1. 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

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 Error Type Form Relative Freq.
single digit error
a → b
79.1%
trans. adj. digits
ab → ba
10.2%
jump trans.
abc → cba
0.8%
twin error
aa → bb
0.5%
phonetic error
a0 ↔ 1a
a = 2, , 9
jump twin error
aca → bcb
0.3%

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)

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:
8 = ( ) (mod 9)
= 35 (mod 9)
= 8

8. Detection Rate
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.

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

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

11. US Postal Money Order Single Digit Errors: Transposition Errors:
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:
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) + a (an-1) + an = 0 (mod 10)
n-odd:
a1 + (a2) + a3 + (a4) (an-1) + an = 0 (mod 10)

20. IBM Scheme
Specific Number: (0)+0+(0)+0+(1)+3+(2)+4+(1)+3+(6)+9 = 0 (mod 10) = 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+98+88+73+68+ 55+ 47+ 32+ 20+ 0 = 0 (mod 11) = 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 PentagonD E C E D A B D C A A B C E B C D B D E C A A E 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* 1 2 3 4 5 6 7 8 9

39. Verhoeff Scheme
General Form: a1a2a 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
AY K1

42. German Bundesbank Scheme
General Form: a1a2a a10a11  = (0,1,5,8,9,4,2,7)(3,6) * = Group Operation D5 A D G K L N S U Y Z (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: a1a2a a9a10 a9 , a10 check digits a1 + a2 + a a9 + a10 = 0 (mod 11) a1 + 2a2 + 3a a9 + 10a10 = 0 (mod 11)

46. An Error Correcting Code
a9a 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) a9+10a10 = 0 (mod 11) a9+10a10 = 0 (mod 11) 5 +9a9+10a10 = 0 (mod 11)

47. An Error Correcting Code
→ = 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) = 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,
Gallian, J. A., Error Detection Methods, ACM Computing Surveys, 28(3), 1996,
Gumm, H. P., Encoding of Numbers to Detect Typing Errors, Inter. J. Applied Eng. Educ., 2, 1986,