1. P versus NP and Cryptography Wabash College Mathematics and Computer Science Colloquium Nov 16, 2010 Jeff Kinne, Indiana State University (Theoretical) Computer Science Formerly student of: Wisconsin, Xavier Other: 3 young kids, 1 math/CS teacher wife

2. Jeff Kinne, Indiana State University 2 Questions? Yes please …

3. Jeff Kinne, Indiana State University 3

4. 4 Secure Communication? Websites are Secure Factoring is “hard” One-way functions exist P not equal to NP

5. Jeff Kinne, Indiana State University 5 NP? Factors of 323? Product of 17 and 19? Factor a 1000 digit number? Multiply 1000 digit numbers? NP: easy to check correct solution (a.k.a. Nondeterministic Polynomial time)

6. Jeff Kinne, Indiana State University 6 NP Can you 3-color the graph/map? easy to check correct solution

7. Jeff Kinne, Indiana State University 7 NP, more examples Is a math claim true “Easy” to check the proof Routing/Scheduling Nash Equilibria in some settings DNA/protein matching Graph problems (vertex cover, clique, TSP, …) Knapsack, subset sum, bin packing … Integer programming …

8. Jeff Kinne, Indiana State University 8 P versus NP P – problems we can solve (efficiently) Clay Math Inst. Millennium Prize Can we solve all NP problems (efficiently)?

9. Jeff Kinne, Indiana State University 9 P versus NP Who cares?

10. Jeff Kinne, Indiana State University 10 If P=NP Optimal scheduling/routing Theorem proving (including all other Clay Math problems!) … No crypto/privacy!

11. Jeff Kinne, Indiana State University 11 P versus NP and cryptography Cryptography “One way” function (e.g., multiplication) If P=NP No one way functions! No cryptography! Encryption: should be easy “Un-encryption”: should be hard

12. Jeff Kinne, Indiana State University 12 If P not equal to NP Not known to imply one way functions (showing that would be major result.) Hard to even approximate many scheduling/routing/etc. problems (Major breakthrough in the 90’s)

13. Jeff Kinne, Indiana State University 13 P versus NP What do we know?

14. Jeff Kinne, Indiana State University 14 Brute force search Factor 1000 digit number Check all ~ 10 1000 possibilities 3-coloring a 1000 vertex graph Try all 3 1000 possibilities Exponential time Can we do better?

15. Jeff Kinne, Indiana State University 15 Better than brute force Is a given number prime? Linear programming Simple example: shortest path Remember where we have been already! Non-trivial algorithms do exist!

16. Jeff Kinne, Indiana State University 16 Conjecture: P not equal NP Need to show “no algorithm can…” No matter how clever… Really hard to show this… So most projections on solution to P versus NP are… 20 years 100 years never

17. Jeff Kinne, Indiana State University 17 One Thing we Do Know!! ____ many many different problems (a.k.a. “NP complete” problems) If could solve ____ then could solve all of NP!

18. Jeff Kinne, Indiana State University 18 NP Complete problems Arbitrary NP problem Circuit SAT 3-coloring Clique Knapsack For more, see http://www.csc.kth.se/~viggo/problemlist/

19. Jeff Kinne, Indiana State University 19 So to settle P versus NP Can just look at 3-coloring or Traveling Salesperson or Knapsack or … Can even focus on a single “universal” algorithm: Try all possible algorithms, each one step at a time…

20. Jeff Kinne, Indiana State University 20 In conclusion …

21. Jeff Kinne, Indiana State University 21 P versus NP “NP-complete” problems Hard to show P not equal NP (there are lots of non-trivial algorithms) Want privacy? You need even more than P not equal to NP Conjectures: P not equal to NP One way functions exist (multiplication) Cryptography exists

22. Jeff Kinne, Indiana State University 22 Thank you!