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The art of number theory v7b[1]
 

The art of number theory v7b[1]

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Feb 5, 2013 Cafe Scientifique Arlington

Feb 5, 2013 Cafe Scientifique Arlington

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    The art of number theory v7b[1] The art of number theory v7b[1] Presentation Transcript

    • Dr. Bob HummelPotomac Institute for Policy Studies hummel@PotomacInstitute.org ORCON: Ask permission before redistributing Not intended for publication
    • STEM: Science, Technology,Engineering, & Mathematicsž  STEM includes mathematicsž  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 TheoryFebruary 4, 2013 Dr. Bob Hummel 2
    • The Phenomenon of Math Phobiaž  Math is cumulativež  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
    • What we teachž  Arithmetic 2+2= 7X6= 4–6=–2 0+5=5ž  Word problems Sally has 23 cents. She…ž  Algebra 5+x=8ž  Geometryž  Graphing, pre-calcž  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
    • Why do we teach these?They are useful…ž  Arithmetic in daily lifež  Word problems are about thinkingž  Calculus for engineeringFebruary 4, 2013 The Art of Number Theory 5
    • But few actually ever use highermathematicsž  Riemann Surfacesž  Category Theoryž  Homotopy Theory ž  Lipschitz Functions ž  Riemann-Roch Theorem ž  Algebraic Topology The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 6
    • Some higher math ends up beingvery importantž  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
    • But the real point is to teachlogical thinkingž  We justify math education as a route to logical thinkingž  Proofsž  Analysis, vice argumentsž  Brain exercises The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 8
    • And those STEM fields benefit frommathematical thinkingž  Mathematics is about analytic thinkingž  Proofsž  Intuition: What is provable? The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 9
    • Let us thinkDeeply of Simple Thingsž  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
    • Outcome of the Ross Programž  A rather large percentage of graduates became practicing mathematicians —  Also some famous physicistsž  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
    • 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
    • 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
    • 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
    • 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
    • Under multiplication, Up is anAbeilian Groupž  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 inversesž  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
    • Greatest common divisor, also calledthe greatest common factorž  gcd(6,9)=3ž  gcd(55,121)=11ž  gcd(35,49)=7ž  In general, a common divisor larger than every other common divisorž  a and b are “relatively prime” if gcd(a,b)=1ž  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
    • Diophantine Equationž  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 Equationž  Examples —  6X(-1)+9X(1)=3 —  55X(-2)+121X(1)=11 —  35X(3)+49X(-2)=7February 4, 2013 The Art of Number Theory 18
    • A lovely math theoremž  LetUn = { x | gcd(x,n)=1}, under multiplication mod nž  Then Un is an Abelian Groupž  n a prime is a special casež  The proof is constructive!February 4, 2013 The Art of Number Theory 19
    • Some examplesž  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, 20ž  Check it outž  How come this works?ž  And, incidentally, this will be important for encryption February 4, 2013 The Art of Number Theory 20
    • 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
    • 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
    • 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
    • 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
    • Fermat’s Theoremž  For any a other than 0 mod p, a p = a mod pž  Equivalently a p–1 ≡ 1 mod p b. 1601 (or maybe 1607)February 4, 2013 The Art of Number Theory 25
    • Euler’s Theoremž  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
    • This would seem to have little todo with encryptionž  After all, the simplest encryption is a letter cipher: A→N B→O C→P D→Q … M→Zž  This encryption method, indeed, any simple cipher, is easily brokenFebruary 4, 2013 The Art of Number Theory 27
    • Public key encryption iscompletely different conceptž  I tell you how to encrypt a message to me Encryption key Me Youž  You encrypt, and send the message to me Encrypted message Me Youž  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
    • RSA Public Key encryptionž  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 YouFebruary 4, 2013 The Art of Number Theory 29
    • Quick Asidež  Finding primes p and q is quick and easy —  Uses a probabilistic algorithm —  Works even if p and q involve hundreds of digitsž  Also, choosing an E is quick and easyFebruary 4, 2013 The Art of Number Theory 30
    • RSA Public Key encryptionž  Next,you encrypt the messagež  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
    • Quick aside 2ž  Computing xE mod N is easy and fast, by repeatedly squaringFebruary 4, 2013 The Art of Number Theory 32
    • In order to decrypt, I need to use thealgorithm to find inversesž  Recall: E satisfies gcd(E, (p–1)·(q–1)) = 1ž  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
    • And now I can decrypt the messagež  To decrypt: I compute niD mod N for each niž  Amazingly, mi ≡ niD mod N for each niž  But if someone else doesn’t know D, they can’t decryptFebruary 4, 2013 The Art of Number Theory 34
    • How to factor Nž  Given N=p·q, find p and qž  I.e., factorizationž  Believed to be “hard”ž  But no one knows for sureFebruary 4, 2013 The Art of Number Theory 35
    • So, the big outstanding question:How to factor large numbers that area product of two primes?ž  As of right now, there is no good wayž  There is also no proof that it can’t be doneFebruary 4, 2013 The Art of Number Theory 36
    • 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) The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 37
    • Quantum Computer role inbreaking RSAž  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 The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 38
    • Prognosisž  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
    • Greater prognosisž  Can we get over math phobia? Yes, I hope so. Enthusiastic, energetic teachers Who encourage thinking deeply of simple thingsž  But maybe not today The Art of Number TheoryFebruary 4, 2013 Dr. Bob Hummel 40