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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
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!
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
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
Kurt GödelWhat are the limits of computation?
Can problems that are solved by systematic search be solved instead by some clever, fast method?
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
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:
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
- Solution gives more than a yes-no answer
When the challenge is met:
- Understanding
- New tools
- Unexpected consequences