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{{Short description|Simple checksum formula}}

The '''Luhn algorithm''' or '''Luhn formula''', also known as the "[[modular arithmetic|modulus]] 10" or "mod 10" [[algorithm]], is a simple [[checksum]] formula used to validate a variety of identification numbers, such as [[credit card number]]s, [[IMEI|IMEI number]]s, [[National Provider Identifier|National Provider Identifier numbers]] in the US, and [[Canada|Canadian]] [[Social Insurance Number]]s. It was created by [[IBM]] scientist [[Hans Peter Luhn]] and described in [http://www.google.com/patents?id=Y7leAAAAEBAJ U.S. Patent No. 2,950,048], filed on January 6, 1954, and granted on August 23, 1960.

{{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 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=[http[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/isostandard/iso_catalogue/catalogue_tc/catalogue_detail70484.htm?csnumberhtml|chapter=39698Annex ISO/IEC 7812-1B:2006 IdentificationLuhn cardsformula --for Identificationcomputing of issuersmodulus-10 “double-add-double” Partcheck 1: Numbering system]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:

The formula verifies a number against its included [[check digit]], which is usually appended to a partial account number to generate the full account number. This number must pass the following test:

# 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 ===

# From the rightmost digit, which is the check digit, moving left, double the value of every second digit; if the product of this doubling operation is greater than 9 (e.g., 8 × 2 = 16), then sum the digits of the products (e.g., 16: 1 + 6 = 7, 18: 1 + 8 = 9) or alternatively subtract 9 from the product (e.g., 16: 16 - 9 = 7, 18: 18 - 9 = 9).

# Take the sum of all the digits.

# If the total [[modular arithmetic|modulo]] 10 is equal to 0 (if the total ends in zero) then the number is valid according to the Luhn formula; else it is not valid.

Assume an example of an account number "7992739871"1789372997 that(just willthe have a"payload", check digit added, making it of thenot formyet 7992739871xincluded):

{| class="wikitable" style="text-align:center;border:none;background:transparent;"

! style="width:1.5em" | Digits reversed

! Account number

| style="width:1.5em" | 7

| style="width:1.5em" | 9

Line 24 ⟶ 28:

| style="width:1.5em" | 7

| style="width:1.5em" | 1

| style="width:1.5em" | '''x'''

| rowspan="2" style="border:none;background:transparent;"|

|-

! Multipliers

! Double every other

| 72

| 1

| style="background: #FFA;" | '''18'''

| 92

| 1

| style="background: #FFA;" | '''4'''

| 72

| 1

| style="background: #FFA;" | '''6'''

| 2

| 1

| 2

| 1

|-

!

| =

| =

| =

| =

| =

| =

| =

| =

| =

| =

|-

!

| style="background: #FFA; color: #000;" | '''14'''

| 9

| style="background: #FFA; color: #000;" | '''1618'''

| 72

| style="background: #FFA; color: #000;" | '''214'''

| '''x'''3

| style="background: #FFA; color: #000;" | '''18'''

| 8

| style="background: #FFA; color: #000;" | '''14'''

| 1

|-

! Sum digits

|'''5''' <br> (1+4)

|7

|9 <br> &nbsp;

|'''9'''

|'''9''' <br> (1+8)

|9

|2 <br> &nbsp;

|4

|'''5''' <br> (1+4)

|7

|3 <br> &nbsp;

|6

|'''9''' <br> (1+8)

|9

|8 <br> &nbsp;

|'''7'''

|'''5''' <br> (1+4)

|7

|1 <br> &nbsp;

|2

|'''x'''

|}

The sum of all the resulting digits in the third row is '''67+x'''56.

The check digit is equal to <math>(10 - (56 \bmod 10))\bmod 10 = 4</math>.

The check digit (x) is obtained by computing the sum of the non-check digits then computing 9 times that value modulo 10 (in equation form, (67 × 9 mod 10)). In algorithm form:

This makes the full account number read 17893729974.

# Compute the sum of the non-check digits (67).

# Multiply by 9 (603).

# The last digit, 3, is the check digit. Thus, '''x=3'''.

=== Example for validating check digit ===

(Alternative method)

The check digit (x) is obtained by computing the sum of the other digits then subtracting the units digit from 10 (67 => Units digit 7; 10 − 7 = check digit 3). In algorithm form:

# Drop the check digit (last digit) of the number to validate. (e.g. 17893729974 &rarr; 1789372997)

# Compute the sum of the digits (67).

# TakeCalculate the unitscheck digit (7see above).

# Compare your result with the original check digit. If both numbers match, the result is valid. {{nowrap|1=(e.g. (givenCheckDigit = calculatedCheckDigit) &hArr; (isValidCheckDigit)).}}

# Subtract the units digit from 10.

# The result (3) is the check digit. In case the sum of digits ends in 0, 0 is the check digit.

This makes the full account number read 79927398713.

Each of the numbers 79927398710, 79927398711, 79927398712, 79927398713, 79927398714, 79927398715, 79927398716, 79927398717, 79927398718, 79927398719 can be validated as follows.

#Double every second digit, from the rightmost: (1×2) = 2, (8×2) = 16, (3×2) = 6, (2×2) = 4, (9×2) = 18

<!--

THERE HAVE BEEN MULTIPLE ERRONEOUS "CORRECTIONS" TO THIS ALGORITHM! PLEASE READ:

Are you about to delete the "+" from "(1+6)" and "(1+8)"? PLEASE DON'T! The algorithm sums the INDIVIDUAL DIGITS, not numbers. Thus, "(1+6)" is more illustrative of what is being done.

If you leave it as "1 + (2) + 7 + (16) + 9 + (6) + 7 + (4) + 9 + (18) + 4" it sums to 88, not 70. This is just plain wrong.

Another formulation:

[ 4 + (2×9) mod 9 + 9 + (2×1) mod 9 + 7 + (2×3) mod 9 + 9 + (2×8) mod 9 + 7 + (2×1) mod 9 ] mod 10

except that each 'mod 9' produces 9 rather than zero if it divides evenly.

The description I found here was *also* incorrect however. It said you take the result of the sum, and divide it by 10 and then *truncate* it. Truncation is basically the *opposite* of taking the modulus. It happens to work out in the case because the sum is 64, but if the sum were 74, you'd get: 74 / 10 = 7.4, which truncates to 7. I've changed it so the sum is 65, which accordingly to the previous description I read would mean the check digit is still a 6, but in fact it should be a 5. I did my best to rephrase it to something that was mathematically correct and provided my poor attempt at a layman's instructions on how to do the computation. Feel free to correct it, but don't put it back to "truncation". That'd be most bad. I would actually advocate using a different number whose sum isn't a multiple of 9 to avoid this mistake being reintroduced.

-->

#Sum all the ''individual'' digits (digits in parentheses are the products from Step 1): x (the check digit) + (2) + 7 + (1+6) + 9 + (6) + 7 + (4) + 9 + (1+8) + 7 = x + 67.

#If the sum is a multiple of 10, the account number is possibly valid. Note that '''3''' is the only valid digit that produces a sum (70) that is a multiple of 10.

#Thus these account numbers are all invalid except possibly 79927398713 which has the correct check digit.

Alternately, you can use the same checksum creation algorithm, ignoring the checksum already in place as if it had not yet been calculated. Then calculate the checksum and compare this calculated checksum to the original checksum included with the credit card number. If the included checksum matches the calculated checksum, then the number is valid.

==Strengths and weaknesses==

The Luhn algorithm will detect anyall single-digit errorerrors, 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 7most of the 10 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.

Line 101 ⟶ 97:

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.

Prepending a 0 to odd-length numbers makes it possible to process the number from left to right rather than right to left, doubling the odd-place digits.

== Pseudocode implementation ==

The algorithm appeared in a US Patent<ref>[http://www.google.com/patents/US2950048 US Patent 2,950,048 - ''Computer for Verifying Numbers'', Hans P Luhn, August 23, 1960]</ref> for a hand-held, mechanical device for computing the checksum. It was therefore 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.

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.

==See also==

* [[Bank card number]]

'''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==

* [httphttps://rosettacode.org/wiki/Luhn_test_of_credit_card_numbers ImplementationLuhn intest 88of 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}}

* [https://github.com/taylorgibb/algorithms Open Source implementation in PowerShell]

* [http://sites.google.com/site/abapexamples/javascript/luhn-validation Luhn implementations in JavaScript]

* [https://gist.github.com/1287893 Validation of Luhn in PHP]

* [https://github.com/adilek/luhn-implementation Implementation in C]

* Ruby: [https://gist.github.com/1182499 Luhn validation], [https://gist.github.com/1409815 Luhn generation]

* [https://bitbucket.org/multipetros/validation/src/e8f43334e067/Validation/Luhn.cs Luhn validation class in C#]

* [https://github.com/nishan/luhn_java/blob/master/src/org/luhn/Luhn.java Luhn validation implementation in Java]

* [http://www.sqlteam.com/forums/topic.asp?TOPIC_ID=76195 Luhn validation in SQL]

* [https://wiki.openmrs.org/display/docs/Check+Digit+Algorithm Luhn algorithms for non-numeric characters]

{{DEFAULTSORT:Luhn Algorithm}}

[[Category:Modular arithmetic]]

[[Category:Checksum algorithms]]

[[Category:Articles with example Python code]]

[[Category:Error detection and correction]]

[[Category:1954 introductions]]

[[Category:Articles with example pseudocode]]