Identification Numbers and Check Digit Schemes: Using AbstractAlgebra in Your High School Mathematics Class

Joseph KirtlandDepartment of MathematicsMarist 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 0.5%

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)

•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)

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

error make to waysof #

detected iserror waysof#dr

•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:

%9890

88dr

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

US Postal Money Order

Single Digit Errors:

Transposition Errors:

%9890

88dr

%090

0dr

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

UPC and EAN

UPC Version AGeneral 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

30+5+33+6+30+0+31+0+30+5+34+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:

%10090

90dr

%8990

80dr

IBM Scheme

Permutations

S10 - permutations of the set {0, 1, 2, …, 9}

- one-to-one & onto mappings

6850721934

9876543210

)8)(7,5)(3,1)(6,9,2,4,0(

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 SchemeTransposition 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:

%10090

90dr

%9890

88dr

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 SchemeGeneral 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

100+98+88+73+68+ 55+ 47+ 32+ 20+ 0 = 0 (mod 11)

0 + 72+64 + 21+48 + 25 + 28 + 6 + 0 + 0 = 0 (mod 11)

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

C

BA

E

B A

CE

D

CDEAB

EDCBA

Symmetries of the Pentagon

Rotations

5

3

D

C

BA

E

C D

BE

A

CBAED

EDCBA

Symmetries of the Pentagon

5

3

A B

C

D

E

C B

A

E

D

A E

D

C

B

A B

C

D

E

A E

D

C

B

Symmetries of the Pentagon

CDEAB

EDCBADEABC

EDCBA

EABCD

EDCBA

ABCDE

EDCBABCDEA

EDCBA

DCBAE

EDCBA

CBAED

EDCBABAEDC

EDCBA

AEDCB

EDCBA

EDCBA

EDCBA

9

876

543

210

Symmetries of the Pentagon

8 * 3 = 5

3 * 8 = 6

NOT COMMUTATIVE!

The Multiplication Table of D5

* 0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 0 6 7 8 9 5

2 2 3 4 0 1 7 8 9 5 6

3 3 4 0 1 2 8 9 5 6 7

4 4 0 1 2 3 9 5 6 7 8

5 5 9 8 7 6 0 4 3 2 1

6 6 5 9 8 7 1 0 4 3 2

7 7 6 5 9 8 2 1 0 4 3

8 8 7 6 5 9 3 2 1 0 4

9 9 8 7 6 5 4 3 2 1 0

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 Z0 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 Code62150334a9a10

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)

16+22+31+45+50+63+73+84+9a9+10a10 = 0 (mod 11)

6 + 4 + 3 + 20 + 0 + 18+ 21 + 32+9a9+10a10 = 0 (mod 11)

104 +9a9+10a10 = 0 (mod 11)

5 +9a9+10a10 = 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 Code16+22+31+48+50+63+73+84+97+102

= 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.