# Luhn algorithm

The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm, named after its creator, IBM scientist Hans Peter Luhn, is a simple checksum formula used to validate a variety of identification numbers, such as credit card numbers, IMEI numbers, National Provider Identifier numbers in the United States, Canadian Social Insurance Numbers, Israeli ID Numbers, South African ID Numbers, Swedish National identification numbers, Swedish Corporate Identity Numbers (OrgNr), Greek Social Security Numbers (ΑΜΚΑ), SIM card numbers and survey codes appearing on McDonald's, Taco Bell, and Tractor Supply Co. receipts. It is described in U.S. Patent No. 2,950,048, granted on August 23, 1960.[1]

The algorithm is in the public domain and is in wide use today. It is specified in ISO/IEC 7812-1.[2] It is not intended to be a cryptographically secure hash function; it was designed to protect against accidental errors, not malicious attacks. Most credit cards 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. If the number already contains the check digit, drop that digit to form the "payload." The check digit is most often the last digit.
2. With the payload, start from the rightmost digit. Moving left, double the value of every second digit (including the rightmost digit).
3. Sum the digits of the resulting value in each position.
4. Sum the resulting values from all positions, ${\displaystyle s}$ .
5. The check digit is calculated by ${\displaystyle ((10-s\operatorname {mod} 10)\operatorname {mod} 10)}$ .

### Example for computing check digit

Assume an example of an account number "7992739871" (just the "payload", check digit not yet included):

 Multipliers Sum digits 7 9 9 2 7 3 9 8 7 1 1 2 1 2 1 2 1 2 1 2 = = = = = = = = = = 7 18 9 4 7 6 9 16 7 2 7 9 (1+8) 9 4 7 6 9 7 (1+6) 7 2

The sum of the resulting digits is 67.

The check digit is equal to ${\displaystyle ((10-67\operatorname {mod} 10)\operatorname {mod} 10)=3}$ .

This makes the full account number read 79927398713.

### Example for validating check digit

1. Drop the check digit (last digit) of the number to validate. (e.g. 79927398713 -> 7992739871)
2. Calculate the check digit (see above)
3. Compare your result with the original check digit. If both numbers match, the result is valid. (e.g.${\displaystyle (givenCheckDigit=calculatedCheckDigit)\Leftrightarrow (isValidCheckDigit)}$ ).

## Strengths and weaknesses

The Luhn algorithm will detect any single-digit error, as well as almost all transpositions of adjacent digits. It will not, however, detect transposition of the two-digit sequence 09 to 90 (or vice versa). It will detect most of the possible twin errors (it will not detect 2255, 3366 or 4477).

Other, more complex check-digit algorithms (such as the Verhoeff algorithm and the Damm algorithm) 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 (by converting 1234 to 0001234 for instance) can perform Luhn validation before or after the padding and achieve the same result.

The algorithm appeared in a United States Patent[1] for a hand-held, mechanical device for computing the checksum. Therefore, it was required to be rather simple. 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

```function checkLuhn(string purportedCC) {
int nDigits := length(purportedCC)
int sum := integer(purportedCC[nDigits-1])
int parity := (nDigits-1) modulus 2
for i from 0 to nDigits - 2 {
int digit := integer(purportedCC[i])
if i modulus 2 = parity
digit := digit × 2
if digit > 9
digit := digit - 9
sum := sum + digit
}
return (sum modulus 10) = 0
}
```

## References

1. ^ a b US patent 2950048A, Luhn, Hans P., "Computer for verifying numbers", published 1960-08-23
2. ^ "Annex B: Luhn formula for computing modulus-10 "double-add-double" check digits". Identification cards — Identification of issuers — Part 1: Numbering system (Standard). International Organization for Standardization, International Electrotechnical Commission. January 2017. ISO/IEC 7812-1:2017.