Grand Challenges in
Mathematics
Are there any?
1. Is Factoring Hard?
Internet security
2. P vs NP?
$1,000,000 Clay Prize
3. Riemann Hypothesis
Hilbert problems (1900)
Grand Challenge (def.)
Problem which has long resisted solution,
whose solution is expected to have
(turns out to have)
far-reaching consequences
Is the parallel postulate independent of the
other four postulates? Euclid, 300 BC
YES: it is independent!
Gauss
never published solution
Bolyai
1823, 1832
Lobachevski
1830
- Non-Euclidean geometry 1830
- Riemannian Geometry 1860
- General Theory of
Relativity 1906 - 15
- GPS 1978
Problem 1.
Is factoring hard?
Multiplying A and B
5 x 9 = 45
23 x 29 = 667
2371 x 2938 =
T ≈ digits(A,B)2
6,965,998
Polynomial time
Factoring N
15 = 3 x 5
prime numbers
1271 = 31 x 41
6,965,997 = 3 x 2321999
T(N) ≈ N ≈ 10digits(N) exponential time
Prime numbers
How many? Infinitely many?
Euclid, 300 BC
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Are there infinitely many twin primes?
Polynomial time
Exponential time
T = (digits)2
T = 10digits
Digits
Time (P)
Time (E)
1
1
10
2
4
100
4
16
10,000
3 hours
8
256
4 min
108
3 years
16
65,536
18 hr
1016
317 mill
years
How fast can we factor?
- Trial division
T(digits)
~
10digits
T(digits) ~ 10(digits/2)
Sloooooow !!!
~
3.1digits
- Elliptic Curve Factorization
T(digits)
~
10√digits
Faster !!!
It is advances in theory, more than anything
else, that lead to dramatic improvements in
computation.
- H. Lenstra
Can we do even better
than Prof. Lenstra?
Can we find a polynomial time
factoring algorithm?
WE DON’T KNOW!
But ...
Who cares?
Besides the mathematicians?
We all do!
Internet Security
RSA Algorithm
Ron Rivest
Adi Shamir
Len Adleman
encode( plaintext ) = ciphertext
decode( ciphertext ) = plaintext
Julius Caesar:
encode( ATTACK ) = BUUBDL
decode( BUUBDL ) = ATTACK
Encoding key: +1: shift forward by 1 letter
Decoding key: -1: shift backwards by 1 letter
RSA:
encode( x ) = xE mod N
decode( y ) = yD mod N
15 mod 7 = 1
N = pq
p and q are prime
Encoding key: E, N
Decoding key: D, N
D is computed from E, p, and q
To break the code: factor N
The secret behind RSA
- It is easy to find large primes p, q
- It is hard to factor large numbers
N
Main tool
312
Theorem, about 1650: 7 2187
21
p
a = a mod p
87
37 = 2187
7
17
37 = 3 mod p
14
3
Fermat
The Annals of
Cryptograhy
A True Story
Main character: a number called RSA-129
N=
1143816257578888676692357799761466120102182967212423625625
61842935706935245733897830597123563958705058989075147599
290026879543541
- August, 1977 Scientific American
- September 3, 1993 Project begins
- April 27, 1994
Message decoded!
THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE Martin Gardner
500+ computers, 8 months, 7500-mips-years
Matrix of 569,466 rows and 524,338 columns
http://www.math.okstate.edu/~wrightd/numthry/rsa129.html
To conclude ...
The factoring problem is unsolved.
So we think RSA is secure.
But we cannot prove this.
We need a theorem!
ARE YOU WORRIED?
Problem 2.
P vs NP problem
One of the 7 Clay Millennium
Prize Problems
$1,000,000 each
www.claymath.org
Alan Turing
What is computation?
Bletchley Park, WW II
Kurt Gödel
What 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. “feasible”
- multiplying x and y
- finding the gcd of x and y
- inverting a matrix
Class NP
Problems whose solution can be
checked in polynomial time.
- sum-subset problem
{ -7, -3, -2, 5, 8 } { -3, -2, 5 } is a certificate
- can N be factored?
N = 25,150,949
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
Obvious fact
P is contained in NP
The Million Dollar Question
P = NP?
Equivalent question
Show that one NP-complete
problem is in P
Is the traveling salesman
problem is in P?
NP
P
A big surprise, 2002
Is a number factorizable?
An NP problem.
In fact: in class P!
Agrawal, Kayal, Saxena
Is a number prime?
Same story
If P = NP, then factoring is in class P
This would be bad news!
Problem 3
The Riemann Hypothesis
Bernhard Rieman
David Hilbert
Paris, 1900
Grand Challenge of the 1850’s
What is the number of primes < N?
p(2)
p(3)
p(4)
p(5)
p(6)
p(7)
p(8)
=
=
=
=
=
=
=
1
2
2
3
3
4
4
Gauss:
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%
Stockmarket?
Li(x) - p(x)
Riemann’s idea
Study the roots of the equation
z(s) = 1 + 1/2s + 1/3s + 1/4s + ... = 0
Where are its complex roots?
Complex plane
y
---
critical strip
- - - - -critical line
Riemann showed all
complex roots lie in the
critical strip
x=0
x=1
Riemann hypothesis (RH): complex roots of z(s)
are on the critical line
x
RH gives p(N)
~
Li(N) in a very strong form
Statistics of the primes
The primes have the smallest possible
randomness (standard deviation).
Why take on a challenge?
Curiosity
Adventure
When the challenge is met:
- Solution gives more than a yes-no
answer
- Understanding
- New tools
- Unexpected consequences
The Grand Challenges
Are Yours!