NPComplete Problems Coloring is complete In particular we

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.

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

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

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)

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?

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

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?

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.

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

Security and Cryptography

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.

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).

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

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

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

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 (De. CSS), Cell phones in Europe (GSM), encoding of fonts by Adobe, many more

More sophisticated approach • Use theory of hard search problems and 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!

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

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

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

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

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

Public-Key Encryption Encrypted Message E(m) Alice Bob • 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

Public-Key Encryption Encrypted Message E(m) Alice • • • Eve the eavesdropper Bob 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.

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

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, …

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.

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!

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

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.

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

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

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

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

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

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

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

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: d e (m ) = m (Modulo N) • Bob reveals e, N as his Public Key • Bob retains d as his Private Key

The RSA Encryption Scheme • Recall Fact: Because Bob knows p and q, he can pick numbers e and d such that: • For all m: d e (m ) = 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)

The RSA Encryption Scheme • Recall Fact: Because Bob knows p and q, he can pick numbers e and d such that: • For all m: d e (m ) = 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: • d d e m = (E(m)) (Modulo N) = (m ) (Modulo N)

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

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

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?

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

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

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

A Magic Trick (cont. ) • 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! It’s Red! I knew he would say that.

Zero Knowledge • 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! Simulation Abracadabra, Goobedy goo! It is Red!

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

Mind Reading • You tell Magician what you were thinking. I was thinking of a banana.

Mind Reading (cont. ) • Magician tells you to open box, and read piece of paper in box. • Magician proves he can predict what you will say. Banana How did he do that!!

Mind Reading (cont. ) • Again, you knew what was going to happen. If you believed magician was indeed a magician, you learn nothing new Zero-Knowledge • Magician just shows you that he can do it • you can simulate it yourself Banana Simulation I was thinking of a banana.

Mind Reading (cont. ) • But why do you believe the magician is more special than you? • Because Magician committed to his guess before you told him. • Commitment is the key.

Cryptographic Commitment • Public Key Encryption Scheme • To commit to a string x, I send y = E(x). • like putting it in the box • To “open” the commitment (box), I use my secret key. • Commitment is secret. • And I can’t change my mind about x once I’ve sent the encryption.

ZK Proofs for NPComplete Problems • I tell you I have a solution to an NPComplete problem, but I won’t tell you what the solution is • You can ask me questions, and my answers will reveal that I have a solution, but they won’t reveal what the solution is

NP-Complete Problems • Remember we can reduce any search problem to Coloring.

ZK Proof for Coloring • Input: Collection of circles. • Magician Claims to Know: Coloring using R, B, G • Multi-step protocol. In each step: • First, Magician picks random permutation : R, B, G , and applies to coloring:

ZK Proof (cont. ) Magician presents encrpyted permuted graph

ZK Proof (cont. ) Verifier (Alice) chooses a random pair of nodes to verify

ZK Proof (cont. ) Magician decrypts those nodes and shows they’re colored in valid way

Can Simulate Magician Each Time Only difference is that a simulator won’t know a valid coloring (NP-Complete problem)

Simulated ZK Proof Interaction

ZK Proof: Analysis • Suppose Magician is an imposter: does not know a valid coloring. • Then at least one pair of connected circles will have colors equal (color equality survives permutation). Alice catches Magician cheating with probability at least 1/n 2. • Repeat protocol 100 n 2 times, Alice catches Magician cheating almost always!

ZK Proof: Analysis (cont. ) • Simulator/imposter can produce the same results each time (just needs to produce different colorings for the two edges) • Since magician permutes randomly each time: • Each result from simulator (or imposter) is in itself indistinguishable from what magician sends • Verifier cannot put together graph from the pairs (cannot gain knowledge; so zero-knowledge)

ZK Proof: Analysis (cont. ) • Key difference between magician & imposter: • Magician knows valid coloring, can commit to it, and won’t be caught in pairwise game • Simulator/imposter does not, and will ultimately be caught • But commitments are secret, by security of encryption scheme. Simulator output and real life are indistinguishable to an eavesdropper, who cannot tell if magician is imposter or not

Wrap-up • Today we saw some examples illustrating techniques from modern cryptography: • Encryption • Zero Knowledge Proofs