The art of number theory v7b[1]

Dr. Bob Hummel
Potomac Institute for Policy Studies

hummel@PotomacInstitute.org

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STEM: Science, Technology,
Engineering, & Mathematics
ž  STEM includes mathematics
ž  But when you call it STEM, do
you think “mathematics”?
ž  Math is at the tail end

Are these people learning math?

The Art of Number Theory
February 4, 2013 Dr. Bob Hummel 2

The Phenomenon of Math Phobia
ž  Math is cumulative
ž  For most of the math curriculum
—  If you fall behind, you remain
behind
—  Answers in math are generally right or wrong

Why do we even bother to teach math?
Don’t calculators and computers obviate math?


What we teach
ž  Arithmetic 2+2= 7X6= 4–6=–2 0+5=5

ž  Word problems Sally has 23 cents. She…

ž  Algebra 5+x=8

ž  Geometry
ž  Graphing, pre-calc
ž  Calculus f ' (x )dx = f (b) − f (a )
∫
Mostly builds one topic to the next.
Parents reinforce children phobia.

Why do we teach these?
They are useful…
ž  Arithmetic in daily life

ž  Word problems are about thinking

ž  Calculus for
engineering

February 4, 2013 The Art of Number Theory 5

But few actually ever use higher
mathematics
ž  Riemann Surfaces
ž  Category Theory
ž  Homotopy Theory

ž  Lipschitz Functions
ž  Riemann-Roch
Theorem
ž  Algebraic Topology

Some higher math ends up being
very important
ž  Riemannian geometry is the
key to General Relativity
Partial Differential Equations
leads to Computational Fluid
Bernard Riemann
Dynamics, and then flight b. 1826
control
And number
theory, and
theory of
primes, leads
to
cryptography

Lehmer Sieve

But the real point is to teach
logical thinking
ž  We justify math education as
a route to logical thinking
ž  Proofs
ž  Analysis, vice arguments
ž  Brain exercises


And those STEM fields benefit from
mathematical thinking
ž  Mathematics is about analytic
thinking
ž  Proofs
ž  Intuition: What is provable?


Let us think
Deeply of Simple Things
ž  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


Outcome of the Ross Program
ž  A rather large percentage of
graduates became practicing
mathematicians
—  Also some famous physicists
ž  The big advantage of number theory:
—  After some basics, many topics are
independent of one another
—  And the basics are simple


Clock Arithmetic
Example: 10:00 + 3hr = 1:00

10+3≡1 mod 12
10+2≡0 mod 12 10+9=7 mod 12

Clock Arithmetic with a different clock

7
6
1 4:00 mod 7
Add 5 “hours”
5
2
7
4 3 6
1

5
2
4+5≡2 mod 7
4 3
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


But what about multiplication?

X
0
1
2
3
4
5
6
Examples:
0
0
0
0
0
0
0
0

1
0
1
2
3
4
5
6
3X5≡1 mod 15
2
0
2
4
6
1
3
5
2X5≡3 mod 7
3
0
3
6
2
5
1
4

4
0
4
1
5
2
6
3
5X5≡4 mod 7
5
0
5
3
1
6
4
2
6X6≡1 mod 7
6
0
6
5
4
3
2
1


Under multiplication, Up is an
Abeilian Group
ž  Zp, p a prime, = {0,1,2,3,… p–1}
ž  Up, p a prime, = {1,2,3,… p–1}

ž  All group properties inherited from R,
except:
—  Multiplicative inverses
ž  For any a in Up, other than 0, find a–1
such that a X a–1=1 mod p
Z7, Z17, Z213466917–1

Greatest common divisor, also called
the greatest common factor
ž  gcd(6,9)=3
ž  gcd(55,121)=11
ž  gcd(35,49)=7
ž  In general, a common divisor larger than
every other common divisor
ž  a and b are “relatively prime” if gcd(a,b)=1
ž  If p is prime, then gcd(a,p)=1 unless a=np
—  I.e., unless a ≡ 0 mod p


Diophantine Equation
ž  Given a, b nonzero integers, find x, y
such that ax+by=gcd(a,b)
ž  Theorem: There always exist an x and
y, integers, that solve the Diophantine
Equation
ž  Examples
—  6X(-1)+9X(1)=3
—  55X(-2)+121X(1)=11
—  35X(3)+49X(-2)=7


A lovely math theorem
ž  LetUn = { x | gcd(x,n)=1}, under
multiplication mod n
ž  Then Un is an Abelian Group

ž  n a prime is a special case

ž  The proof is constructive!


Some examples
ž  U21 = { 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20}
ž  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

ž  Check it out
ž  How come this works?
ž  And, incidentally, this will be important for
encryption


Euclid’s Algorithm to find gcd’s
1
gcd( 35, 49) = 7
35 49
35 2
14 35
28
2
b.325 BC
7 14

14
This is the gcd And 1, 2, 2 are the partial quotients
0
1
1+ Continued fraction!
2+ 1
2


Another Example
2 p=563

230 563 a=230
460 2
103 230 2 2 4 3 2 3
206 4
24 103
563 1
96 3 = 2+
7 24 230 1
21 2 2+
1
3 7 4+
1
6 3 3+
1
2+
1 3 3
3

0 2, 5/2, 22/9, 71/29, 164/67, 563/230

So, what is the inverse of 230 mod 563
p=563
Answer: −164 = 399
a=230

Diophantine: – 164 · 230 + 67 · 563 = 1
2 2 4 3 2 3

So (– 164) · 230 = 1 mod 563 1
2+
1
I.e., 230 – 1 = 399 mod 563 2+
1
4+
1
Check: 230 · 399 = 91770 = 563 · 163 + 1 3+
1
2+
3

2, 5/2, 22/9, 71/29, 164/67, 563/230

A faster way of computing partial
quotients
p=563
a=230

2 2 4 3 2 3

0 1 2 5 22 71 164 563
1
1 0 1 2 9 29 67 230 2+
1
2+
1
4+
1
3+
1
2+
Inverse is either 164 or –164 3

2, 5/2, 22/9, 71/29, 164/67, 563/230

Fermat’s Theorem
ž  For any a other than 0 mod p,
a p = a mod p
ž  Equivalently
a p–1 ≡ 1 mod p

b. 1601 (or maybe 1607)


Euler’s Theorem
ž  Generalizes Fermat’s Theorem

If gcd(a,n) = 1
a φ(n) ≡ 1 mod n
where φ(n) is the number in the set Un

b. 1707


This would seem to have little to
do with encryption
ž  After all, the simplest encryption is a
letter cipher:
A→N
B→O
C→P
D→Q
…
M→Z

ž  This encryption method, indeed, any
simple cipher, is easily broken


Public key encryption is
completely different concept
ž  I tell you how to encrypt a message to
me Encryption key
Me You

ž  You encrypt, and send the message to
me Encrypted message
Me You

ž  Only I know how to decrypt A variation allows
one to “sign”
Me You messages to
prove authentication
Decrypt


RSA Public Key encryption
ž  Uses number theory!
ž  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

I send you N and E

Me
N and E You


Quick Aside
ž  Finding primes p and q is quick and easy
—  Uses a probabilistic algorithm
—  Works even if p and q involve hundreds of
digits

ž  Also, choosing an E is quick and easy


RSA Public Key encryption
ž  Next,you encrypt the message
ž  You have N and E
—  As does everyone else
Your message is m1, m2, m3, … Converted to numbers

You compute ni ≡ miE mod N for each mi

You send me ni for each i

ni
Me You


Quick aside 2
ž  Computing xE mod N is easy and fast, by
repeatedly squaring


In order to decrypt, I need to use the
algorithm to find inverses
ž  Recall: E satisfies gcd(E, (p–1)·(q–1)) = 1
ž  So I can use the continued fraction
algorithm to find D such that:

ED ≡ 1 mod (p–1)·(q–1)


And now I can decrypt the message

ž  To decrypt: I compute

niD mod N for each ni
ž  Amazingly,

mi ≡ niD mod N for each ni

ž  But if someone else doesn’t know D,
they can’t decrypt


How to factor N

ž  Given N=p·q, find p and q
ž  I.e., factorization
ž  Believed to be “hard”
ž  But no one knows for sure


So, the big outstanding question:
How to factor large numbers that are
a product of two primes?

ž  As of right now, there is no good way

ž  There is also no proof that it can’t be done


But if we had a quantum computer,
there is a reasonably fast way
ž  Based on Shor’s Algorithm
—  A probabilistic algorithm, specifically for a
quantum computer
ž  Uses number theory:
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

4. Compute gcd(ar/2 – 1, N), which be a divisor of N I.e., 1, p, or q

5. If it is 1, then try again (at step 1)


Quantum Computer role in
breaking RSA
ž  Powers of a form a periodic series:

a, a2, a3, a4, a5, …, ar, a, a2, … ar, a, a2, …

ž  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


Prognosis
ž  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


Greater prognosis
ž  Can we get over math phobia?

Yes, I hope so.
Enthusiastic, energetic teachers

Who encourage thinking deeply of simple
things

ž  But maybe not today

The art of number theory v7b[1]

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The art of number theory v7b[1]