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{{Short description|Simple checksum formula}}
{{redirects|Luhn|people named Luhn|Luhn (surname)}}
The '''Luhn algorithm''' or '''Luhn formula''', also known as the "[[modular arithmetic|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.
The '''Luhn algorithm''' or '''Luhn formula''' (creator: [[IBM]] scientist [[Hans Peter Luhn]]), also known as the "[[modular arithmetic|modulus]] 10" or "mod 10" [[algorithm]], is a simple [[check digit]] formula used to validate a variety of identification numbers. {{efn|It is described in U.S.[[United PatentStates|US]] No.patent 2,950,0482950048A, granted on August 23, {{date|1960-08-23|DMY}}.<ref name="patentUS2950048A">{{cite patent|title=Computer for Verifying Numbers|country=US |number=2950048A |status=patent |titlepubdate=Computer for verifying numbers {{date|pubdate=1960-08-23 |fdateDMY}}|gdate=1954{{date|1960-0108-06 23|DMY}}|invent1=Luhn|inventor1-first=Hans P. Peter|inventor1-lastfdate=Luhn 1954-01-06|inventorlink=Hans Peter Luhn}}</ref>}}
 
The algorithm is in the [[public ___domain]] and is in wide use today. It is specified in [[ISO/IEC 7812-1]].<ref>{{cite tech report |title=Identification cards —{{mdash}} Identification of issuers —{{mdash}} Part 1: Numbering system |number=[[ISO/IEC 7812]]-1:{{date|2017 |DMY}}|institution=[[International Organization for Standardization]], & [[International Electrotechnical Commission]] |date=January{{date|Jan 2017 |DMY}}|type=Standard standard|url=https://www.iso.org/standard/70484.html |chapter=Annex B: Luhn formula for computing modulus-10 “double-add-double” check digits}}</ref> It is not intended to be a [[cryptographic hash function|cryptographically secure hash function]]; it was designed to protect against accidental errors, not malicious attacks. Most [[credit cardscard number]]s 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:
# If the number already containsDrop the check digit, dropfrom thatthe digitnumber to(if formit's thealready "payload"present). TheThis check digit is most oftenleaves the last digitpayload.
# WithStart with the payload digits. Moving from right to left, startdouble every second digit, starting from the rightmostlast digit. MovingIf left,doubling doublea thedigit results in a value of> every9, secondsubtract digit9 from it (includingor thesum rightmostits digitdigits).
# Sum all the valuesresulting ofdigits (including the resultingones digitsthat were not doubled).
# The check digit is calculated by <math>(10 - (s \bmod 10)) \bmod 10</math>, where s is the sum from step 3. This is the smallest number (possibly zero) that must be added to <math>s</math> to make a multiple of 10. Other valid formulas giving the same value are <math>9 - ((s + 9)\bmod 10)</math>, <math>(10 - s)\bmod 10</math>, and <math>10\lceil s/10\rceil - s</math>. Note that the formula <math>(10 - s)\bmod 10</math> will not work in all environments due to differences in how negative numbers are handled by the [[modulo]] operation.
 
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Assume an example of an account number 1789372997 (just the "payload", check digit not yet included):
 
{| class="wikitable" style="text-align:center;border:none;background:transparent;"4353464434047422| style="width:1.5em" | 4283039836977353| style="width:1.5em" | 9
|! style="width:1.5em" | Digits reversed
| style="width:1.5em" | 7
| style="width:1.5em" | 9
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|-
!
| style="background: #FFA; color: #000;" | '''14'''
| 9
| style="background: #FFA; color: #000;" | '''18'''
| 2
| style="background: #FFA; color: #000;" | '''14'''
| 3
| style="background: #FFA; color: #000;" | '''18'''
| 8
| style="background: #FFA; color: #000;" | '''14'''
| 1
|-
! Sum digits
|'''5''' <br> (1+4)
|9 <br> &nbsp;
|9
|'''9''' <br> (1+8)
|2 <br> &nbsp;
|2
|'''5''' <br> (1+4)
|3 <br> &nbsp;
|3
|'''9''' <br> (1+8)
|8 <br> &nbsp;
|8
|'''5''' <br> (1+4)
|1 <br> &nbsp;
|1
|}
 
The sum of the resulting digits is 56.
 
The check digit is equal to <math>(10 - (56 \operatorname{mod}bmod 10))\bmod 10 = 4</math>.
 
This makes the full account number read 17893729974.
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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<ref name="patentUS2950048A" /> 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 ==
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sum := 0
parity := length mod 2
'''for''' i from 1 to (length - 1) '''do'''
'''if''' i mod 2 !== parity '''then'''
sum := sum + cardNumber[i]
'''elseif''' cardNumber[i] > 4 '''then'''
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'''end if'''
'''end for'''
'''return''' cardNumber[length] == ((10 - (sum mod 10)) mod 10)
'''end function'''
 
== Code implementation ==
 
=== [[C Sharp (programming language)|C#]] ===
<syntaxhighlight lang="c#" line="1">
bool IsValidLuhn(in int[] digits)
{
int check_digit = 0;
for (int i = digits.Length - 2; i >= 0; --i)
check_digit += ((i & 1) is 0) switch
{
true => digits[i] > 4 ? digits[i] * 2 - 9 : digits[i] * 2,
false => digits[i]
};
 
return (10 - (check_digit % 10)) % 10 == digits.Last();
}
</syntaxhighlight>
 
=== [[Java (programming language)|Java]] ===
<syntaxhighlight lang="java" line="1">
public static boolean isValidLuhn(String number) {
int n = number.length();
int total = 0;
boolean even = true;
// iterate from right to left, double every 'even' value
for (int i = n - 2; i >= 0; i--) {
int digit = number.charAt(i) - '0';
if (digit < 0 || digit > 9) {
// value may only contain digits
return false;
}
if (even) {
digit <<= 1; // double value
}
even = !even;
total += digit > 9 ? digit - 9 : digit;
}
int checksum = number.charAt(n - 1) - '0';
return (total + checksum) % 10 == 0;
}
</syntaxhighlight>
 
=== [[TypeScript]] ===
<syntaxhighlight lang="typescript" line="1">
function luhnCheck(input: number): boolean {
const number = input.toString();
const digits = number.replace(/\D/g, '').split('').map(Number);
let sum = 0;
let isSecond = false;
for (let i = digits.length - 1; i >= 0; i--) {
let digit = digits[i];
if (isSecond) {
digit *= 2;
if (digit > 9) {
digit -= 9;
}
}
sum += digit;
isSecond = !isSecond;
}
return sum % 10 === 0;
}
</syntaxhighlight>
 
 
=== [[Python (programming language)|Python]] ===
<syntaxhighlight lang="python" line="1">
class LuhnAlgorithm:
"""Class to validate a number using Luhn algorithm.
Args:
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_reversed(self) -> tuple:
""" Returns the last digit and the remaining numbers reversed """
return int(self.input_value[-1]), self.input_value[-2::-1]
def __checksum(self) -> bool:
last_digit, remaining_numbers = self.last_digit_and_remaining_numbers_reversed()
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 == 0
def verify(self) -> bool:
"""Verify a number using Luhn algorithm."""
return self.__checksum()
</syntaxhighlight>
[https://github.com/codeperfectplus/Sanatio/blob/main/sanatio/utils/checksum.py Source Github]
 
== Uses ==
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* [[Payment card number|Credit card numbers]]
* [[International Mobile Equipment Identity|IMEI numbers]]
* [[CUSIP]] numbers for North American financial instruments
* [[National Provider Identifier|National Provider Identifier numbers]] in the United States
* [[Canada|Canadian]] [[social insurance number]]s
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* [[South Africa|South African]] ID numbers
* [[South Africa|South African]] Tax reference numbers
* [[Personal identity number (Sweden)| Swedish]] [[nationalPersonal identificationidentity numbers number]]s
* [[Sweden|Swedish]] Corporate Identity Numbers (OrgNr)
* [[Greece|Greek]] Social Security Numbers (ΑΜΚΑ)
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* [[European Patent Convention|European patent]] application numbers
* Survey codes appearing on [[:File:McDonalds Receipt Luhn Algorithm.png|McDonald's]], [[:File:Taco Bell Receipt Luhn Algorithm.png|Taco Bell]], and [[:File:Tractor Supply Receipt Luhn Algorithm.png|Tractor Supply Co.]] receipts
* [[United States Postal Service]] package tracking numbers use a modified Luhn algorithm<ref>{{citeCite webbook |url=https://postalpro.usps.com/mnt/glusterfs/2023-10/Pub%20199_v28_10102023.pdf |title=Publication 199: Intelligent Mail Package Barcode (IMpb) Implementation Guide for Confirmation Services and Electronic Payment Systems |urldate=https://postalpro.usps.com/pub199{{date|2023-10-10|DMY}} |publisher=[[United States Postal Service]] |edition=28th |___location=[[United States]] |language=en |access-date={{date|2023-11-29|DMY}} November|archive-url=https://web.archive.org/web/20231117004502id_/https://postalpro.usps.com/mnt/glusterfs/2023-10/Pub%20199_v28_10102023.pdf |archive-date={{date|2023-11-17|DMY}} |url-status=live}}</ref>
* Italian VAT numbers ([[VAT identification number#European Union VAT identification numbers|Partita Iva]])<ref>{{Cite web |last=Albanese |first=Ilenia |date={{date|2022-08-10|DMY}} |title=A cosa serve la Partita Iva? Ecco cosa sapere |trans-title=What is a VAT number for? Here's what to know |url=https://www.partitaiva.it/partita-iva-cosa-serve/ |url-status=live |archive-url=https://web.archive.org/web/20240629162018/https://www.partitaiva.it/partita-iva-cosa-serve/ |archive-date={{date|2024-06-29|DMY}} |access-date={{date|2024-06-29|DMY}} |website=Partitaiva.it |language=it}}</ref>
 
==References==
<references/>
 
==Notes==
{{notelist}}
 
==External links==
* [https://rosettacode.org/wiki/Luhn_test_of_credit_card_numbers ImplementationLuhn intest 150of languagescredit oncard thenumbers] on [[Rosetta Code project]]: Luhn algorithm/formula implementation in 160 programming languages {{As of|1=2024|2=07|3=22|lc=y|url=https://rosettacode.org/w/index.php?title=Luhn_test_of_credit_card_numbers&action=history}}
 
{{DEFAULTSORT:Luhn Algorithm}}
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[[Category:1954 introductions]]
[[Category:Articles with example pseudocode]]
[[Category:Management cybernetics]]
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