Fault Tolerance
CDA 5140 Spring 06
Everyday FT
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?
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
UPC continued
a12 chosen such that:
(a1a2a3…a12)(3 1 3 1 3 1 3 1 3 1 3 1)
= 3a1 + a2 + 3a3 + … + a12 = 0 (mod 10)
Example: Kellogg product
2nd Kellogg Product
Nutri Grain Product
Different manufacturer
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)
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:
UPC example continued
(a1a2a3 …ai… a12)(3 1 3 1 3 1 3 1 3 1 3 1) = 0 (mod 10)
(a1a2a3 …ai’…a12)(3 1 3 1 3 1 3 1 3 1 3 1)=0 (mod 10)
Thus
((a1a2a3 …ai…a12) - (a1a2a3…ai’…a12))(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
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)(a1a2a3…a8) = 0 (mod 10)
Version E UPC cont’d
• If a7 is {3} then a8 is chosen such that
(3 1 3 1 1 3 0 1) )(a1a2a3…a8) = 0 (mod 10)
If a7 is {4} then a8 is chosen such that
(3 1 3 1 3 3 0 1) (a1a2a3…a8) = 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)(a1a2a3…a8) = 0 (mod 10)
Version E UPC Example
New Yorker Magazine
Since a7 is 9, use (3 1 3 1 3 1 3 1)
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
manufacturer’s number to identify the country
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
(4) = 8 (5) = 1
(6) = 3
(8) = 7 (9) = 9
(3) = 6
(7) = 5
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?
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
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
Increased Detection cont’d
• 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?
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, (a1a2a3…a10)with weight
vector (10 9 8 7 6 5 4 3 2 1) gives:
(a1a2a3…a10)(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 = ?
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 don’t 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
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
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
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:
(a1a2a3…a10)(3 7 6 1 8 9 4 5 2 1) = 0 (mod 11)
(a1a2a3…a11)(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?
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:
(a1a2a3…a10)(1 1 1 1 1 1 1 1 1 1) = 0 (mod 11)
(a1a2a3…a10)(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.
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.
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?
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