Luhn algorithm


The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm, is a simple check digit formula used to validate a variety of identification numbers. The purpose is to design a numbering scheme in such a way that when a human is entering a number, a computer can quickly check it for errors.
The algorithm is in the public domain and is in wide use today. It is specified in ISO/IEC 7812-1. It is not intended to be a cryptographically secure hash function; it was designed to protect against accidental errors, not malicious attacks. Most credit card numbers and many government identification numbers use the algorithm as a simple method of distinguishing valid numbers from mistyped or otherwise incorrect numbers.

Description

The check digit is computed as follows:
  1. Drop the check digit from the number. This leaves the payload.
  2. Start with the payload digits and double every second digit when numbered from the left.
  3. Process the payload from right-to-left. If a doubled digit exceeds 9, subtract 9 from the digit.
  4. Sum all the resulting digits.
  5. The check digit is calculated by, where s is the sum from step 3. This is the smallest number that must be added to to make a multiple of 10.
  6. Other valid formulas giving the same value are,, and. Note that the formula will not work in all environments due to differences in how negative numbers are handled by the modulo operation.

Example for computing check digit

Assume an example of an account number 1789372997 :
Digits reversed7992739871
Multipliers2121212121
==========
149182143188141
Sum digits5
9
9
2
5
3
9
8
5
1

The sum of the resulting digits is 56.
The check digit is equal to.
This makes the full account number read 17893729974.

Example for validating check digit

  1. Drop the check digit of the number to validate.
  2. Calculate the check digit
  3. Compare your result with the original check digit. If both numbers match, the result is valid.

Strengths and weaknesses

The Luhn algorithm will detect all single-digit errors, as well as almost all transpositions of adjacent digits. It will not, however, detect transposition of the two-digit sequence 09 to 90. It will detect most of the possible twin errors.
Other, more complex check-digit algorithms can detect more transcription errors. The Luhn mod N algorithm is an extension that supports non-numerical strings.
Because the algorithm operates on the digits in a right-to-left manner and zero digits affect the result only if they cause shift in position, zero-padding the beginning of a string of numbers does not affect the calculation. Therefore, systems that pad to a specific number of digits can perform Luhn validation before or after the padding and achieve the same result.
The algorithm appeared in a United States Patent for a simple, hand-held, mechanical device for computing the checksum. The device took the mod 10 sum by mechanical means. The substitution digits, that is, the results of the double and reduce procedure, were not produced mechanically. Rather, the digits were marked in their permuted order on the body of the machine.

Pseudocode implementation

The following function takes a card number, including the check digit, as an array of integers and outputs true if the check digit is correct, false otherwise.
function isValid
sum := 0
parity := length mod 2
for i from 1 to do
if i mod 2 parity then
sum := sum + cardNumber
elseif cardNumber > 4 then
sum := sum + 2 * cardNumber - 9
else
sum := sum + 2 * cardNumber
end if
end for
return cardNumber
'''end function'''

Uses

The Luhn algorithm is used in a variety of systems, including: