Mumbai university > Electronics and telecommunication Engineering > Sem 7 > Data compression and Encryption

Marks: 5

Years: May 2016

1. Security of RSA is increased by increasing key length but in turn it has heavier processing load.

2. Hence elliptic curve cryptography is used which gives equal security for a smaller key size thereby reducing processing achieved.

3. Elliptic curve encryption/ decryption:

• The first task in this system is to encode the plain text message ‘m’ to be sent as an x-y point Pm.

• It is the point Pm that will be encrypted as a cipher text and subsequently decrypted.

• We can’t simply encode the message as the x or y co-ordinate of a point because not all such co-ordinates are in eq(a,b).

• As with the key exchange system, an encryption/ decryption system requires a point G and an elliptic group eq(a,b) as parameters.

• Each user A selects a private key nA and generates a public key PA = nA x G.

• To encrypt and send a message Pm to B, A chooses a random positive integer k and produces the cipher text Cm consisting of the pair of points

Cm = {kG, Pm + k Pn}.

• Note that A has used B’s public key PB. To decrypt the cipher text, B multiplies the first point in the pair by B’s secret key and subtracts the result from the second point

• A has masked the message Pm by adding k PB to it. Nobody but A knows the value of k so even though PB is a public key, nobody can remove the mask k PB. However, A also includes a clue which is enough to remove the mask if one knows the private key nB.For an attacker to recover the message, the attacker would have to compute k given G and kG which is assumed hard.

1. Security of elliptic curve cryptography:

• The security of ECC depends on how difficult is to determine k given kP and P. This is referred to as the elliptic curve logarithm problem.

• The fastest known technique for taking the elliptic curve logarithm is known as the Pollard rho method.

• A considerably smaller key size can be used for ECC compared to RSA. Furthermore, for equal key lengths, the computational effort required for ECC and RSA is comparable. Thus, there is a computational advantage to using ECC with a shorter key length than a comparably secure RSA.

1. Elliptic curves :

• Elliptic curves are not ellipses.

• They are so named because they are described by cubic equations similar to that used for calculating circumference of ellipse.

• In general, cubic equations for elliptic curves take the form :

$y^2$ + axy + by = $x^3$ + c$x^2$ + dx + e

where a, b, c, d, e are real numbers.

Consider an elliptic curve : $y^2$ = $x^3$ + ax + b ---> cubic equation highest power = 3

For given values of a and b say (n = 1 and b = 1) the curve is as shown below :

$y^3$ = $x^3$ + ax + b