Ut' s contain about cryptography

1. Public Key
Cryptography
Bryan Pearsaul

2. Outline
• What is Cryptology?
• Symmetric Ciphers
• Asymmetric Ciphers
• Diffie-Hellman
• RSA (Rivest/Shamir/Adleman)
• Moral Issues

3. Outline
• Summary
• References

4. What is Cryptology?
• The science of keeping data secure
• Two transformation algorithms:
Enciphering and Deciphering
• Symmetric ciphers
• Asymmetric ciphers

5. Symmetric Ciphers
• Also known as private key
• Both parties must agree on the key
in advance
• D_K(E_K(P)) = P
• Not very computationally intensive
• Key must be securely sent to both parties

6. Symmetric Cipher Example
• k = 4
Enciphering
E
E_K(X)
Deciphering
D
D_K(E_K(X)) = X
X
K
• Turn plaintext SECRET into
ciphertext
• S+4=W, E+4=I, C+4=G, R+4=V, E+4=I,
T+4=X

7. Symmetric Cipher Example
• Much more elaborate transformations
are available
• Some that are so complicated that
even if the transformation was
public a key would still be needed
• Still require a distributed key

8. Asymmetric cipher
• Also known as public key
Enciphering
E
E_K(X)
Deciphering
D
D_K’(E_K(X)) = X
X
K’
K
• Two keys: public k, private k’
• Private key not required for both
parties
• More computationally intensive

9. Diffie-Hellman
• One of the first public key
cryptographic systems
• Developed by Martin Hellman, Ralph
Merkle, and Whitfield Diffie at
Stanford University in 1976

10. Diffie-Hellman
• Based on a special case of the
subset-sum, or knapsack, problem
Subset-sum Problem
5
8
4
11
6
20

11. Diffie-Hellman Example
• Block cipher
• Block size of 7 bits. Possible 27 combinations
• Private key (a’1, a’2, … , a’n) of 7 integers: (1, 2, 5, 11, 32, 87, 141)
• Chose two special integers, w and m, such that w and m are relatively prime,
meaning gcd(w,m) = 1: w = 901, m = 1234
• Public key (a1, a2, … , an) of 7 integers using the equation: ai = w * a’i mod m:
(901, 568, 803, 39, 450, 645, 1173)
• Partition SECRET into 7 bit blocks each block consisting of xn bits (x1, x2, …, xn)
S
1010011
E
1000101
C
1000011
R
1010010
E
1000101
T
1010100
• Bx = ∑ xiai
i=1
n
• S = 1 X (901) + 0 X (568) + 1 X (803) + 0 X (39) + 0 X (450) + 1 X (645) + 1 X (1173)
• S = 3522

12. Diffie-Hellman Example
• Encrypted blocks Bx received. Special version of subset-sum problem
• Which subset of (a’1, a’2, … , a’n) sums to B’x where B’x = Bx * w-1 mod m
• w-1 is the modular inverse of w for m, w * w-1 mod m = 1
• B’x = 3522 X (901)-1 mod 1234
• B’x = 3522 X 1171 mod 1234
• B’x = 234
1. sum ← 0
2. for i = n step -1 until 1 do
if ai + sum <= B’x
then sum ← sum + ai;
subset(i) ← 1
else subset(i) ← 0
3. if sum = B’x then exit with subset
else exit with “failure”
• Private key (1, 2, 5, 11, 32, 87, 141), B’x = 234, find subset (1, 0, 1, 0, 0, 1, 1) = S

13. Diffie-Hellman
• An algorithm that solves the
particular problem on which a
cryptographic system is based.
• An algorithm which solves NP-
complete problems quickly
• Two possible points of vulnerability

14. RSA
• Factorization so far is unsolvable in
polynomial-time
• Based on the difficulty of factoring
large numbers
• Developed by Ron Rivest, Adi
Shamir, and Leonard Adleman at
MIT in 1977.

15. RSA Example
• Find two large prime integers, p and q, and form product n = pq
• Find a random integer, e, that is relatively prime to Ф(n) = (p-1)(q-1)
• p and q are kept private, (n,e) are the public key
• Message is partitioned into blocks, b, such that b < n
• Each block is encrypted using the equation: c = be mod n
• For the private key, calculate integer d which is the modular inverse of e
for Ф(n), or e * d mod Ф(n) = 1
• Once d is calculated it becomes your private key and all records of
p and q should be destroyed
• Each encrypted block, c, is decrypted using the equation: b = cd mod n
• p = 61, q = 53, n = 3233, Ф(n) = 3120, e = 17, d = 2753
• encrypt(123) = 12317 mod 3233 = 855
• decrypt(855) = 8552753 mod 3233 = 123

16. RSA
• Factorization cannot be done in
polynomial-time
• Factoring is required to break the
system
• Security of RSA relies on two
assumptions

17. Moral Issues
• Information Theft
• Who does the data belong to?
• Privacy

18. Summary
• Diffie-Hellman and RSA
• Symmetric and Asymmetric ciphers
– Pros and Cons
• Cryptology
• Moral Issues

19. References
• A. Shamir, “A Polynomial-Time Algorithm for Breaking the Basic Merkle-Hellman
Cryptosystem", Advances in Cryptology - CRYPTO '82 Proceedings, pp. 279-288,
Plenum Press, 1983. IEEE Transactions on Information Theory, Vol. IT-30, pp. 699-
704, 1984.
• A.K. Dewdney, The New Turning Omnibus, pp. 250-257, Henry Holt and Company,
2001.
• RSA Cryptosystem, http://primes.utm.edu/glossary/page.php?sort=RSA.
• Cryptology FAQ, http://www.faqs.org/faqs/cryptography-faq/part06/.
• The Extended Euclidian Algorithm,
http://www.grc.nasa.gov/WWW/price000/pfc/htc/zz_xeuclidalg.html.