1. CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

2. Today’s Topics: 1. Prime factorization 2. Primality testing 2

3. 1. Prime factorizations Primes are the atoms of integers 3

4. Primes 4

5.  Which of the following is prime? A. 1 B. 6 C. 7 D. 21 5

6. Prime factorization  Basic theorem of number theory: any integer can be factored as a product of primes (we will prove this soon)  Moreover, this factorization is unique  Example: 36=2*2*3*3 6

7. Existence of prime factorization 7

8. 8

9. How to find the prime factorization?  What is the prime factorization of 100? A. 2*5 B. 2*2*5 C. 10*10 D. 2*2*5*5 E. Other 9

10. How to find the prime factorization?  What is the prime factorization of 82768328638217? 10

11. How to find the prime factorization?  What is the prime factorization of 82768328638217?  Not so easy anymore…  The security of RSA (which for example, protects your bank accounts) is based on the assumption that it is hard to factor numbers  We cannot prove it.  In fact, maybe somebody knows how to factor numbers fast (lets wait for more Snowden revelations…) 11

12. 2. Primality testing 12

13. Primality testing  If it’s hard to factor numbers into primes, lets focus on an easier problem  Test if a given integer is prime  How can you do it? Try and come up with the best algorithm you can 13

14. Primality testing  Algorithm 1: isPrime(n): 1. For i=2..n-1 1.1 If n MOD i=0: return False 2. Return True 14

15. Primality testing  Algorithm 1: isPrime(n): 1. For i=2..n-1 1.1 If n MOD i=0: return False 2. Return True 15 How many steps? A. ~n B. ~n 2 C. ~log(n) D. Doesn’t depend on n E. Other

16. Better primality testing? 16

17. Primality testing  Algorithm 1: isPrime(n): 1. For i=2..n-1 1.1 If n MOD i=0: return False 2. Return True 17

18. Primality testing (II) 18

19. Primality testing (II) 19 How many steps? A. ~n B. ~n 2 C. ~log(n) D. Doesn’t depend on n E. Other

20. An even faster algorithm?  The prime numbers used in RSA are very big, with 100s of digits  These algorithms will take forever  There are much faster algorithms, which run in time ~log(n); they are randomized algorithm (e.g. Miller-Rabin) 20

21. Log vs poly time 21