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# The art of number theory v7b[1]

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Feb 5, 2013 Cafe Scientifique Arlington

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### The art of number theory v7b[1]

1. 1. Dr. Bob HummelPotomac Institute for Policy Studies hummel@PotomacInstitute.org ORCON: Ask permission before redistributing Not intended for publication
2. 2. STEM: Science, Technology,Engineering, & Mathematics  STEM includes mathematics  But when you call it STEM, do you think “mathematics”?  Math is at the tail end Are these people learning math? The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 2
3. 3. 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 Why do we even bother to teach math? Don’t calculators and computers obviate math? The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 3
4. 4. What we teach  Arithmetic 2+2= 7X6= 4–6=–2 0+5=5  Word problems Sally has 23 cents. She…  Algebra 5+x=8  Geometry  Graphing, pre-calc  Calculus f (x )dx = f (b) − f (a ) ∫ Mostly builds one topic to the next. Parents reinforce children phobia. The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 4
5. 5. Why do we teach these?They are useful…  Arithmetic in daily life  Word problems are about thinking  Calculus for engineeringFebruary 4, 2013 The Art of Number Theory 5
6. 6. But few actually ever use highermathematics  Riemann Surfaces  Category Theory  Homotopy Theory   Lipschitz Functions   Riemann-Roch Theorem   Algebraic Topology The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 6
7. 7. Some higher math ends up beingvery important  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 The Art of Number Theory February 4, 2013 Dr. Bob Hummel 7
8. 8. But the real point is to teachlogical thinking  We justify math education as a route to logical thinking  Proofs  Analysis, vice arguments  Brain exercises The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 8
9. 9. And those STEM fields benefit frommathematical thinking  Mathematics is about analytic thinking  Proofs  Intuition: What is provable? The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 9
10. 10. Let us thinkDeeply 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 The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 10
11. 11. 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 The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 11
12. 12. Clock Arithmetic Example: 10:00 + 3hr = 1:00 10+3≡1 mod 12 10+2≡0 mod 12 10+9=7 mod 12 The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 12
13. 13. 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 The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 13
14. 14. 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   The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 14
15. 15. 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   The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 15
16. 16. Under multiplication, Up is anAbeilian 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, Z213466917–1February 4, 2013 The Art of Number Theory 16
17. 17. Greatest common divisor, also calledthe 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 pFebruary 4, 2013 The Art of Number Theory 17
18. 18. 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)=7February 4, 2013 The Art of Number Theory 18
19. 19. A lovely math theorem  LetUn = { 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
20. 20. 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
21. 21. 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 2February 4, 2013 The Art of Number Theory 21
22. 22. 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, 563/230February 4, 2013 The Art of Number Theory 22
23. 23. So, what is the inverse of 230 mod 563 p=563 Answer: −164 = 399 a=230Diophantine: – 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 February 4, 2013 The Art of Number Theory 23
24. 24. A faster way of computing partialquotients 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, 563/230February 4, 2013 The Art of Number Theory 24
25. 25. 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)February 4, 2013 The Art of Number Theory 25
26. 26. Euler’s Theorem  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. 1707February 4, 2013 The Art of Number Theory 26
27. 27. This would seem to have little todo with encryption  After all, the simplest encryption is a letter cipher: A→N B→O C→P D→Q … M→Z  This encryption method, indeed, any simple cipher, is easily brokenFebruary 4, 2013 The Art of Number Theory 27
28. 28. Public key encryption iscompletely different concept  I tell you how to encrypt a message to me Encryption key Me You  You encrypt, and send the message to me Encrypted message Me You  Only I know how to decrypt A variation allows one to “sign” Me You messages to prove authentication DecryptFebruary 4, 2013 The Art of Number Theory 28
29. 29. 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 I send you N and E Me N and E YouFebruary 4, 2013 The Art of Number Theory 29
30. 30. 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 easyFebruary 4, 2013 The Art of Number Theory 30
31. 31. RSA Public Key encryption  Next,you encrypt the message  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 YouFebruary 4, 2013 The Art of Number Theory 31
32. 32. Quick aside 2  Computing xE mod N is easy and fast, by repeatedly squaringFebruary 4, 2013 The Art of Number Theory 32
33. 33. In order to decrypt, I need to use thealgorithm to find inverses  Recall: E satisfies gcd(E, (p–1)·(q–1)) = 1  So I can use the continued fraction algorithm to find D such that: ED ≡ 1 mod (p–1)·(q–1)February 4, 2013 The Art of Number Theory 33
34. 34. And now I can decrypt the message  To decrypt: I compute niD mod N for each ni  Amazingly, mi ≡ niD mod N for each ni  But if someone else doesn’t know D, they can’t decryptFebruary 4, 2013 The Art of Number Theory 34
35. 35. How to factor N  Given N=p·q, find p and q  I.e., factorization  Believed to be “hard”  But no one knows for sureFebruary 4, 2013 The Art of Number Theory 35
36. 36. So, the big outstanding question:How to factor large numbers that area product of two primes?  As of right now, there is no good way  There is also no proof that it can’t be doneFebruary 4, 2013 The Art of Number Theory 36
37. 37. 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: 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) The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 37
38. 38. Quantum Computer role inbreaking RSA  Powers of a form a periodic series: a, a2, a3, a4, a5, …, ar, a, a2, … ar, a, a2, …  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 The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 38
39. 39. 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 The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 39
40. 40. Greater prognosis  Can we get over math phobia? Yes, I hope so. Enthusiastic, energetic teachers Who encourage thinking deeply of simple things  But maybe not today The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 40