Elliptic Curve Crypto &

ECC Diffie-Hellman

Presenter: Le Thanh Binh

Outline

1. What is Elliptic Curve ?

2. Addition on an elliptic curve

3. Elliptic Curve Crypto (ECC)

4. ECC Diffie–Hellman

Lets start with a puzzle…• What is the number of balls that may be

piled as a square pyramid and also rearranged into a square array?

Lets start with a puzzle…• What is the number of balls that may be

piled as a square pyramid and also rearranged into a square array?

• Sol: Let x be the height of the pyramid… Thus,

We also want this to be a square: Hence,

(4900 balls) y=70

Graphical Representation

Curves of this nature are called ELLIPTIC

CURVES

What is an Elliptic Curve ?

• An elliptic curve E is the graph of an equation of the form

y2 = x3 + ax + b

“Elliptic curve” is not a cryptosystem

Addition on an elliptic curve mod p

The Elliptic Curve E

Addition on an elliptic curve mod p

Start with two points P and Q on E

Addition on an elliptic curve mod p

Draw the line L through P and Q.

Addition on an elliptic curve mod pThe line L intersects the cubic curve E in a third

point. Call that third point R.

Addition on an elliptic curve mod pDraw the vertical line through R.

It hits E in another point.

Addition on an elliptic curve mod pWe define the sum of P and Q on E to be the

reflected point. We denote it by P ⊕ Q or just P + Q

Addition on an elliptic curve mod p

If P1 and P2 are on E, we can define P3 = P1 + P2

P1

P2

P3

x

y

Addition on an elliptic curve mod pSuppose that we want to add the points

on the elliptic curve

Let the line connecting P1 to P2 be

L : y = mx + vExplicitly, the slope and y-intercept of L are given by

Addition on an elliptic curve mod pWe find the intersection of

by solving

We already know that x1 and x2 are solutions, so we can find the third solution x3 by comparing the two sides of

Equating the coefficients

and hence

Then we compute y3

And finally

Addition on an elliptic curve mod p

ExampleConsider y2 = x3 + 2x + 3 (mod 5)

x = 0 y2 = 3 no solution (mod 5)x = 1 y2 = 6 = 1 y = 1,4 (mod 5)

x = 2 y2 = 15 = 0 y = 0 (mod 5)

x = 3 y2 = 36 = 1 y = 1,4 (mod 5)x = 4 y2 = 75 = 0 y = 0 (mod 5)

Then points on the elliptic curve are(1,1) (1,4) (2,0) (3,1) (3,4) (4,0) and the

point at infinity:

What is (1,4) + (3,1) = P3 = (x3,y3)?

Consider y2 = x3 + 2x + 3 (mod 5)What is (1,4) + (3,1) = P3 = (x3,y3)?

P1

P2

P3

x

y

y=mx+v

y2=x3+Ax+B

Addition on an elliptic curve mod p

m = (1-4)(3-1)-1 = -32-1

((a mod n)(b mod n)) mod n = ab mod n

= (2)(3) mod 5

= 6 mod 5

= 1

= ((-3 mod 5)(2-1 mod 5))mod5

Consider y2 = x3 + 2x + 3 (mod 5)What is (1,4) + (3,1) = P3 = (x3,y3)?

Addition on an elliptic curve mod p

m = 1 x3 = 1 - 1 - 3 = 2 (mod 5)

y3 = 1(1-2) - 4 = 0 (mod 5)

On this curve, (1,4) + (3,1) = (2,0)

P1

P2

P3

x

y

y=mx+v

y2=x3+Ax+B

Elliptic Curve Crypto (ECC)

Elliptic curve cryptography [ECC] is a

public-key cryptosystem (just like RSA)

Public Public

Private Private

Public-key cryptosystem

Public PublicPublic-key cryptosystem

???

Secret SecretPublic-key cryptosystem

Addition on an elliptic curve

ECC Diffie-Hellman• Public: Elliptic curve and point (x,y) on curve• Private: Alice’s A and Bob’s B

Alice, A Bob, B

A(x,y)

B(x,y)

Alice computes A(B(x,y)) Bob computes B(A(x,y)) These are the same since AB = BA

• Public: Curve y2 = x3 + 7x + b (mod 37) and point (2,5) b = 3

• Alice’s private: A = 4• Bob’s private: B = 7• Alice sends Bob: 4(2,5) = (7,32)• Bob sends Alice: 7(2,5) = (18,35)• Alice computes: 4(18,35) = (22,1)• Bob computes: 7(7,32) = (22,1)

ECC Diffie–Hellman - Example

Addition 4 times

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http://www-cs-students.stanford.edu/~tjw/jsbn/ecdh.html

Demonstration