1. RSA Public Key Algorithm

2. RSA Algorithm history  Invented in 1977 at MIT  Named for Ron Rivest, Adi Shamir, and Len Adleman  Based on 2 keys, 1 public and 1 private

3. Basic Algorithm  A private and a public key are generated by a user.  The public key is given to a sender, who encrypts the message using that key.  The user then decrypts the message using the private key

4. Key generation  2 large prime numbers P and Q are randomly chosen  N = P * Q  S = (P-1) (Q-1)  Value E is chosen where 1<E<N and the greatest common denominator of E and S is 1  A value D is calculated where (D*E) modS = 1  D is the private key  (N,E) is published as the public key

5. Encryption  To send a message a person uses the public key  The plaintext message is broken into binary segments no larger than N  Each segment is encrypted with the algorithm ciphertext = plaintext E mod N

6. Decryption  To decrypt the message, the recipient will calculate each segment with plaintext = ciphertext D mod N

7. Confused?

8. Alfred chooses 2 primes  P = 19 and Q = 31  N = P*Q = 589

9. Alfred finds e  (P-1)(Q-1) = 540  E needs to have no GCD with 540  Can be found with Euclid’s algorithm (Example 27.6, p614 Automata)  E = 49

10. Alfred finds D  D is computed using an extension of Euclid’s algorithm  D = 1069  1069 is the private key  (589,49) is the public key

11. Batman wants to send message  The message will be “A”  ASCII code for A is 65  The encryption algorithm is then: 65 49 mod 589  However, don’t actually need to compute 65 49

12. Batman exploits 2 facts  N i+j = N i * N j  (N*M) mod K = (N mod K)(M mod K) mod K  This is called modular exponentiation  Note that 65 49 = 65 1+16+32

13. Batman encrypts message and sends  65 49 mod 589 = 65 1+16+32 mod 589  (65 1 *65 16 *65 32 ) mod 589  (65 1 mod 589)(65 16 mod 589)(65 32 mod 589) mod 589  (65 * 524 * 102) mod 589  3474120 mod 589  = 198  Batman sends Alfred the message 198

14. Alfred decrypts message  Alfred uses the private key 1069 and computes 198 1069 mod 589  This is done with the same process used in encryption to retrieve the message “A”

15. Why it’s effective  Larger integers increase effectiveness  Encryption and decryption are inverses of each other  User A can find E and D efficiently  Modular exponentiation allows both users to compute encryption and decryption efficiently

16. Why it’s effective  Any eavesdropper will not be able to recreate the enciphered text because the modular exponentiation is not an invertible function  Also, the private key D cannot be calculated from the public keys N and E

17. References  Automata, Computability, and Complexity. Elaine Rich. Pearson Prentice Hall, 2008.  RSA Laboratories' Frequently Asked Questions About Today's Cryptography, Version 4.1. RSA Laboratories Inc, 2000. http://www.rsa.com/rsalabs/node.asp?id =2152# http://www.rsa.com/rsalabs/node.asp?id =2152#