Identification Numbers and Error Detection

1
Identification Numbers and Error Detection

• Meredith Wachs

2
Where do you find identification numbers?

• Checks
• Credit cards
• Drivers licenses
• VIN Numbers
• Zip Codes
• SSN
• ISBN Numbers
• Bar Codes/UPCs (Universal Product Codes)

3
Check Digits

• Used to reduce error
• Grocery items, credit cards, overnight mail,
  magazines, personal checks, travelers checks,
  soft drink cans, and automobiles have check
  digits (COMAP)

4
Congruence mod n

• a is congruent to b, mod n, if n divides
• a-b (Stillwell, Elements of Number Theory)
• For example, 27 is congruent to 6 mod 3 because
  27-621, which is divisible by three.
• Similarly, a number is congruent (mod n) to its
  remainder after being divided by n for example,
  20/36 R2, so 202 (mod 3)

5
Money Order

• COMAP example- The identification number on a
  money order is 63024383845.
• The last digit is the check digit, so the first
  10 digits should be 5 mod 9.
• How do we divide such a large number by 9?
• How do we know this?

6
Calculating Divisibility Rules

• Take a number zabcdefg
• This could also be written as z g100 f101
  e102 d103 c104 b105 a106
• Lets replace each 10n by its congruence mod 9

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Divisibility by 9 continued

• 10011 (mod 9)
• 101101 (mod 9)
• 102111 (mod 9)
• By analogy, we see that every place will have a
  weight of 1, so a(1)b(1)c(1)d(1)e(1)f(1)g(1)
  leaves the same remainder (mod 9) as abcdefg, so
  this is our divisibility rule.

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Divisibility by 11

• Well proceed in the same way
• 1001 mod 11
• 101-1 mod 11
• 102 -1-11 mod 11
• 103 1021011-1-1 mod 11
• In this way, we see that abcdefg(geca)-(fdb)
  (mod 11)
• Ex. 3458679(9653)-(784)4 (mod 11), so is
  not divisible by 11.

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Money Order Revisited

• If the check digit only checks the divisibility
  of the sum, what can be overlooked?

10
Potential Problems with the Check Digit

• Any number can be replaced with its congruence
  mod n (so 0 can be replaced with 9 in this
  example)
• Also, any digits can be easily transposed.

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Other Uses of the Check Digit

• Travelers checks and Euro banknotes use the
  check digit, but the entire number, including the
  check digit, should be divisible by 9.
• If you know the divisor rule, you can easily
  figure out what the check digit should be. For
  example, if a travelers check ID number is
  3487956321 and it has a divisor rule of 9, what
  should the check digit be?

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Answer

• The sum of the digits is 48, and 48654, which
  is divisible by 9, so the check digit should be 6.

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More Sophisticated- UPCs

• A 12-digit number that is found along the bottom
  of a barcode
• A BBBBB CCCCC D
• A is the type of good, B is the
  manufacturers code, C is the product code, D
  is the check digit (COMAP)

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UPC Error Prevention

• Take the code A BCDEF GHIJK L. Going from left
  to right, calculate 3AB3CD3EF3GH3IJ3KL.
  If it isnt divisible by 10, it is an incorrect
  UPC.
• This detects all single-position errors and
  about 89 of other errors (COMAP 326).
• Try this on your own!

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The U.S. Banking System

• To identify a certain bank, it has an
  identification number (the first string on the
  bottom of the check). The string is ABCDEFGH,
  and the check digit is I. I must be the last
  digit of the resulting sum of 7A3B9C7D3E9F7G
  3H

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Here, we can see that 7(1)3(2)9(1)0007(2)3(
4) 48. Since 8 is the check digit, this is a
valid number.
http//blog.wellsfargo.com/GuidedByHistory/images/
Earth_Check_large.jpg
17
Credit Card Numbers

• A more effective algorithm, which is used for
  credit cards, is called Codabar, and is also used
  by libraries, blood banks, photofinishing
  companies, German banks, and the South Dakota
  drivers license department (COMAP 327).
• Codabar allows computers to detect 100 of
  single-position errors and about 98 of other
  common errors (COMAP 328).
• The Codabar algorithm was developed by Hans Peter
  Luhn (1896-1964) and was patented in 1960
  (http//www.merriampark.com/anatomycc.htm).

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The Algorithm

• Assume a 16-digit card number, with the final
  number being the check digit. Add every digit in
  the odd-numbered spaces (going from left to
  right) and multiply by two.
• Then add the number of digits in odd-numbered
  spaces that gt4.
• Finally, add all the digits in the even-numbered
  spaces (except for the check digit).
• The check digit will need to be whatever will
  make the total divisible by 10.
• Ex. What is the check digit for card number
  124785943967210?

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Solution

• Ex. What is the check digit for card number
  124785943967210?
• 1489362033266
• 66369
• 692754971 104
• 1046110, so the check digit is 6.

20
ISBN Numbers

• The 10-digit International Standard Book Number
  detects 100 of single errors and 100 of
  transposition errors (COMAP 328)
• An ISBN of A-BCDE-FGHI-J, with J as the check
  digit, is valid if 10A9B8C7D6E5F4G3H2IJ
  is divisible by 11.

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ISBNs continued

• Example 0-387-95587-9
• 10(0)9(3)8(8)7(7)6(9)5(5)4(5)3(8)2(7)928
  6
• 286 is divisible by 11 because (62)-80, which
  is divisible by 11.

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Why does this work every time?

• COMAP proof
• Let there be an error in the B slot called B.
• Then both calculations (with and without errors)
  must be divisible by 11 in order for B to not be
  detected. Then the difference between the
  calculations must be divisible by 11.

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Proof continued

• So (10A9B8C7D6E5F4G3H2IJ)
  (10A9B8C7D6E5F4G3H2IJ) 9(B-B).
• Since B,B lt9, B-B cannot be 11, and so 9(B-B)
  cannot be divisible by 11 unless BB.

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Another Example

• What is the check digit for the ISBN 0-7167-1910?

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Solution

• 10(0)9(7)8(1)7(6)6(7)5(1)4(9)3(1)2(0)
  199
• The next number divisible by 11 is 19910209.
• To represent 10, ISBN numbers have an X.

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Code 39

• This uses the digits 0-9 and letters A-Z (which
  correspond to 10-35)
• Code 39 is used by the DoD, automotive companies,
  and the health industry (COMAP 329).
• A 15-character string is validated by whether
  15a14b13c.1o is divisible by 36 (with o as
  the check digit).
• The VIN system is a more complicated alphanumeric
  system.

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Bar Codes

• To decode the information in a bar code, a beam
  of light is passed over the bars and spaces via a
  scanning device, such as a handheld wand of a
  fixed-beam device. The dark bars reflect very
  little light back to the scanner, whereas the
  light spaces reflect much light. The differences
  in reflection intensities are detected by the
  scanner and converted to strings of 0s and 1s
  that represent specific numbers and letters.
  (COMAP 334)

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Postnet Codes

• A bar code for a ZIP4 code (1 check digit)
• There are 52 long or short bars, with one guard
  bar on either side and the remaining 50 bars
  grouped into 10 groups of 5, with 2 long bars and
  3 short bars each.
• The check digit makes the sum of all ten numbers
  divisible by 10.

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http//www-math.cudenver.edu/wcherowi/jcorner/bar
codes.html
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UPC Bar Codes

• Each digit is made up of seven modules
• There are guard bars, a center division, and a
  difference between manufacturer and product
  numbers to make reading the codes as accurate as
  possible.
• Bar codes for UPCs have been in use since a pack
  of Wrigley Juicy Fruit gum was scanned on June
  26, 1974 in Marshs Supermarket in Troy, Ohio.
  The first barcodes were made by National Cash
  Register, but smearing ink problems soon made IBM
  the top contender in the market.
  (http//en.wikipedia.org/wiki/Barcode)

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http//en.wikipedia.org/wiki/Universal_Product_Cod
e Is the UPC above valid? (3013363512
60 0 mod 10, so yes, it is valid.)
32
Illinois Drivers License Numbers

• In contrast to Social Security numbers, an
  Illinois drivers license number can help
  reconstruct a persons surname (by sound, not by
  spelling), first and middle initials, date of
  birth, and gender.
• These forms of ID numbers are also used in the
  National Archives, the Library of Congress, and
  in genealogy research. (COMAP 341-2).

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Soundex Coding System for Surnames

• Delete all occurrences of h and w. (so Wachs
  becomes acs)
• Assign number as follows a,e,i,o,u,y 0
  b,f,p,v 1 c,g,j,k,q,s,x,z 2 d,t 3 l 4
  m,n 5 r 6 (so acs 022)
• If two or more letters with the same numeric
  value are adjacent, omit all but the first (so
  were left with ac)

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Soundex continued

• 4. Delete the first character of the original
  name if still present.
• 5. Delete all occurrences of a,e,i,o,u,y. (so
  were left with c)
• 6. Retain only the first three digits
  corresponding to the remaining letters append
  trailing 0s if fewer than three letters remain
  precede the three digits with the first letter of
  the surname (so we have W200)
• Because of the way this is coded, many errors in
  spelling are taken into account.
• taken directly from COMAP 342

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The Middle Digits-First Initial

• InitialCodeInitialCodeInitialCodeInitialCode
• A 0 H 320 O 640 V 860
• B 60 I 400 P 660 W 880
• C 100 J 420 Q 700 X 940
• D 160 K 500 R 720 Y 960
• E 200 L 520 S 780 Z 980
• F 240 M 540 T 800
• G 280 N 620 U 840

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The Middle Digits-Middle Initial

• InitialCodeInitialCodeInitialCodeInitialCode
• A 1 H 8 O 14 V 18
• B 2 I 9 P 15 W
  19
• C 3 J 10 Q 15 X 19
• D 4 K 11 R 16 Y 19
• E 5 L 12 S 17 Z
  19
• F 6 M 13 T 18
• G 7 N 14 U 18

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Calculating the Middle Digits

• Add the code for the first initial to the code
  for the middle initial. For my initials, MJ, you
  have 54010550.
• So far, my number is W200-550.
• All middle digits information taken from
  http//www.highprogrammer.com/alan/numbers/dl_us_s
  hared.html

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The Last Five Digits

• In Illinois, the last five digits retain the
  birth date and sex of the person.
• Each month is assumed to have 31 days (starting
  with January 1 as 001). My birthday, March 2, is
  therefore 231622064. If male, you are done.
  If female, add 600 to this number (so I am 664).
• Put the last two digits of your birth year before
  this number. Thus, the last five digits of my
  Illinois drivers license is 8-8664. My complete
  number is W200-5508-8664. As the website author
  points out, IL not having overflow numbers makes
  it likely to have multiple people with the same
  number printed on their license.
• COMAP 343

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Conclusions

• We have seen several ways of identifying objects
  or people with numbers and the likelihood of
  error and the ways error is caught with each one.
  As COMAP points out on p. 329, Like many
  practices in the real world, historical
  accident and lack of knowledge about existing
  methods seem to be the explanation for having so
  many means of identification. We see this
  especially in the difference between drivers
  license numbers and SSN, which were assigned
  prior to computers and the development of many of
  the systems used (like Soundex).