1. Introduction to Cryptography Lecture 9

2. Public – Key Cryptosystems Each participant has a public key and a private key. It should be infeasible to determine the private key from knowledge of the public key.

3. Bob Alice message Public – Key Cryptosystems Bob encrypts message using Alice’s public key Alice decrypt message using her private key

4. Prime Numbers Definition: A prime number is an integer number that has only two divisors: one and itself. Example: 1, 2,17, 31. Prime numbers distributed irregularly among the integers There are infinitely many prime numbers

5. Factoring The Fundamental Theorem of Arithmetic tells us that every positive integer can be written as a product of powers of primes in essentially one way. Example:

6. The RSA Public – Key Cryptosystem In 1978, Ronald Rives, Adi Shamir, and Leonard Adelman wrote a paper called “A Method for Obtaining Digital Signatures and Public Key Cryptosystem”. They described a cipher system in which senders encrypt message using a method and a key that are publicly distributed.

7. The RSA Public – Key Cryptosystem Alice: Selects two prime numbers p and q. Calculates m = pq and n = (p - 1)(q - 1). Selects number e relatively prime to n Finds inverse of e modulo n Publishes e and m

8. The RSA Public – Key Cryptosystem To encrypt the message x: Bob computes:. Bob sends y to Alice. To Decrypt the message y: Alice computes:.

9. The RSA Public – Key Cryptosystem Example: p =127, q = 223. Then m = 28321 and n = 27972 Let e = 5623, check gcd(n,e) = 1. Then using Extended Euclidean Algorithm d = 22495. Public Key: (5623, 28321).

10. The RSA Public – Key Cryptosystem Example: Let the message be x = 3620. Then Alice gets one and decrypts it Then

11. The RSA Public – Key Cryptosystem Why does this method work?  Last step is a little bit more complicate How secure is RSA? Can opponent deduce d and n from (m,e)?  The opponent can find n and d only if he can factor m.

12. Factoring Problem of factoring a number is very hard Fermat’s factoring method sometimes can be used to find any large factors of a number fair quickly (pg.251) Want to make sure Fermat’s factoring method does not work for your key p and q should be at least 155 decimal digits each

13. Homework Read pg.286-293. Exercises: 2(a), 4(c), 5(a) on pg.294. Those questions will be a part of your collected homework.