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 check digit formula used to validate a variety of identification numbers. It is described in US patent 2950048A, granted on .[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.
The check digit is computed as follows:
(10-(s\bmod10))
s
9-((s+9)\bmod10)
(10-s)\bmod10
10\lceils/10\rceil-s
(10-s)\bmod10
Assume an example of an account number 1789372997 (just the "payload", check digit not yet included):
| 4283039836977353 | 9 | 7 | 9 | 9 | 2 | 7 | 3 | 9 | 8 | 7 | 1 | |
| Multipliers | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| = | = | = | = | = | = | = | = | = | = | |||
| 14 | 9 | 18 | 2 | 14 | 3 | 18 | 8 | 14 | 1 | |||
| Sum digits | 5 (1+4) | 9 | 9 (1+8) | 2 | 5 (1+4) | 3 | 9 (1+8) | 8 | 5 (1+4) | 1 |
The sum of the resulting digits is 56.
The check digit is equal to
(10-(56\operatorname{mod}10))=4
This makes the full account number read 17893729974.
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 (or vice versa). It will detect most of the possible twin errors (it will not detect 22 ↔ 55, 33 ↔ 66 or 44 ↔ 77).
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 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.
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(cardNumber[1..length]) sum := 0 parity := length mod 2 for i from 1 to length do if i mod 2 != parity then sum := sum + cardNumber[i] elseif cardNumber[i] > 4 then sum := sum + 2 * cardNumber[i] - 9 else sum := sum + 2 * cardNumber[i] end if end for return cardNumber[length]
class LuhnAlgorithm(BaseChecksumAlgorithm): """ Class to validate a number using Luhn algorithm.
Arguments: input_value (str): The input value to validate.
Returns: bool: True if the number is valid, False otherwise. """ def __init__(self, input_value: str) -> None: self.input_value = input_value.replace(' ', )
def last_digit_and_remaining_numbers(self) -> tuple: """Returns the last digit and the remaining numbers""" return int(self.input_value[-1]), self.input_value[:-1]
def __checksum(self) -> int: last_digit, remaining_numbers = self.last_digit_and_remaining_numbers nums = [int(num) if idx % 2 != 0 else int(num) * 2 if int(num) * 2 <= 9 else int(num) * 2 % 10 + int(num) * 2 // 10 for idx, num in enumerate(reversed(remaining_numbers))]
return (sum(nums) + last_digit) % 10
def verify(self) -> bool: """Verify a number using Luhn algorithm""" return self.__checksum
The Luhn algorithm is used in a variety of systems, including: