p a prime number
E(a,b) where 
	0 < a and b < p
	-(4a3+27b2) != 0
	y^2 = x^3 + ax + b
P a point generator in E
n random integer => secret
compute nP

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

m random integer => secret
compute mP 
send mP to 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 generator in E
n random integer => private key
nP => public key

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

want to send a M point to Bob
m random integer => private key
mP  => public key
    map message to a set of points (M) in E
send (mP, M + nmP) to Bob

nmP = n * mP
M = M + nmP - nmP = M

- 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 dided by 3 (need to add 1 : $ for padding)
size = 6
Coding:
map msg characters to list of points in E: 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: Ci = Q + m(nP)
C = [
		C1	C2	C3
		..	..	..
		..	..	Cn
	]
send(mP, C)

receive(mP, C)
bob decrypt the points using his private key n : Di = Ci - n(mP) = Qi + m(nP) - n(mP) = Qi
D = [
		Q1	Q2	Q3
		..	..	..
		..	..	Qn
	]
Decoding:
M = D * A-1

+ [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
+ [5] Stallings, W. Cryptography and Network Security, (2003) Prentice Hall, 3rd Edition.
+ [6] V. S. Miller. Use of Elliptic Curves in Cryptography. Advances in Cryptology CRYPTO’85, pp. 417-426, 1986.
+ [7] N. Koblitz. Elliptic Curve Cryptosystems. Mathematics of Computation, Vol. 48, No. 177, pp. 203-209, 1987.
+ [8] Zhu Yufei, Zhang Yajuan. Introduction to elliptic curve cryptosystem. Beijing: Science press, 10, 130 (in chinese), 2006.