P versus NP and Cryptography Wabash College Mathematics and Computer Science Colloquium Nov 16, 2010

1. P versus NP and CryptographyWabash CollegeMathematics and Computer Science ColloquiumNov 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. Questions? • Yes please … Jeff Kinne, Indiana State University

3. Jeff Kinne, Indiana State University

4. Secure Communication? • Websites are Secure • Factoring is “hard” • One-way functions exist • P not equal to NP 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) Jeff Kinne, Indiana State University

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

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

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

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

14. Exponential time Brute force search • Factor 1000 digit number • Check all ~ 101000 possibilities • 3-coloring a 1000 vertex graph • Try all 31000 possibilities • Can we do better? Jeff Kinne, Indiana State University

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

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

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

20. In conclusion … 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) Conjectures: P not equal to NP One way functions exist (multiplication) Cryptography exists • Want privacy? You need even more than P not equal to NP Jeff Kinne, Indiana State University

22. Thank you! Jeff Kinne, Indiana State University