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Dr. Bob Hummel Potomac Institute for Policy Studies
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STEM: Science, Technology, Engineering, & Mathematics � STEM includes mathematics � But when you call it STEM, do
you think “mathematics”? � Math is at the tail end
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 2
Are these people learning math?
The Phenomenon of Math Phobia
� Math is cumulative � For most of the math curriculum
� If you fall behind, you remain behind
� Answers in math are generally right or wrong
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 3
Why do we even bother to teach math? Don’t calculators and computers obviate math?
What we teach
� Arithmetic � Word problems � Algebra � Geometry � Graphing, pre-calc � Calculus
2+2=
Sally has 23 cents. She…
7X6= 4–6=–2 0+5=5
5+x=8
( ) ( ) ( )afbfxx'f −=∫ d
4 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
Mostly builds one topic to the next. Parents reinforce children phobia.
Why do we teach these? They are useful… � Arithmetic in daily life � Word problems are about thinking
� Calculus for engineering
5 The Art of Number Theory February 4, 2013
But few actually ever use higher mathematics � Riemann Surfaces � Category Theory � Homotopy Theory
� Lipschitz Functions � Riemann-Roch
Theorem � Algebraic Topology
6 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
Some higher math ends up being very important
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 7
Bernard Riemann b. 1826
� Riemannian geometry is the key to General Relativity
Partial Differential Equations leads to Computational Fluid Dynamics, and then flight control
Lehmer Sieve
And number theory, and
theory of primes, leads
to cryptography
But the real point is to teach logical thinking � We justify math education as
a route to logical thinking � Proofs � Analysis, vice arguments � Brain exercises
8 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
And those STEM fields benefit from mathematical thinking � Mathematics is about analytic
thinking � Proofs � Intuition: What is provable?
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 9
Let us think Deeply of Simple Things � Arnold E. Ross
� The Ross Math Program ○ 1957 to 2000 ○ Dan Shapiro continues the program
� The Ohio State Math program for High School Students
� Based on Number Theory
10 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
Outcome of the Ross Program
� A rather large percentage of graduates became practicing mathematicians � Also some famous physicists
� The big advantage of number theory: � After some basics, many topics are
independent of one another � And the basics are simple
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 11
Clock Arithmetic Example: 10:00 + 3hr = 1:00
10+3≡1 mod 12
12 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
10+2≡0 mod 12 10+9=7 mod 12
Clock Arithmetic with a different clock
7
1
2
3 4
5
6 4:00 mod 7
7
1
2
3 4
5
6
Add 5 “hours”
4+5≡2 mod 7
13 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
4+3≡0 mod 7
Addition table mod 7
+ 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 2 3 4 5 6 0
2 2 3 4 5 6 0 1
3 3 4 5 6 0 1 2
4 4 5 6 0 1 2 3
5 5 6 0 1 2 3 4
6 6 0 1 2 3 4 5
14 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
But what about multiplication?
X 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6 0 6 5 4 3 2 1
Examples: 3X5≡1 mod 15 2X5≡3 mod 7 5X5≡4 mod 7 6X6≡1 mod 7
15 The Art of Number Theory
Dr. Bob Hummel February 4, 2013
Under multiplication, Up is an Abeilian Group � Zp, p a prime, = {0,1,2,3,… p–1} � Up, p a prime, = {1,2,3,… p–1} � All group properties inherited from R,
except: � Multiplicative inverses
� For any a in Up, other than 0, find a–1 such that a X a–1=1 mod p
Z7, Z17, Z213466917–1
16 The Art of Number Theory February 4, 2013
Greatest common divisor, also called the greatest common factor � gcd(6,9)=3 � gcd(55,121)=11 � gcd(35,49)=7 � In general, a common divisor larger than
every other common divisor � a and b are “relatively prime” if gcd(a,b)=1 � If p is prime, then gcd(a,p)=1 unless a=np
� I.e., unless a ≡ 0 mod p
17 The Art of Number Theory February 4, 2013
Diophantine Equation
� Given a, b nonzero integers, find x, y such that ax+by=gcd(a,b)
� Theorem: There always exist an x and y, integers, that solve the Diophantine Equation
� Examples � 6X(-1)+9X(1)=3 � 55X(-2)+121X(1)=11 � 35X(3)+49X(-2)=7
18 The Art of Number Theory February 4, 2013
A lovely math theorem
� Let Un = { x | gcd(x,n)=1}, under multiplication mod n
� Then Un is an Abelian Group
� n a prime is a special case
� The proof is constructive!
February 4, 2013 The Art of Number Theory 19
Some examples � U21 = { 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20} � Inverses:
� 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20 � 1, 11, 16, 17, 8, 19, 2, 13, 4, 5, 10, 20
� Check it out � How come this works? � And, incidentally, this will be important for
encryption
February 4, 2013 The Art of Number Theory 20
Euclid’s Algorithm to find gcd’s
b.325 BC
35 49
1
35
14 35
2
28
7 14
2
14
0
This is the gcd
gcd( 35, 49) = 7
And 1, 2, 2 are the partial quotients
1 + 1
2 + 1 2
Continued fraction!
21 The Art of Number Theory February 4, 2013
Another Example
230 563
2
460 103 230
2
206
24 4
103 96
7
p=563
a=230
24
3
21
3 7 2
6
1 3
3
3
0
2 2 4 3 2 3
2 + 1
2 + 1
4 + 1
3 + 1
2 + 1
3
2, 5/2, 22/9, 71/29, 164/67, 563/230 22 The Art of Number Theory February 4, 2013
563
230 =
So, what is the inverse of 230 mod 563 p=563
a=230
2 2 4 3 2 3
2 + 1
2 + 1
4 + 1
3 + 1
2 + 1
3
2, 5/2, 22/9, 71/29, 164/67, 563/230
Diophantine: – 164 · 230 + 67 · 563 = 1
So (– 164) · 230 = 1 mod 563
I.e., 230 – 1 = 399 mod 563
Check: 230 · 399 = 91770 = 563 · 163 + 1
23 The Art of Number Theory February 4, 2013
Answer: −164 = 399
A faster way of computing partial quotients
p=563
a=230
2 2 4 3 2 3
2 + 1
2 + 1
4 + 1
3 + 1
2 + 1
3
2, 5/2, 22/9, 71/29, 164/67, 563/230
0 1 2 5 22 71 164 563
1 0 1 2 9 29 67 230
Inverse is either 164 or –164
24 The Art of Number Theory February 4, 2013
Fermat’s Theorem
� For any a other than 0 mod p, a p = a mod p
� Equivalently a p–1 ≡ 1 mod p
b. 1601 (or maybe 1607)
25 The Art of Number Theory February 4, 2013
Euler’s Theorem
� Generalizes Fermat’s Theorem
a φ(n) ≡ 1 mod n where φ(n) is the number in the set Un
If gcd(a,n) = 1
b. 1707
26 The Art of Number Theory February 4, 2013
This would seem to have little to do with encryption � After all, the simplest encryption is a
letter cipher:
� This encryption method, indeed, any simple cipher, is easily broken
A→N B→O C→P D→Q
M→Z …
27 The Art of Number Theory February 4, 2013
Public key encryption is completely different concept � I tell you how to encrypt a message to
me
� You encrypt, and send the message to me
� Only I know how to decrypt
Me You Encryption key
Me You Encrypted message
Me You
Decrypt
A variation allows one to “sign” messages to prove authentication
28 The Art of Number Theory February 4, 2013
RSA Public Key encryption
� Uses number theory! � First, I need to tell you how to encrypt a
message I choose two prime numbers, p and q
Set N = p·q
Choose any E such that gcd(E, (p–1)·(q–1)) = 1
Me You N and E
I send you N and E
29 The Art of Number Theory February 4, 2013
Quick Aside
� Finding primes p and q is quick and easy � Uses a probabilistic algorithm � Works even if p and q involve hundreds of
digits
� Also, choosing an E is quick and easy
30 The Art of Number Theory February 4, 2013
RSA Public Key encryption
� Next, you encrypt the message � You have N and E
� As does everyone else
Your message is m1, m2, m3, …
You compute ni ≡ miE mod N for each mi
You send me ni for each i
Converted to numbers
Me You ni
31 The Art of Number Theory February 4, 2013
Quick aside 2
� Computing xE mod N is easy and fast, by repeatedly squaring
32 The Art of Number Theory February 4, 2013
In order to decrypt, I need to use the algorithm to find inverses � Recall: � So I can use the continued fraction
algorithm to find D such that:
E satisfies gcd(E, (p–1)·(q–1)) = 1
ED ≡ 1 mod (p–1)·(q–1)
33 The Art of Number Theory February 4, 2013
And now I can decrypt the message
� To decrypt: I compute
� Amazingly,
� But if someone else doesn’t know D, they can’t decrypt
niD mod N for each ni
mi ≡ niD mod N for each ni
34 The Art of Number Theory February 4, 2013
How to factor N
� Given N=p·q, find p and q � I.e., factorization � Believed to be “hard” � But no one knows for sure
35 The Art of Number Theory February 4, 2013
So, the big outstanding question: How to factor large numbers that are a product of two primes?
� As of right now, there is no good way
� There is also no proof that it can’t be done
February 4, 2013 The Art of Number Theory 36
But if we had a quantum computer, there is a reasonably fast way
� Based on Shor’s Algorithm � A probabilistic algorithm, specifically for a
quantum computer � Uses number theory:
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 37
1. Choose any a in UN (mod N)
2. Find r = o(a) mod N Smallest r such that ar ≡ 1 mod N
3. If r is odd, go back to 1, and try again
5. If it is 1, then try again (at step 1)
4. Compute gcd(ar/2 – 1, N), which be a divisor of N I.e., 1, p, or q
Quantum Computer role in breaking RSA � Powers of a form a periodic series:
� A quantum computer can quickly do an FFT
to find the period of a periodic series � The periodic series can be held in log2N qubits
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 38
a, a2, a3, a4, a5, …, ar, a, a2, … ar, a, a2, …
Prognosis
� Bob’s opinion: � Breakthrough’s are coming too fast to
believe there won’t be a practical quantum computer soon
� RSA will get broken, but some time later ○ Needs a lot of qubits ○ Needs control and a good programming ability
� Quantum computers will mostly be used to break RSA ○ And for quantum key distribution
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 39
Greater prognosis
� Can we get over math phobia?
� But maybe not today
February 4, 2013 The Art of Number Theory
Dr. Bob Hummel 40
Yes, I hope so. Enthusiastic, energetic teachers
Who encourage thinking deeply of simple things