Introduction to Elliptic Curves

What is an Elliptic Curve?

An Elliptic Curve is a curve given by an equation

E : y2 = f(x)

Where f(x) is a square-free (no double roots) cubic or a quarticpolynomial

After a change of variables it takes a simpler form:

E : y2 = x3 + Ax + B 0274 23 ≠+ BA

So y2 = x3 is not an elliptic curve but y2 = x3-1 is

Why is it called Elliptic?

Arc Length of an ellipse =

∫− −−

−1

1 222

22

)1)(1(1 dx

xkxxka

( ) dxxa

xabaa

a∫− −−−

22

2222 /1

Let k2 = 1 – b2/a2 and change variables x ax.

Then the arc length of an ellipse is

dxy

xka∫−−

=1

1

221Length Arc

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

Graph of y2 = x3-5x+8

0>∆

Elliptic curves can have separate components

E : Y2 = X3 – 9X

0<∆

Addition of two Points P+Q

P+Q

R

Q

P

Doubling of Point P

P

2*P

RTangent Line to E at P

Point at Infinity

P

Q

O

Addition of Points on E

1. Commutativity. P1+P2 = P2+P1

2.Existence of identity. P + O = P

3. Existence of inverses. P + (-P) = O

4. Associativity. (P1+P2) + P3 = P1+(P2+P3)

Addition FormulaSuppose that we want to add the points

P1 = (x1,y1) and P2 = (x2,y2) on the elliptic curve

E : y2 = x3 + Ax + B.

21 xx ≠ 21 xx =IfIf

12

12

xxyym

−−

=1

21

23

yAxm +

=

212

3 xxmx −−=

1313 )( yxxmy −−=

Note that when P1, P2 have rational coordinates and A and B are rational, then P1+P2 and 2P also have rational coordinates

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).

The many uses of elliptic curves.

Really Complicated first…

Elliptic curves were used to prove Fermat’s Last Theorem

Ea,b,c : y2 = x (x – ap) (x + bp)

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.

Elliptic Curves and String TheoryIn 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!Points of E with coordinates in the complex numbers C form a torus, that is, the surface of a donut.

Congruent Number Problem

Which positive rational n can occur as areas of right triangles with rational sides?

Ex: 5 is a congruent number because it is the area of 20/3, 3/2, 41/6 triangle

This question appears in 900A.D. in Arab manuscripts

A theorem exists to test the numbers but it relies on an unproven conjecture.

Congruent Number Problem cont….

nab=

2222 cba =+Suppose a, b and c satisfy

bcanx )( +

=2

2 )(2yb

can +=Then set

xnxy 232 −=A Calculation shows that

ynxb 2

=y

nxc22 +

=ynxa /)( 22−=Conversely:

A positive rational number n is congruent if and only if the elliptic curvehas a rational point with y not equal to 0

Congruent Number Problem cont…xxy 2532 −=Continuing with n = 5

We have Point (-4,6) on the curve

172862279

14416812 ==− yxisPfindWe

We can now find a, b and c

Factoring Using Elliptic Curves

Ex: We want to factor 4453

Step 1. Generate an elliptic curve with point P mod n

)3,1()4453(mod21032 =−+= PletxxyStep 2. Compute BP for some integer B.

)4453(mod37136

132

10322

≡=+y

xfirstPcomputeLets

)4453(mod371161)4453,6gcd( 1 ≡= −findtothatfacttheusedWe

32303)1(371323713),(2 2 ≡−−−≡−≡= xyxwithyxPthatfindwe

)3230,4332(2 isP

Factoring Continued..

Step 3. If step 2 fails because some slope does not exist mod n, the we have found a factor of n.

PandPaddwePcomputeTo 23

43313227

1433233230=

−−isslopeThe

)4453(mod4331161)4453,4331gcd( 1−≠= findnotcanweBut

445361, offactorthefoundhaveweHowever

CryptographySuppose that you are given two points P and Q in E(Fp).

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

Q = P + P + … + P = mP.

m summands

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

Elliptic Curve Diffie-Hellman Key ExchangePublic 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

Send QBob to Alice

to Bob Send QAlice

Compute bQAlice Compute aQBob

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

Can you solve this?

Suppose a collection of cannonballs is piled in a square pyramidwith one ball on the top layer, four on the second layer, nine on the third layer, etc.. If the pile collapses, is it possible to rearrange the balls into a square array (how many layers)?

...)())()(( 23 +++−=−−− xcbaxcxbxax32 PPFind +

solutionstrivialarePandP 21Hint:

Solution

6)12)(1(...321 2322 ++

=++++xxxx

6)12)(1(2 ++

=xxxy This is an elliptic curve

)1,1()0,0( 21 PPWe know two points

The line through these points is y = x

6236)12)(1( 23

2 xxxxxxx ++=++

=

021

23 23 =+− xxx

Solution cont…

)21,

21(

2310 3 −=++ isPthereforex

2332 −= xyisPandPthroughlineThe

6)12)(1()23( 2 ++

=−xxxx

0...2

51 23 =+− xx

2511

21

=++ x 24=x 70=y

22222 7024...321 =++++

References

Elliptic Curves Number Theory and CryptographyLawrence C. Washington

http://www.math.vt.edu/people/brown/doc.htmlhttp://www.math.brown.edu/~jhs/