CPE5002/CSE5210 Advanced Network Security  Advanced Cryptography and Information Security Lecture 12. Outline. Background Group Field Principle of public key systems Discrete Logarithm Problem (DLP) Elliptic Curve Cryptography EC with real numbers EC with finite groups
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
(You can find many more from the Web)
CPE5021  Advanced Network Security
defined on S is a group if satisfies the following properties:
 Closure:
 Identity:
 Associativity:
 Inverses:
For each
If a group S satisfies the commutative law the group is said to be Abelian.
CPE5021  Advanced Network Security
 (F; +) is an Abelian group;
 (F \ {0}, *) is an Abelian group, where {0}is the identity of addition and zero of multiplication;
 distributive: x; y; z F
x * (y + z) = x * y + x* z
(x + y) * z = x * z + y * z
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
(in bits)
Example algorithm
DLP key size for equivalent security
(p in bits)
RSA key size for equivalent security
(n in bits)
ECC key size for equivalent security
(n in bits)
Key size ratio of RSA to ECC (approx)
56

512
512
112
5:1
80
SKIPJACK22
1024
1024
160
6:1
112
Triple DES
2048
2048
224
9:1
128
AES128
3072
3072
256
12:1
192
AES192
7680
7680
384
20:1
256
AES256
15360
15360
512
30:1
Elliptic curve cryptosystem (ECC)Extract from my student’s Thesis – Markku N.M. Pekkarinen
Key Size Equivalence Against Best Known Attacks
(Based on López and Dahab, 2000 and Fibíková, 2002)
CPE5021  Advanced Network Security
Progress in Integer Factorisation (Certicom 1997)
CPE5021  Advanced Network Security
Progress in solving ECDLP Certicom (2002)
CPE5021  Advanced Network Security
Given group elements,
find an integer x such that
xis calledthe discrete logof to the base .
 It is easy to compute
 It is hard to find x, knowing and
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
y2 = x3 + ax + b, where x, y, a and b are real numbers, where 4a3 + 27b20 – condition for distinct single roots (smooth curve).
CPE5021  Advanced Network Security
An EC over a group (G,+) is defined with the following:
1. Addition: If P and Q are distinct, and P Q, define P+Q as follows:
2. For every P, define P + (P) = O
3. If P=(x,0), then P+P = O, (a vertical line)
Otherwise, draw a tangent line through P, the intersected point is defined as –R, then P+P =2P =R.
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
1. Adding distinct points P and Q (1)When P = (xP,yP) and Q = (xQ,yQ) and P Q, P Q,P + Q = R(xR, yR) with xR = s2  xP  xQ and yR = s(xP  xR)  yPwhere s = (yP  yQ) / (xP  xQ)
2. Doubling the point P(2)When yP is not O,2P = R(xR, yR) with xR = s2  2xP and yR = s(xP  xR) yP
where s = (3xP2 + a) / (2yP )
3. P + (P) =O (3)
4. If P = (xP,yP) and yP =0, then P + P = 2P = O (4)
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
Let Pand Qon E(Z11,mod)
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
m + AsBp  Bs(Asg) = m + As Bsg  BsAsg = m
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
{g=(2,7), 2g=(5,2), 3g=(8,3), 4g=(10,2)
5g=(3,6), 6g=(7,9), 7g=(7,2), 8g=(3,5)
9g=(10,9), 10g=(8,8),11g=(5,9),12g=(2,4)}
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security
CPE5021  Advanced Network Security