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{{Short description|Simple checksum formula}}
{{redirects|Luhn|people named Luhn|Luhn (surname)}}
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 [[United States|US]] patent 2950048A, granted on {{date|1960-08-23|DMY}}.<ref name="US2950048A">{{cite patent|title=Computer for Verifying Numbers|country=US|number=2950048A|status=patent|pubdate={{date|1960-08-23|DMY}}|gdate={{date|1960-08-23|DMY}}|invent1=Luhn|inventor1-first=Hans Peter|fdate=1954-01-06|inventorlink=Hans Peter Luhn}}</ref>}} 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]].<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={{date|Jan 2017|DMY}}|type=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 card 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:
# Drop the check digit from the number (if it's already present). This leaves the payload.
# Start with the payload digits. Moving from right to left, double every second digit, starting from the last digit. If doubling a digit results in a value > 9, subtract 9 from it (or sum its digits).
# Sum all the resulting digits (including the ones that 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.

=== Example for computing check digit ===

Assume an example of an account number 1789372997 (just the "payload", check digit not yet included):

{| class="wikitable" style="text-align:center;border:none;"
! style="width:1.5em" | Digits reversed
| style="width:1.5em" | 7
| style="width:1.5em" | 9
| style="width:1.5em" | 9
| style="width:1.5em" | 2
| style="width:1.5em" | 7
| style="width:1.5em" | 3
| style="width:1.5em" | 9
| style="width:1.5em" | 8
| style="width:1.5em" | 7
| style="width:1.5em" | 1
|-
! Multipliers
| 2
| 1
| 2
| 1
| 2
| 1
| 2
| 1
| 2
| 1
|-
!
| =
| =
| =
| =
| =
| =
| =
| =
| =
| =
|-
!
| 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''' (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 <math>(10 - (56 \bmod 10))\bmod 10 = 4</math>.

This makes the full account number read 17893729974.

=== Example for validating check digit ===

# Drop the check digit (last digit) of the number to validate. (e.g. 17893729974 → 1789372997)
# Calculate the check digit (see above)
# Compare your result with the original check digit. If both numbers match, the result is valid. {{nowrap|1=(e.g. (givenCheckDigit = calculatedCheckDigit) ⇔ (isValidCheckDigit)).}}

==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'' (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<ref name="US2950048A" /> 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(cardNumber[1..length])
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'''
sum := sum + 2 * cardNumber[i] - 9
'''else'''
sum := sum + 2 * cardNumber[i]
'''end if'''
'''end for'''
'''return''' cardNumber[length] == ((10 - (sum mod 10)) mod 10)
'''end function'''

== Uses ==
The Luhn algorithm is used in a variety of systems, including:
* [[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
* [[Israeli identity card|Israeli]] ID numbers
* [[South Africa|South African]] ID numbers
* [[South Africa|South African]] Tax reference numbers
* [[Personal identity number (Sweden)| Swedish Personal identity numbers ]]
* [[Sweden|Swedish]] Corporate Identity Numbers (OrgNr)
* [[Greece|Greek]] Social Security Numbers (ΑΜΚΑ)
* [[SIM_card#ICCID|ICCID]] of SIM cards
* [[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>{{Cite book |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 |date={{date|2023-10-10|DMY}} |publisher=[[United States Postal Service]] |edition=28th |location=[[United States]] |language=en |access-date={{date|2023-11-29|DMY}} |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 Luhn test of credit card numbers] on [[Rosetta Code]]: 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}}
[[Category:Modular arithmetic]]
[[Category:Checksum algorithms]]
[[Category:Error detection and correction]]
[[Category:1954 introductions]]
[[Category:Articles with example pseudocode]]

Luhn algorithm - Revision history

imported>JJMC89 bot III: Removing :Category:Eponymous algorithms of mathematics per Wikipedia:Categories for discussion/Log/2025 October 27#Eponyms in mathematics round 2