1. Cryptography Deffie hellman

2. organization Foundations Symmetric key Symmetric key weaknesses Assymmetric key Deffie hellman – key exchange RSA – public key, private key Applications of new idea

3. Foundation Encryption: S k [P] -> [C] Polynomial if k is given Decryption: S k -1 [C] -> [P] Polynomial if k is given K – key P – plain text C – Cypher text Hacker? Figure out S k Unconditionally secure No matter what Computationally secure Summumb to unlimited computation

4. Symmetric key encryption

5. DES/ AES Data encryption standard Advanced encryption standard DES - Data encryption standard - 56 bit key size - 1999 - broken in 22 hrs - insecure protocol AES - Advanced encryption standard - 128 bit, 192 bit and 256 bit No successful attack till now

6. Problem of symmetric key exchange

7. Problem - motivation 1874, William Stanley Jevons Trapdoor functions in cryptography Function f and its trapdoor t can be generated in polynomial time (f,t) = Gen() – generator function Computation of f is easy (polynomial time) but inverse is very hard unless you have the trapdoor

8. Deffie Hellmann (1976) Key exchange - Motivation Colors - two basic assumptions Generating new colors (mixing) -Easy to mix two colors to make a third color G+R = Y Separating colors from mixture -Given a third color, its hard to find the exact original colors Y = G+R One way function One way lock Easy in one direction Hard in the other Realizations - colors - Modulus (math func)

10. Colors - example

14. Deffie hellman – primitive root modulo n Mod function - % We use a prime modulus - 7 We find a primitive root of prime number 7 Why primitive root: This has a property of coverage ->> 3 is one example (this is known as generator)

15. Deffie hellman - one way lock Primitive root Prime modulo – 17 Primitive root – 3 (generator) Coverage - If 3 is raised to an integer x, then solution (3^x%17) is equally likely to be any integer between 1,17 One way function Finding modulus is easy 3^29%17 = 12 (easy) Reverse problem is hard 3^x mod 17 = 12; only way to find x is brute(hard) (this is discrete log problem)

16. Primitive root module n – one way function

17. Primitive root modulo n

20. Problem with deffie hellman key exchange Have to maintain many keys Open problem – public and private keys (introduced by deffie hellman) Solutions 1977 RSA 2002 turing award 1985 Elgamal excyption, based on deffie hellman

21. Fundamental issues in crypto Privacy No shall be able to read messages Authentication/ non repudiation The sender should be verifiable Integrity The message is same (no modification)

22. RSA (Rivest, Shamir, Adleman) – privacy (y) 1977 2002 Turing One way function Prime factors

23. Authentication – non repudiation (y)

24. Digital signature algorithm(DSA) - integrity (y)

25. An overview of techniques

26. Common attacks – not exhaustive Replay attack Problem of public key distribution Breaking the math – (P=NP) Open problems – may be? Unconditionally secure Timed keys

27. Thanks