Fault Tolerance CDA 5140 Spring 06

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Fault ToleranceCDA 5140 Spring 06

• Everyday FT

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Background

• Use of check digits for error detection on
  everyday applications used extensively but most
  people unaware of it
• Examples airline tickets, credit cards, bank
  accounts, library books, grocery products,
  drivers licenses, passports, rental cars, UPS,
  express mail, UPC bar codes, etc.
• Not used for SSNs, telephone numbers some
  serial numbers on currency - why not?

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Universal Product Codes (UPC)

• First used on groceries, now on most products
  some products now have Radio Frequency
  Identification (RFID) which is much more
  complicated
• UPC has 12 digits, a1, a2, a12 each of which
  is from the set 0, 1, , 9 and
• a1 is the product type
• a2 to a6 identifies the manufacturer
• a7 to a11 identifies the product
• a12 is the check digit
• In some products a12 is not printed but is on bar
  code

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UPC continued

• a12 chosen such that
• (a1a2a3a12)(3 1 3 1 3 1 3 1 3 1 3 1)
• 3a1 a2 3a3 a12 0 (mod 10)

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Example Kellogg product
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2nd Kellogg Product
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Nutri Grain Product

• Different manufacturer

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UPC Examples

• Note for all 3 products that first digit is 0
  indicating a household product
• First 2 products from same manufacturer, so next
  5 digits same (38000) but 3rd is different
  manufacturer so different digits (16000)
• First 2 products are different, so different next
  5 digits distinct (66330 and 01302)

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UPC examples

• To determine if first example is a valid code,
  calculate
• 3(0) 1(3) 3(8) 1(0) 3(0) 1(0) 3(6)
  1(6) 3(3) 1(3) 3(0) 1(7) 70 0(mod 10)
• This code is able to detect any single error, for
  example, typed in incorrectly.
• Assume a digit ai is incorrectly entered as ai
  and if this were not detected, then would have

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UPC example continued

• (a1a2a3 ai a12)(3 1 3 1 3 1 3 1 3 1 3 1) 0
  (mod 10)
• (a1a2a3 aia12)(3 1 3 1 3 1 3 1 3 1 3 1)0 (mod
  10)
• Thus
• ((a1a2a3 aia12) - (a1a2a3aia12))(3 1 3 1 3
  1 3 1 3 1 3 1) 0
• which implies either (ai - ai) 0 or 3 (ai -
  ai) 0 (mod 10)
• which is a contradiction, and hence a single
  error must be detected

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Version E UPC

• Used for special products such as round soda
  cans, small items, magazines
• Only 8 digits long
• Uses one of 4 formulae depending on last
  non-check digit
• If a7 is 0, 1, 2 then a8 is such that
• (3 1 3 3 1 3 1 1)(a1a2a3a8) 0 (mod 10)

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Version E UPC contd

• If a7 is 3 then a8 is chosen such that
• (3 1 3 1 1 3 0 1) )(a1a2a3a8) 0 (mod 10)
• If a7 is 4 then a8 is chosen such that
• (3 1 3 1 3 3 0 1)(a1a2a3a8) 0 (mod 10)
• If a7 is 5,6,7,8,9 then a8 is chosen such that
• (3 1 3 1 3 1 3 1)(a1a2a3a8) 0 (mod 10)

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Version E UPC Example

• New Yorker Magazine
• Since a7 is 9, use (3 1 3 1 3 1 3 1)

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Other UPC Versions

• Shipping container version has 2 check digits,
  one comparable to standard, other using modulus
  103
• European countries have 13-digit version, weight
  vector (1 3 1 3 . . . 1 3 1) with 2 digits ahead
  of the manufacturers number to identify the
  country

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Credit Card Methods

• IBM developed more complex system for use with
  credit cards, libraries, blood banks, some
  vehicle agencies, etc.
• Use permutation ? that maps as
• ?(0) 0 ?(1) 2 ?(2) 4 ?(3) 6
• ?(4) 8 ?(5) 1 ?(6) 3 ?(7) 5
• ?(8) 7 ?(9) 9

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Credit Card Method

• For any string of digits (a1a2a3 . . . an-1), n
  even, check digit an is set such that
• ?(a1) a2 ?(a3) a4 ... ?(an-1) an 0
  (mod 10)
• For n odd, apply ? to even numbered positions.
• Consider a VISA ad with card number
• 4417 1234 5678 9122 - is this valid?

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Error Detection for UPC Credit Card Schemes

• Both schemes can detect all single digit errors.
• Common error of transpositions of two digits
  where ab becomes ba is not detected by UPC
  code if a-b 5, and not by IBM code if
  a-b 9
• For UPC code, why is this true?
• So, for the 90 such possible transpositions for
  UPC code, all except 05, 16, 27, 38, 49 and
  reversals are detected, for IBM not 90 or 90

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Increased Detection Ability

• Next to 2-digit transformation, most common
  transformation is jump transformation which
  changes abc into cba, e.g. ...6163 to
  ...6361...
• Neither of UPC or IBM schemes would detect these
• A weight vector of (7 3 9 7 3 9 7 3) is used with
  8-digit number mod 10 by banks many western
  countries for passport to detect such errors,
  others use (7 3 1 7 3 1 )
• Both schemes detect jumps if a-c is not 5

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Increased Detection contd

• Other classes of errors detected depending on the
  3 weights used
• for example aca to bcb, or, aa to bb can
  be detected with the proper choice of weights
• 4-weight schemes increase percentage of errors
  detected but order of weights becomes very
  important
• for example, if weights are 1, 3, 9, 7, then we
  have
• 1a 3c 9a 10a 3c 3c (mod 10) and
• 1b 3c 9b 10b 3c 3c (mod 10)
• So error not detected. What if change weights to
  1, 3, 7, 9?

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Other Modulus Schemes

• All previous schemes were modulus 10.
• Scheme for ISBN (book) numbers is mod 11
  detects all single errors and transposition
  errors.
• 9-digit number and one check digit,
  (a1a2a3a10)with weight vector (10 9 8 7 6 5 4 3
  2 1) gives
• (a1a2a3a10)(10 9 8 7 6 5 4 3 2 1) 0 (mod
  11)
• if a10 is 10 then it is written as X
• Our text has ISBN number 0-471-29342-3 and sums
  to 187 ?

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Other Modulus Schemes

• Various schemes developed to avoid use of X but
  not as well known as ISBN.
• Most common error is single digit but many
  schemes dont cover such as choice is based on
  simplicity, e.g. USPS money orders, FedEx, UPS
  packages simply assign check digits based on mod
  9 or 7

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Other Modulus Schemes

• Airline companies, FedEx, UPS divide number by 7
  take remainder as check digit
• For example, UPS number 601 399 957 divided by 7
  gives a remainder of 4 so on the package is
• 601 399 957 4
• division by 7 not as effective at detecting
  single errors but is fairly effective at
  detecting transpositions

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Alphanumeric Schemes

• Include alphanumeric characters, for example the
  3-out-of-9 code or 39 Code, allows digits 0,
  1, 9 and letters A, B, , Z and characters -
  and . and space, used by DoD, automotive and
  health industries
• Letters assigned numbers 10 through 35, and other
  3 characters, 36,37 and 38 respectively
• Multiply (a1a2a3 an)(n n-1 3 2 1) mod 39 and
  take remainder as check digit, converted to
  alphanumeric if necessary
• Can add ,/,, as 39 through 42 to detect all
  single digit errors all transposition errors

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Error Correcting Approaches

• Some schemes add second check digit to give more
  power
• Example Norwegian citizen registration numbers
  with 9 digits plus 2 check digits calculated as
• (a1a2a3a10)(3 7 6 1 8 9 4 5 2 1) 0 (mod 11)
• (a1a2a3a11)(5 4 3 2 7 6 5 4 3 2 1) 0 (mod
  11)
• Detects all single digit errors and all double
  errors except those where difference between
  correct version incorrect one is of form (0 0 0
  a 0 0 0 0 0 11-a 0) Why?

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Error Correcting

• Numbers such that check digits would be 10 not
  used
• More powerful version detects all double errors
  and corrects all single errors
• The information is 8 digits and check is 2
  digits
• (a1a2a3a10)(1 1 1 1 1 1 1 1 1 1) 0 (mod 11)
• (a1a2a3a10)(1 2 3 4 5 6 7 8 9 10) 0 (mod 11)
• Note that omitting numbers with check digit 10
  still gives over 82 million numbers.

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Error Correcting

• First calculation determines magnitude m of any
  single error, since if there were no errors, sum
  would be 0 modulus 11
• Second calculation, assuming single error, will
  then identify position i of error since if sum s
  is non-zero, have mi s, so determine that
  position i is too large by m (use modulus
  properties to get an even multiple of i)
• Thus, knowing magnitude position, single error
  can be corrected.

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Error Correcting

• Given 8 information digits 73245018 can calculate
  a9 and a10
• (7 3 2 4 5 0 1 8 a9 a10)(1 1 1 1 1 1 1 1 1 1)
• 7 3 2 4 5 1 8 a9 a10 30
  a9 a10
• 8 a9 a10 0 (mod 11)
• (7 3 2 4 5 0 1 8 a9 a10)(1 2 3 4 5 6 7 8 9 10)
• 7 6 6 16 25 7 64 9a9 10a10
• 131 9a9 10a10 10 9a9 10a10 0
  (mod 11)
• 2 equations in 2 unknowns - what are values of a9
  and a10?

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Summary

• Various schemes discussed have many uses when
  used to detect errors
• Techniques based on Number Theory properties and
  used sum of zero but any digit could be used
• Techniques likely to be extended to much more
  powerful but simple-to-calculate schemes