Grand Challenges in Mathematics

Are there any?

1. Is Factoring Hard?

2. P vs NP?

3. Riemann Hypothesis

Internet security

$1,000,000 Clay Prize

Hilbert problems (1900)

Problem which has long resisted solution,whose solution is expected to have

(turns out to have) far-reaching consequences

Grand Challenge (def.)

never published solutionGauss

Bolyai

Lobachevski

1823, 1832

1830

Is the parallel postulate independent of the other four postulates? Euclid, 300 BC

YES: it is independent!

- Riemannian Geometry

- General Theory of Relativity

- Non-Euclidean geometry 1830

1860

1978

1906 - 15

- GPS

Problem 1.

Is factoring hard?

23 x 29 =

Multiplying A and B

1271 = 31 x 41

Factoring N

Polynomial time

exponential time

6672371 x 2938 = 6,965,998

5 x 9 = 45

T ≈ digits(A,B)2

15 = 3 x 5 prime numbers

6,965,997 = 3 x T(N) ≈ N ≈ 10digits(N)

2321999

Prime numbersEuclid, 300 BC

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

Are there infinitely many twin primes?

How many? Infinitely many?

Polynomial time Exponential time

T = (digits)2 T = 10digits

1 10

4 100

16 10,000 3 hours

256 108 3 years

Digits Time (P) Time (E)

1

2

4

8

16 65,536 1016 317 mill years

4 min

18 hr

- Trial division

T(digits) ~ 10digits

T(digits) ~ 10(digits/2)

- H. Lenstra

Sloooooow !!!

- Elliptic Curve Factorization

It is advances in theory, more than anything else, that lead to dramatic improvements in computation.

T(digits) ~ 10√digits Faster !!!

~ 3.1digits

How fast can we factor?

Can we do even betterthan Prof. Lenstra?

Can we find a polynomial time factoring algorithm?

WE DON’T KNOW!

But ...Who cares?

We all do!

Besides the mathematicians?

Internet SecurityRSA Algorithm

encode( plaintext ) = ciphertextdecode( ciphertext ) = plaintext

encode( ATTACK ) = BUUBDLdecode( BUUBDL ) = ATTACK

Julius Caesar:

Encoding key:Decoding key:

+1: shift forward by 1 letter-1: shift backwards by 1 letter

Ron Rivest Adi Shamir Len Adleman

encode( x ) = xE mod Ndecode( y ) = yD mod N

15 mod 7 = 1

RSA:

Encoding key:Decoding key:

E, ND, N

N = pqp and q are prime

To break the code: factor N

D is computed from E, p, and q

- It is easy to find large primes

- It is hard to factor large numbers

The secret behind RSA

Main tool

Theorem, about 1650:

37 = 3 mod p

Fermatap = a mod p

p, q

N

37 = 2187

218773

2187

1

717

2

143

The Annals of Cryptograhy

A True Story

Main character: a number called RSA-129

Martin Gardner

114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541

N =

- August, 1977 Scientific American

http://www.math.okstate.edu/~wrightd/numthry/rsa129.html

- September 3, 1993 Project begins- April 27, 1994 Message decoded!

Matrix of 569,466 rows and 524,338 columns500+ computers, 8 months, 7500-mips-years

THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE

The factoring problem is unsolved.So we think RSA is secure.But we cannot prove this.

To conclude ...

ARE YOU WORRIED?

We need a theorem!

Problem 2.

P vs NP problem

One of the 7 Clay Millennium Prize Problems

$1,000,000 each

www.claymath.org

Alan TuringWhat is computation?

Bletchley Park, WW II

Kurt GödelWhat are the limits of computation?

Can problems that are solved by systematic search be solved instead by some clever, fast method?

Letter of Gödel to von Neumann, 1956

Class P

Problems that can be solved in polynomial time.

- multiplying x and y- finding the gcd of x and y- inverting a matrix

“feasible”

Class NP

Problems whose solution can be checked in polynomial time.

- sum-subset problem{ -7, -3, -2, 5, 8 }

- can N be factored?N = 25,150,949

{ -3, -2, 5 } is a certificate

4513, 5573 is a certificate

- traveleing salesman problem

NP complete

Size of the search space

Subsets of an N-element set

Size = 2N Grows exponentially in N

Possible factors of an d-digit number

Size = 10d Grows exponentially in d

P = NP?

Obvious fact

P is contained in NP

The Million Dollar QuestionP

NP

Equivalent questionShow that one NP-complete problem is in PIs the traveling salesman problem is in P?

A big surprise, 2002

Agrawal, Kayal, Saxena

In fact: in class P!

Is a number factorizable?

An NP problem.

Is a number prime?Same story

If P = NP, then factoring is in class P

This would be bad news!

Paris, 1900

Problem 3

The Riemann Hypothesis

Bernhard Rieman David Hilbert

Grand Challenge of the 1850’sWhat is the number of primes < N?

p(2) = 1p(3) = 2p(4) = 2p(5) = 3p(6) = 3p(7) = 4p(8) = 4

p(N) is approximately N divided by the number of digits in N,times 2.302...

p(N) ~ Li(N) = integral of 1/log(x) from 2 to N

N p(N) Li(N) - p(N) R.Err.106 78,498 129 1.6%109 5.085x106 1,700 .003%1012 3.761x1010 38,262 10-4%

Gauss:

Stockmarket?

Li(x) - p(x)

Riemann’s idea

Study the roots of the equation

1 + 1/2s + 1/3s + 1/4s + ... = 0

Where are its complex roots?

z(s) =

x=1

y

xx=0

Complex plane

Riemann hypothesis (RH): complex roots of z(s) are on the critical line

- - - - -critical line

Riemann showed allcomplex roots lie in the critical strip

- - - critical strip

The primes have the smallest possible randomness (standard deviation).

Statistics of the primes

RH gives p(N) ~ Li(N) in a very strong form

Why take on a challenge?

Curiosity

Adventure

- Solution gives more than a yes-no answer

When the challenge is met:

- Understanding

- New tools

- Unexpected consequences

The Grand ChallengesAre Yours!