NP-Complete Problems

1. NP-Complete Problems • Coloring is complete • In particular, we can reduce solving any search problem to finding a valid coloring for some collection of circles! • So, if we could solve Coloring quickly, then P = NP • That’s why we believe Coloring can’t be solved quickly by any computer. • We call such problems NP-Complete.

2. NP-complete problems • Coloring • Traveling Salesman Problem • Knapsack problem • Partition Problem

3. Knapsack problem • We are given a set of items each having an integer weight • We are given an integer capacity for the knapsack • We ask if we can exactly pack the knapsack using a subset of these items

4. Sample Knapsack problem • Item weights 2,4,9,13,17,23,32,70,123,157 • Capacity is 228 • Packing 157 + 32 + 17 + 13 + 9 • Capacity is 226 • Packing (there are none)

5. Partition problem • We are given a set of items each having an integer weight • We are asked if we can divide all the items into 2 groups with equal total weight • Is this NP-complete?

6. Partition problem • We are given a set of items each having an integer weight • We are asked if we can divide all the items into 2 groups with equal total weight • Like knapsack problem • Capacity of knapsack is half the total weight

7. Sample Partition problem • Item weights 2,4,9,13,17,23,32,70,123,157 • Total weight is 450 • Packing 123 + 70 + 32 = 225 • Packing 157 + 23 + 17 + 13 + 9 + 4 + 2 = 225 • Why is this different from the PCP?

8. Other Hard Problems? • There are other problems besides NP-Complete Problems that we also believe are hard. • Can we be sure? • No. • But people have been trying to solve certain mathematical problems for centuries. • So it seems reasonable to assume that nobody will figure out how to solve them soon • Then again, what about Fermat’s Last Theorem? … • People continue to work on NP-completeness.

9. Wrap-up • We’ve seen problems that computers can’t solve at all, and also problems that computers probably can’t solve in our lifetimes. • Too much bad news? • Could it ever be useful to have hard problems? • Yes! • Cryptography

10. Security and Cryptography

11. Cryptography • Why do we care so much about hard problems? • Because sometimes we want to make things hard. • Protecting Privacy, Authenticity • Want to make it hard for adversaries to: • Steal our credit cards • Impersonate us • Etc. • Makes it possible for companies to protect intellectual property.

12. Cryptography • Science of making things hard for adversaries = Cryptography • Dates back to Julius Caesar • Caesar cipher – shift each character by a few places • "UHWXUA WR URPH" encodes “RETURN TO ROME“ • Used extensively during WW 2 (and every other war) • Used to encode passwords • Used to prevent copying of software and data (e.g. DVD).

13. Requirements of a cryptosystem • Easy to encode messages • Hard to decode messages

14. One Approach... It’s so complicated! It must be secure! Cryptosystem XYZ (Patent Pending)

15. Extra!: Cryptosystem XYZ Broken 2 Days After Release! One Approach...

16. One Approach... • Unfortunately, this approach is often used in real life. • This is one of the reasons why you hear about so many security systems being broken! • Examples: DVD encryption (DeCSS), Cell phones in Europe (GSM), encoding of fonts by Adobe, many many more

17. More sophisticated approach • Use the theory of hard search problemsand the notion of reducing one problem to another. • Show that if you break this security system, you do so by solving some of the world’s greatest unsolved problems first!

18. Encryption • The most basic problem in Cryptography is Encryption: Private Message m Bob Alice

19. Encryption • The most basic problem in Cryptography is Encryption: Private Message m Bob Alice Eve the eavesdropper

20. Encryption • The most basic problem in Cryptography is Encryption: Encrypted Message E(m) Bob Alice Eve the eavesdropper

21. Encryption • Have to make it easy for Bob to recover m • But hard for Eve to learn anything about m Encrypted Message E(m) Bob Alice Eve the eavesdropper

22. Public-Key Cryptography[Diffie-Hellman 1976] Bob’s Public Key Bob’sSecret Key Bob • Everybody knows Bob’s published Public Key. • Only Bob knows his private key.

23. Public-Key Encryption Encrypted Message E(m) Bob Alice • Alice uses Bob’s public key to encrypt m. • Bob uses his private key to recover (decrypt) m. • Relationship between public and private key is such that encryption with former enables decryption with latter

24. Public-Key Encryption Encrypted Message E(m) Bob Alice Eve the eavesdropper • Alice and Eve both know Bob’s public key. • Eve must not be able to “break” the encryption even though she knows the public key. • Discovering private key from public key must be VERY hard.

25. Basic Math Review • Let’s recall some basic mathematics: • A number p is called prime if its only factors are 1 and itself. • Examples:

26. Basic Math Review • Let’s recall some basic mathematics: • A number p is called prime if its only factors are 1 and itself. • Examples: 2, 3, 5, 7, 11, 13, 17, 19, …

27. Basic Math Review • Let’s recall some basic mathematics: • A number p is called prime if its only factors are 1 and itself. • Examples: 2, 3, 5, 7, 11, 13, 17, 19, … • There are many prime numbers. • Fact: It is known how to check quickly if a number is prime or not. • So, to find a big prime number, we can just keep generating large random numbers until we find a prime.

28. Basic Math Review • Given two primes p and q, it is easy to multiply them together: N = pq • But given N, how do you find p and q quickly?i.e. how do you factor N into primes? • Easy for small numbers (e.g. 6 or 35). • For centuries, mathematicians have been trying to find ways to factor large numbers quickly. No one knows how! • Factoring a 10,000 digit N would take centuries on the fastest computer in existence!

29. How do we know factoring is hard? • Problem has a long history • Prizes are offered and have been for a long time • Factoring progress happens slowly

30. Basic Math & Crypto • We want to make it so that for Eve the eavesdropper to break our system, she would have to factor a very large number. • We’ll (almost) do that.

31. Modular Arithmetic • Ordinary integers: … -4 -3 -2 -1 0 1 2 3 4 …

32. Modular Arithmetic • Ordinary integers: • Integers Modulo N: … -4 -3 -2 -1 0 1 2 3 4 … 0 1 (N– 1) 2 (N– 2) (N– 3) 3 …

33. 0 1 11 2 10 9 3 … Modular Arithmetic • Example: Arithmetic Modulo 12 (like Arithmetic performed on hours) • 3 + 11 (Modulo 12) = • 2 – 4 (Modulo 12) = • 5 * 4 (Modulo 12) = • 4 * 3 (Modulo 12) =

34. 0 1 11 2 10 9 3 … Modular Arithmetic • Example: Arithmetic Modulo 12 (like Arithmetic on time) • 3 + 11 (Modulo 12) = 2 • 2 – 4 (Modulo 12) = • 5 * 4 (Modulo 12) = • 4 * 3 (Modulo 12) =

35. 0 1 11 2 10 9 3 … Modular Arithmetic • Example: Arithmetic Modulo 12 (like Arithmetic on time) • 3 + 11 (Modulo 12) = 2 • 2 – 4 (Modulo 12) = 10 • 5 * 4 (Modulo 12) = • 4 * 3 (Modulo 12) =

36. 0 1 11 2 10 9 3 … Modular Arithmetic • Example: Arithmetic Modulo 12 (like Arithmetic on time) • 3 + 11 (Modulo 12) = 2 • 2 – 4 (Modulo 12) = 10 • 5 * 4 (Modulo 12) = 8 • 4 * 3 (Modulo 12) =

37. 0 1 11 2 10 9 3 … Modular Arithmetic • Example: Arithmetic Modulo 12 (like Arithmetic on time) • 3 + 11 (Modulo 12) = 2 • 2 – 4 (Modulo 12) = 10 • 5 * 4 (Modulo 12) = 8 • 4 * 3 (Modulo 12) = 0

38. The RSA Encryption Scheme [Rivest Shamir Adleman 1978] • Bob picks two large primes p and q, and computes: N = pq • Fact: Because Bob knows p and q, he can pick numbers e and d such that: • For all m: (me)d= m (Modulo N) • Bob reveals e, N as his Public Key • Bob retains d as his Private Key

39. The RSA Encryption Scheme • Recall Fact: Because Bob knows p and q, he can pick numbers e and d such that: • For all m: (me)d= m (Modulo N) • To Encrypt a message m, Alice uses Bob’s public key (e, N) to compute the encrypted message: • E(m) = me(Modulo N)

40. The RSA Encryption Scheme • Recall Fact: Because Bob knows p and q, he can pick numbers e and d such that: • For all m: (me)d= m (Modulo N) • To Encrypt a message m, Alice uses Bob’s public key (e, N) to compute the encrypted message: • E(m) = me(Modulo N) • To Decrypt, Bob uses private key (d) to compute: • m = (E(m))d (Modulo N) = (me)d(Modulo N)

41. The RSA Encryption Scheme • To Encrypt a message m, Alice computes: • E(m) = me(Modulo N) • The only known way to compute m from E(m) involves knowing d, which implies knowing p and q, which implies factoring N. • For Eve to break this system, she would have to solve a long-standing open problem in Mathematics • This is probably the most widely used Public-Key Encryption Scheme in the world. • Look at Help on IE

42. Shifting Gears: Proofs… • Bob wants to convince Alice of (prove to her) the validity of some statement (like “I really am Bob!”) • But Bob doesn’t want to reveal any of his secrets to Alice in the process… Bob Alice

43. Zero-Knowledge Proofs • What is the least amount of information Bob can reveal, while still convincing Alice? • Amazingly, it is possible for Bob to convince Alice of something without revealing any new information to Alice at all! • How can that be?

44. Magic Tricks • Magic tricks are like zero-knowledge proofs: • Good magic tricks reveal nothing about how they work. • What makes a magic trick good?

45. Two balls: Purple and Red, otherwise identical • Blindfolded Magician • You give a random ball to magician A Magic Trick

46. Magician tells you the color! • Magician proves he can distinguish balls blindfolded. • You learn nothing except this. A Magic Trick (cont.) Abracadabra, Goobedy goo! It is Red! Wow! He’sso cool!

47. You knew exactly what magician was going to do. • And he did it, i.e. showed you he can do it! • Since you knew to begin the magician was a magician and would get the right answer, you could not have learned anything new! A Magic Trick (cont.) It’sRed! I knew hewould say that.

48. What it means: • Alice “knows” what is going to happen. • She can produce the same answers herself: no need for magician • CS-speak: Alice can simulate it herself! Zero Knowledge Simulation Abracadabra, Goobedy goo! It isRed!

49. Magician asks you to think of either • “Apple” or • “Banana” • Magician then gives you a sealed box. Another Magic Trick

50. You tell Magician what you were thinking. Mind Reading I was thinkingof a banana.