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Joseph Kirtland Department of Mathematics Marist College

Identification Numbers and Check Digit Schemes: Using Abstract Algebra in Your High School Mathematics Class. Joseph Kirtland Department of Mathematics Marist College. Check Digit Schemes. Goal: To catch errors when identification numbers are transmitted.

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Joseph Kirtland Department of Mathematics Marist College

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  1. Identification Numbers and Check Digit Schemes: Using AbstractAlgebra in Your High School Mathematics Class Joseph Kirtland Department of Mathematics Marist College

  2. Check Digit Schemes • Goal: To catch errors when identification numbers are transmitted. • Append an extra digit using mathematical methods. • There are schemes that append two or more digits...error correcting schemes.

  3. Common Error Patterns

  4. Modular Arithmetic x (mod n ) = r where r is the remainder when x is divided by n (n is a positive integer and 0 ≤ r ≤ n-1). x = y (mod n) if x and y have the same remainder when divided by n.

  5. Modular Arithmetic • 51 (mod 9) = 6 (51=5•9+6) • 213 (mod 10) = 3 (213=21•10+3) • 143 (mod 11) = 0 (143=13•11+0) • 57 = 107 (mod 10) • 3 = 43 (mod 10) • 60 = 0 (mod 10)

  6. US Postal Money Order

  7. US Postal Money Order General Form: a1a2a3a4a5a6a7a8a9a10a11 a11 = (a1+ a2+ a3+ a4+ a5+ a6+ a7+ a8+ a9 + a10) (mod 9) Specific Number: 67021200988 8 = (6 + 7 + 0 + 2 + 1 + 2 + 0 + 0 + 9 + 8) (mod 9) = 35 (mod 9) = 8

  8. Single digit error (a→ b): 10 choices for a and 9 choices for b resulting in 90 possible ways. Transposition error (ab→ ba): 10 choices for a and 9 choices for b resulting in 90 possible ways. Detection Rate

  9. US Postal Money Order a11 = (a1+ a2+ a3+ a4+ a5+ a6+ a7+ a8+ a9 + a10) (mod 9)

  10. US Postal Money Order a11 = (a1+ a2+ a3+ a4+ a5+ a6+ a7+ a8+ a9 + a10) (mod 9) Single Digit Errors:

  11. US Postal Money Order a11 = (a1+ a2+ a3+ a4+ a5+ a6+ a7+ a8+ a9 + a10) (mod 9) Single Digit Errors: Transposition Errors:

  12. UPC and EAN

  13. UPC Version A General Form: a1-a2a3a4a5a6-a7a8a9a10a11-a12 a1 - number system char. // a2a3a4a5a6 - company // a7a8a9a10a11 - product // a12 - check digit 3a1+a2+3a3+a4+3a5+a6+3a7+a8+3a9+a10+3a11+a12 = 0 (mod 10) Specific Number: 0-53600-10054-0 30+5+33+6+30+0+31+0+30+5+34+0 = 0 (mod 10) 40 = 0 (mod 10)

  14. UPC Scheme – Single Digit Errors …a… → …b… c + 3a = 0 (mod 10) & c + 3b = 0 (mod 10) (c + 3a) – (c + 3b) = 0 (mod 10) 3a – 3b = 0 (mod 10) 3(a – b) = 0 (mod 10) a – b = 0 (mod 10) a = b

  15. UPC Scheme – Transposition Errors …ab… → …ba… c +3a+b = 0 (mod 10) & c+3b+a = 0 (mod 10) (c + 3a + b) – (c + 3b + a) = 0 (mod 10) 3a + b – 3b – a = 0 (mod 10) 2a – 2b = 0 (mod 10) 2(a – b) = 0 (mod 10) Undetected when |a – b| = 5

  16. UPC Scheme Single Digit Errors: Transposition Errors:

  17. IBM Scheme

  18. Permutations S10 - permutations of the set {0, 1, 2, …, 9} - one-to-one & onto mappings

  19. IBM Scheme General Form: a1a2a3 . . . an-1an  = (0)(1, 2, 4, 8, 7, 5)(3,6)(9) n-even: (a1) + a2 + (a3) + a4 + . . . + (an-1) + an = 0 (mod 10) n-odd: a1 + (a2) + a3 + (a4) + . . . + (an-1) + an = 0 (mod 10)

  20. IBM Scheme Specific Number: 00001324136 9 (0)+0+(0)+0+(1)+3+(2)+4+(1)+3+(6)+9 = 0 (mod 10) 0 + 0 + 0 + 0 + 2 + 3 + 4 + 4 + 2 + 3 + 3 + 9 = 0 (mod 10) 30 = 0 (mod 10)

  21. IBM Scheme – Single Digit Errors …a… → …b… c + σ(a) = 0 (mod 10) & c + σ(b) = 0 (mod 10) (c + σ(a)) – (c + σ(b)) = 0 (mod 10) σ(a) – σ(b) = 0 (mod 10) σ(a) – σ(b) = 0 σ(a) = σ(b) a = b

  22. IBM Scheme Transposition Errors …ab… → …ba… c+σ(a)+b = 0(mod 10) & c+σ(b)+a = 0 (mod 10) (c + σ(a) + b) – (c + σ(b) + a) = 0 (mod 10) σ(a) – σ(b) + b – a = 0 (mod 10) σ(a) – a = σ(b) – b (mod 10) σ designed so this will not occur unless a = 0 and b = 9 or a = 9 and b = 0.

  23. IBM Scheme Single Digit Errors: Transposition Errors:

  24. Theorem (Gumm, 1985) Suppose an error detecting scheme with an even modulus detects all single digit errors. Then for every i and j there is a transposition error involving positions i and j that cannot be detected.

  25. International Standard Book Numbers

  26. ISBN-10……ISBN-13………EAN-13

  27. ISBN-10 Scheme General Form: a1a2a3a4a5a6a7a8a9a10 a1... – group/country number (0,1=English, 3=German, 9978=Ecuador) ai…aj – publisher number aj+1…a9 – serial number a10 – check digit

  28. ISBN-10 Scheme 10a1+9a2+8a3+7a4+6a5+5a6+4a7+3a8+2a9+a10 = 0 (mod 11) Specific Number: 0-88385-720-0 100+98+88+73+68+ 55+ 47+ 32+ 20+ 0 = 0 (mod 11) 0 + 72+64 + 21+48 + 25 + 28 + 6 + 0 + 0 = 0 (mod 11) 264 = 0 (mod 11)

  29. ISBN-10 Scheme? • What if you need a 10?

  30. ISBN-10 Scheme? • What if you need a 10? • X represents 10.

  31. ISBN-10 Scheme? • What if you need a 10? • X represents 10. • Does catch all single digit and transposition of adjacent digit errors, but introduces a new character.

  32. Symmetries of the Pentagon

  33. Symmetries of the Pentagon Reflections D D E E C C B A B A

  34. Symmetries of the Pentagon Rotations D A E E C B B A C D

  35. Symmetries of the Pentagon C D E B A E D D C A A E B C B C D B D E C A E A B

  36. Symmetries of the Pentagon

  37. Symmetries of the Pentagon 8 * 3 = 5 3 * 8 = 6 NOT COMMUTATIVE!

  38. The Multiplication Table of D5

  39. Verhoeff Scheme General Form: a1a2a3 . . . an-1an  = (0)(1,4)(2,3)(5,6,7,8,9) * = Group Operation D5 n-1(a1)*n-2(a2)*n-3(a3)* . . . *(an-1)*an = 0 (a)*b≠ (b)*a - antisymmetric

  40.  = (0)(1,4)(2,3)(5,6,7,8,9)

  41. German Bundesbank Scheme AY7831976K1

  42. German Bundesbank Scheme General Form: a1a2a3 . . . a10a11  = (0,1,5,8,9,4,2,7)(3,6) * = Group Operation D5 A D G K L N S U Y Z 0 1 2 3 4 5 6 7 8 9 (a1)*2(a2)*3(a3)* . . . *10(a10)*a11 = 0

  43. German Bundesbank Scheme This scheme has one major problem…………………………………………………what is it?

  44. The Euro!

  45. An Error Correcting Scheme General Form: a1a2a3 . . . a9a10 a9 , a10 check digits a1 + a2 + a3 + . . . + a9 + a10 = 0 (mod 11) a1 + 2a2 + 3a3 + . . . + 9a9 + 10a10 = 0 (mod 11)

  46. An Error Correcting Code 62150334a9a10 6+2+1+5+0+3+3+4+a9+a10 = 0 (mod 11) 24 +a9+a10 = 0 (mod 11) 2 +a9+a10 = 0 (mod 11) 16+22+31+45+50+63+73+84+9a9+10a10 = 0 (mod 11) 6 + 4 + 3 + 20 + 0 + 18+ 21 + 32+9a9+10a10 = 0 (mod 11) 104 +9a9+10a10 = 0 (mod 11) 5 +9a9+10a10 = 0 (mod 11)

  47. An Error Correcting Code 6215033472 → 6218033472 6+2+1+8+0+3+3+4+7+2 = 0 (mod 11) 36 = 0 (mod 11) 3 = 0 (mod 11)

  48. An Error Correcting Code 16+22+31+48+50+63+73+84+97+102 = 3i (mod 11) 6+4+3+32+0+18+21+32+63+20 = 3i (mod 11) 199 = 3i (mod 11) 1 = 3i (mod 11) i = 4

  49. References • Gallian, J.A., The Mathematics of Identification Numbers, College Math Journal, 22(3), 1991, 194-202. • Gallian, J. A., Error Detection Methods, ACM Computing Surveys, 28(3), 1996, 504-517. • Gumm, H. P., Encoding of Numbers to Detect Typing Errors, Inter. J. Applied Eng. Educ., 2, 1986, 61-65.

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