1. 1 Simple and Unbreakable: The Mathematics of Internet Security Dr. Monica Nevins Department of Mathematics and Statistics University of Ottawa University of Ottawa Day, 2007

2. 2 Cryptography ca. 50 BC Example: VENI, VIDI, VICI Becomes: YHQL, YLGL, YLFL Caesar cipher: Shift each letter forward by 3

3. 3 Second World War : Enigma Secret device Secret settings (rotors and plugs) 10 20 possibilities "Uncrackable" Cracked by mathematicians in early 1940.

4. 4 Today Millions of people need private, secure communication over the internet every day. Everyone has access to every interchange of communication. How can we start secure communications without first having secure communications?

5. 5 A Thought Experiment Say the only secure communication in this room is to lock your message in a box. Anything not in the box can be read or duplicated or stolen. Could you send me a secret message (that I can read but no one else can)?

6. 6 The model for public key cryptography M C C, d …M ! C?? AliceBob Eve d e e

7. 7 We need a one-way function Multiply : 17 x 11 = ?187 Factor : 91 = ? X ?7 x 13 This is a one-way function: Multiplication is easy Factoring is hard

8. 8 How hard is factoring? Say N has 20 digits. To find a factor, you need to search up to:  N ~ 10 digits How many numbers is that? 10 10 = 10,000,000,000 = 10 billion

9. 9 Idea: Find two large prime numbers p and q. Set N = pq. But: isn't finding primes just as hard as factoring? NO! Check out the AKS algorithm, 2003.

10. 10 But how does this give us a cryptosystem?

11. 11 Modular Arithmetic Doing math "mod 10" means taking the remainder after division by 10 4 x 4 = 16 implies 4 x 4 = 6 mod 10 4 x 4 = 16 implies 4 x 4 = 1 mod 5

12. 12 Multiplication Table, mod 5 x01234 000000 101234 202413 303142 404321 Mysterious patterns, but : easy to calculate.

13. 13 More powerful: exponentiation Consider powers of 4 mod 91: 4 1 = 4 4 2 = 16 4 3 = 64 4 4 = 256 = 74 mod 91 4 5 = 1024 = 23 mod 91 …

14. 14 Exponentiation “mod N” is one-way Calculating powers mod N is easy; Calculating roots mod N is hard. Except: it’s easy if you have the secret key:  (N) = (p-1)(q-1) For example: N = 91 = 13 x 7 gives  (N) = 12 x 6 = 72.

15. 15 How the secret key works When e and d satisfy ed = 1 mod  (N), (Example: 5 x 29 = 145 = 1 mod 72) then C = M e mod N if and only if M = C d mod N.

16. 16 RSA Cryptosystem Two primes: p = 7, q = 13. Set N = pq = 91. Choose an e = 5. Public key: (N, e) = (91, 5) Now  = 72 and d = 29, since ed = 5 x 29 = 145 = 1 mod 72. Private key: d = 29.

17. 17 RSA Encryption Get the public key (N,e) = (91,5) Secret message: M = 4 Calculate C = M e mod N: C = 4 5 mod 91 = 1024 mod 91 = 23 mod 91

18. 18 The Cryptogram 23 ??

19. 19 RSA Decryption Given C = 23 and private key d = 29, calculate: C d = 23 29 mod 91 Since 23 6 = 1 mod 91, 23 29 = 23 5 = 6436343 = 4 mod 91 So the secret message was M = 4 !

20. 20 Security of RSA Mathematicians have been studying number theory for ages --- we have confidence that there are no shortcuts. New technologies (quantum computer) Need new cryptosystems built on different mathematical concepts to ensure we stay ahead of technology (elliptic curves, lattice cryptosystems, etc)

21. 21 For more information Come and enjoy undergraduate studies in Pure Mathematics at the University of Ottawa mnevins@uottawa.ca