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 2 7 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 (a 1, a 2, …, a n ) of 7 integers using the equation: a i = w * a’ i mod m: (901, 568, 803, 39, 450, 645, 1173) Partition SECRET into 7 bit blocks each block consisting of x n bits (x 1, x 2, …, x n ) S 1010011 E 1000101 C 1000011 R 1010010 E 1000101 T 1010100 B x = ∑ x i a i 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 B x received. Special version of subset-sum problem Which subset of (a’ 1, a’ 2, …, a’ n ) sums to B’ x where B’ x = B x * 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 a i + sum <= B’ x then sum ← sum + a i ; 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 = b e 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 = c d mod n p = 61, q = 53, n = 3233, Ф(n) = 3120, e = 17, d = 2753 encrypt(123) = 123 17 mod 3233 = 855 decrypt(855) = 855 2753 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.