the ubiquity of elliptic curves

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The Ubiquity of Elliptic Curves Joseph Silverman (Brown University) SUMS – Providence – February 22, 2003

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The Ubiquity of Elliptic Curves. Joseph Silverman (Brown University) SUMS – Providence – February 22, 2003. Elliptic Curves Geometry, Algebra, Analysis and Beyond…. What is an Elliptic Curve?. An elliptic curve is a curve that’s also naturally a group. - PowerPoint PPT Presentation

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Page 1: The Ubiquity of  Elliptic Curves

The Ubiquity of Elliptic Curves

Joseph Silverman (Brown University)SUMS – Providence – February 22, 2003

Page 2: The Ubiquity of  Elliptic Curves

Elliptic CurvesGeometry, Algebra, Analysis and

Beyond…

Page 3: The Ubiquity of  Elliptic Curves

• An elliptic curve is a curve that’s also naturally a group.

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

• Geometrically using intersection theory• Algebraically using polynomial equations• Analytically using complex analytic functions

• 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 -

Page 4: The Ubiquity of  Elliptic Curves

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 }

Page 5: The Ubiquity of  Elliptic Curves

A Typical Elliptic Curve E

E : Y2 = X3 – 5X + 8

- 5 -

Surprising Fact: We can use geometry to make the points of an elliptic curve into a group. Surprising Fact: We can use geometry to make the points of an elliptic curve into a group.

Page 6: The Ubiquity of  Elliptic Curves

The Group Law on anElliptic Curve

Page 7: The Ubiquity of  Elliptic Curves

Adding Points P + Q on E

P

Q

P+Q

R

- 7 -

Page 8: The Ubiquity of  Elliptic Curves

Doubling a Point P on E

P

2*P

RTangent Line to E at P

- 8 -

Page 9: The Ubiquity of  Elliptic Curves

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

- 9 -

Page 10: The Ubiquity 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 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 other words, the addition law + makes the points of E into a commutative group.

All of the group properties are trivial 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.

- 10 -

Page 11: The Ubiquity of  Elliptic Curves

Algebraic Formulas for Addition on E

Suppose that we want to add the points

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

on the elliptic curve

E : y2 = x3 + Ax + B.

- 11 -

. 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 in a field K and if P1 and P2 have coordinates in K,

then P1+ P2 and 2P1 have coordinates in K.

Page 12: The Ubiquity of  Elliptic Curves

The Group of Points on E with Coordinates in a Field K

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

Theorem (Poincaré, 1900): Let K be a field and suppose that an elliptic curve E is given by an equation of the form

y2 = x3 + A x + B with A,B K.

Let E(K) be the set of points of E with coordinates in K,

E(K) = { (x,y) E : x,y K } { O }.

Then E(K) is a subgroup of E.

- 12 -

Page 13: The Ubiquity of  Elliptic Curves

A Numerical 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.

- 13 -

Page 14: The Ubiquity of  Elliptic Curves

Elliptic Curves and Complex Analysis

Or…How the Elliptic Curve Acquired Its Unfortunate Moniker

Page 15: The Ubiquity of  Elliptic Curves

The Arc Length of an Ellipse

- 15 -

The arc length of a (semi)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 (semi)ellipse

x2/a2 + y2/b2 = 1

-a

b

a

Page 16: The Ubiquity of  Elliptic Curves

An Elliptic Curve!

The Arc Length of an Ellipse

- 16 -

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

Page 17: The Ubiquity of  Elliptic Curves

Elliptic Integrals and Elliptic Functions

- 17 -

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.

Page 18: The Ubiquity of  Elliptic Curves

Elliptic Functions and Elliptic Curves

- 18 -

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

Page 19: The Ubiquity of  Elliptic Curves

The Complex Points on an Elliptic Curve

E(C) =

- 19 -

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

Page 20: The Ubiquity of  Elliptic Curves

Elliptic Curves andNumber Theory

Rational Points on Elliptic Curves

Page 21: The Ubiquity of  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 other words, E(Q) is a finitely generated group.- 21 -

Page 22: The Ubiquity of  Elliptic Curves

E(Q) : The Group of Rational Points

The elements of finite order in the group E(Q) are quite well understood.

- 22 -

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: There is an elliptic curve with

Number of generators for E(Q) 24.

Page 23: The Ubiquity of  Elliptic Curves

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

- 23 -

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.

Page 24: The Ubiquity of  Elliptic Curves

Elliptic Curves andCryptography

Page 25: The Ubiquity of  Elliptic Curves

The (Elliptic Curve) Discrete Log ProblemLet A be a group and let P and Q be known elements of A.

- 25 -

• There are many cryptographic constructions based on the difficulty of solving the DLP in various finite groups.

• The first group used for this purpose (Diffie-Hellman 1976) was the multiplicative group Fp* in a finite field.

• Koblitz and Miller (1985) independently suggested using the group E(Fp) of points modulo p on an elliptic curve.

• At this time, the best algorithms for solving the elliptic curve discrete logarithm problem (ECDLP) are much less efficient than the algorithms for solving DLP in Fp* or for factoring large integers.

The Discrete Logarithm Problem (DLP) is to find an integer m satisfying

Q = P + P + … + P = mP.

m summands

Page 26: The Ubiquity of  Elliptic Curves

Elliptic Curve Diffie-Hellman Key Exchange

- 26 -

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

Page 27: The Ubiquity of  Elliptic Curves

Elliptic Curves andClassical Physics

Page 28: The Ubiquity of  Elliptic Curves

The Elliptic Curve and the Pendulum

- 28 -

Page 29: The Ubiquity of  Elliptic Curves

The Elliptic Curve and the Pendulum

- 29 -

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

Page 30: The Ubiquity of  Elliptic Curves

How to Solve the Pendulum Equation

- 30 -

)sin( d

d 22

2

k

t

d )sin(d d

d 22

2

kt

d )sin(

d

dd

2

1 22

kt

0) (taking )cos( d

d

2

1 22

Ckt

tk d 2)cos(

d

.1 withd

21

d2

)cos(

d 42

4xy

y

x

x

x

.2

tan substituteNow

x

Page 31: The Ubiquity of  Elliptic Curves

How to Solve the Pendulum Equation

- 31 -

.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!!!

Page 32: The Ubiquity of  Elliptic Curves

Elliptic Curves andModern Physics

Page 33: The Ubiquity of  Elliptic Curves

Elliptic Curves and String Theory

- 33 -

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.

Page 34: The Ubiquity of  Elliptic Curves

Elliptic Curves andNumber Theory

Fermat’s Last Theorem

Page 35: The Ubiquity of  Elliptic Curves

Fermat’s Last Theorem and Fermat Curves

- 35 -

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?

Page 36: The Ubiquity of  Elliptic Curves

Elliptic Curves and Fermat’s Last Theorem

- 36 -

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).

Page 37: The Ubiquity of  Elliptic Curves

Elliptic Curves and Fermat’s Last Theorem

- 37 -

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?

Page 38: The Ubiquity of  Elliptic Curves

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

- 38 -

E is modular if it is parameterized by modular forms!

There are many equivalent definitions, none of them particularly intuitive. Here’s one:

).(mod0 satisfying )(SL matrices all for 2 Ncdc

ba

Z

)()( 2

fdcdc

baf

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

Page 39: The Ubiquity of  Elliptic Curves

Conclusion

- 39 -

Page 40: The Ubiquity of  Elliptic Curves

The Ubiquity ofElliptic Curves

Joseph Silverman (Brown University)SUMS – Providence, RI – February 22, 2003