ECC shared key

Bob

p a prime number

E(a,b) where

0 < a and b < p

-(4a3+27b2) != 0

y^2 = x^3 + ax + b

P point in E

n random integer => secret

compute nP

send E(a,b), P, nP to Alice

Alice

m random integer => secret

compute mP

send mP to Bob

shared key = mnP = nmP

ECC public key

Bob

p a prime namber

E(a,b) where

0 < a and b < p

-(4a3+27b2) != 0

y^2 = x^3 + ax + b

P point in E

n random integer => private key

nP => public key

share E(a,b), P, nP to Alice

Alice

want to send a M point to Bob

m random integer => private key

mP => public key

send (mP, M + nmP) to Bob

Bob

nmP = n * mP

M = M + nmP - nmP = M

Chararters Mapping to Points

- E(): eleptic curve

- P: point generator

- S: set of mapping points

- A: a non singular matrix

- A-1: inverse of matrix A

- m: bob's secret key

- n: alice's secret key

send msg = "hello"

size = 5 not divisible by 3 (need to add 1 : $ for padding)

size = 6

Coding:

map msg character using S => [P(1,0), P(1,7), P(2,3), P(2,3), P(2,0), $]

create a matrix M of 3 rows like so

M = [

P(1,0) P(1,7)

P(2,3) P(2,3)

P(2,0) $

]

choose A as a non singular matrix of 3*3

result S = AM

S = [

Q1(x,y) Q2(x2,y2) Q3(x3,y3)

... ... ...

... ... Qn(xn,yn)

]

encryption points: C = (mP, Q + m(nP))

C = [

C1 C2 C3

.. .. ..

.. .. Cn

]

send(mP, Q + m(nP))

decryption points: D = Q + m(nP) - n(mP) = Q

D = [

Q1 Q2 Q3

.. .. ..

.. .. Qn

]

Decoding:

M = D * A-1

Biobliographie

+ [1] http://en.wikipedia.org/wiki/Elliptic_curve_cryptography

+ [2] https://www.certicom.com/index.php/ecc-tutorial

+ [3] http://www.eccworkshop.org/

books:

+ [4] William Stalings