elliptic curve E to

be modular?

The variable represents the elliptic curve E whose lattice is L = {n1+n2 : n1,n2 Z }.

So just as in string theory, the space of all elliptic curves makes an unexpected appearance.

Elliptic Curves and Modularity

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E is modular if it is parameterized by modular forms!

There are many equivalent definitions, all of them rather complicated and technical. Here’s one:

).N(mod0c satisfying )Z(SLdc

ba matrices all for 2

)()( 2

fdcdc

baf

A modular form is a function f() with the property

Conclusion

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The Ubiquity ofElliptic Curves

Joseph Silverman (Brown University)Public Lecture

Dublin – September 4, 2007