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. Append an extra digit using mathematical methods. There are schemes that append two or more digits...error correcting schemes.

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%

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.

Modular Arithmetic 51 (mod 9) = 6 (51=5•9+6)

US Postal Money Order

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

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.

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

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

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:

UPC and EAN

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)

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

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

UPC Scheme Single Digit Errors: Transposition Errors:

IBM Scheme

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

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)

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)

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

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.

IBM Scheme Single Digit Errors: Transposition Errors:

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.

International Standard Book Numbers

ISBN-10……ISBN-13………EAN-13

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

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 + 0 264 = 0 (mod 11)

ISBN-10 Scheme? What if you need a 10?

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

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.

Symmetries of the Pentagon

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

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

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

Symmetries of the Pentagon

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

The Multiplication Table of D5 * 1 2 3 4 5 6 7 8 9

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

 = (0)(1,4)(2,3)(5,6,7,8,9)

German Bundesbank Scheme AY7831976K1

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

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

The Euro!

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)

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)

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)

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

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.