Luhn algorithm

Các tài liệu về thuật toán Luhn 2008

Các tài liệu về thuật toán Luhn
Table of Contents
1. Luhn algorithm .................................................................................................................. 2
1.1. Strengths and weaknesses....................................................................................................... 2
1.2. Informal explanation................................................................................................................... 2
1.3. Example ....................................................................................................................................... 3
1.4. Implementation ........................................................................................................................... 4
2. Credit Card Validation - Check Digits .............................................................................. 6
2.1. Prefix, Length, and Check Digit Criteria.................................................................................. 6
2.2. LUHN Formula (Mod 10) for Validation of Primary Account Number ................................ 7
3. How To Generate *Valid* Credit Card Numbers .............................................................. 9
3.1. Typical credit card anatomy ...................................................................................................... 9
3.2. What is the “Luhn” or “Mod 10″ check? ................................................................................ 10
3.3. Credit card numbers valid or invalid? .................................................................................... 10
3.4. Closing remarks ........................................................................................................................ 12
3.5. Resources and References .................................................................................................... 13
4. Actionscript Credit Card Validation - Luhn Check.........................................................14

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1. Luhn algorithm

The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm,
is a simple checksum formula used to validate a variety of identification numbers, such as credit
card numbers and Canadian Social Insurance Numbers. It was created by IBM scientist Hans
Peter Luhn and described in U.S. Patent 2,950,048 , filed on January 6, 1954, and granted on
August 23, 1960.

The algorithm is in the public domain and is in wide use today. 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 collections of random digits.

1.1. 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). Other, more complex check-digit algorithms (such as the Verhoeff 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 only affect
the result if they cause shift in position, zero-padding the beginning of a string of numbers does
not affect the calculation. Therefore, systems that normalize to a specific number of digits by
converting 1234 to 00001234 (for instance) can perform Luhn validation before or after the
normalization and achieve the same result.

The algorithm appeared in a US Patent 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.

1.2. Informal explanation

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 account number must pass
the following test:

1. Counting from rightmost digit (which is the check digit) and moving left, double the value
of every even-positioned digit. For any digits that thus become 10 or more, take the two
numbers and add them together. For example, 1111 becomes 2121, while 8763
becomes 7733 (from 2×6=12 → 1+2=3 and 2×8=16 → 1+6=7).
2. Add all these digits together. For example, if 1111 becomes 2121, then 2+1+2+1 is 6;
and 8763 becomes 7733, so 7+7+3+3 is 20.
3. If the total ends in 0 (put another way, if the total modulus 10 is congruent to 0), then the
number is valid according to the Luhn formula; else it is not valid. So, 1111 is not valid
(as shown above, it comes out to 6), while 8763 is valid (as shown above, it comes out
to 20).



1.3. Example

Consider the example identification number 446-667-651. The first step is to double every other
digit, counting from the second-to-last digit and moving left, and sum the digits in the result. The
following table shows this step (highlighted rows indicating doubled digits):

Digit Double Substitute Result
1 1 1
5 10 1+0 1
6 6 6
7 14 1+4 5
6 6 6
6 12 1+2 3
6 6 6
4 8 8 8
4 4 4
Total Sum: 40

The sum of 40 is divided by 10; the remainder is 0, so the number is valid.

A lookup table (i.e. calculate Double, Reduce, and Sum of digits only once and for all) can be
used (0123456789 is mapped to 0246813579)

Digit Double Substitute Result
0 0 0 0
1 2 2 2
2 4 4 4
3 6 6 6
4 8 8 8
5 10 1+0 1
6 12 1+2 3
7 14 1+4 5
8 16 1+6 7
9 18 1+8 9



1.4. Implementation

This C# function implements the algorithm described above, returning true if the given array of
digits represents a valid Luhn number, and false otherwise.

bool CheckNumber(int[] digits)
{
int sum = 0;
bool alt = false;
for(int i = digits.Length - 1; i >= 0; i--)
{
if(alt)
{
digits[i] *= 2;
if(digits[i] > 9)
{
digits[i] -= 9;
}
}
sum += digits[i];
alt = !alt;
}
return sum % 10 == 0;
}

The following is an algorithm (in C#) to generate a number that passes the Luhn algorithm. It
fills an array with random digits then computes the sum of those numbers as shown above and
places the difference 10-sum (modulo 10) in the last element of the array.

int[] CreateNumber(int length)
{
Random random = new Random();
int[] digits = new int[length];
// For loop keeps default value of zero for last slot in array
for(int i = 0; i < length - 1; i++)
{
digits[i] = random.Next(10);
}
int sum = 0;
bool alt = true;
for(int i = length - 2; i >= 0; i--)
{
if(alt)
{
int temp = digits[i];
temp *= 2;
if(temp > 9)
{
temp -= 9;
}
sum += temp;
}
else
{
sum += digits[i];
}
alt = !alt;
}
int modulo = sum % 10;



if(modulo > 0)
{
digits[length-1] = 10 - modulo;
}
// No else req'd - keep default value of zero for digits[length-1]
return digits;
}



2. Credit Card Validation - Check Digits

This document outlines procedures and algorithms for Verifying the accuracy and validity of
credit card numbers. Most credit card numbers are encoded with a "Check Digit". A check digit
is a digit added to a number (either at the end or the beginning) that validates the authenticity of
the number. A simple algorithm is applied to the other digits of the number which yields the
check digit. By running the algorithm, and comparing the check digit you get from the algorithm
with the check digit encoded with the credit card number, you can verify that you have correctly
read all of the digits and that they make a valid combination.

Possible uses for this information:

 When a user has keyed in a credit card number (or scanned it) and you want to validate
it before sending it our for debit authorization.
 When issuing cards, say an affinity card, you might want to add a check digit using the
MOD 10 method.

2.1. Prefix, Length, and Check Digit Criteria

Here is a table outlining the major credit cards that you might want to validate.

CARD TYPE Prefix Length Check digit Algorithm
MASTERCARD 51-55 16 mod 10
VISA 4 13, 16 mod 10
34
AMEX 15 mod 10
37
300-305
Diners Club/ Carte Blanche 36 14 mod 10
38
Discover 6011 16 mod 10
2014
enRoute 15 any
2149
JCB 3 16 mod 10
2131
JCB 15 mod 10
1800



2.2. LUHN Formula (Mod 10) for Validation of Primary Account Number

The following steps are required to validate the primary account number:

Step 1: Double the value of alternate digits of the primary account number beginning with the
second digit from the right (the first right--hand digit is the check digit.)

Step 2: Add the individual digits comprising the products obtained in Step 1 to each of the
unaffected digits in the original number.

Step 3: The total obtained in Step 2 must be a number ending in zero (30, 40, 50, etc.) for the
account number to be validated.

For example, to validate the primary account number 49927398716:

Step 1:

49927398716
x2 x2 x2 x2 x2
------------------------------
18 4 6 16 2

Step 2: 4 +(1+8)+ 9 + (4) + 7 + (6) + 9 +(1+6) + 7 + (2) + 6

Step 3: Sum = 70 : Card number is validated

Note: Card is valid because the 70/10 yields no remainder.

The great folks at ICVERIFY are the original source of this data, I only formatted it in HTML.

If you are in the market, I wrote a set of FoxPro modules for Windows/Dos that interface nicely
with ICVERIFY in a multi-user LAN setup. You just set up ICVERIFY on a single station, and all
stations on the LAN can authorize credit cards with a single FOXBASE function call. Of course,
you have to license ICVERIFY by the node, but it is very reasonable. I also wrote a couple of
simple functions to perform pre-authorization, card screening, etc.

Here is a Microsoft Excel worksheet that will validate a number for you (useful for understanding
the algorithm, it is in a .ZIP compressed format)

Horace Vallas made a NeoWebScript (Tcl really) procedure that implements it.
Check it out at https://enterprise.neosoft.com/secureforms/hav/

Because I get at least a letter a week regarding this routine, here are some additional helpful
notes:

Make sure that you:



1. have started with the rightmost digit (including the check digit) (figure odd and even
based upon the rightmost digit being odd, regardless of the length of the Credit Card.)
ALWAYS work right to left.
2. the check digit counts as digit #1 (assuming that the rightmost digit is the check digit)
and is not doubled
3. double every second digit (starting with digit # 2 from the right)
4. remember that when you double a number over 4, (6 for example) you don't add the
result to your total, but rather the sum of the digits of the result (in the above example
6*2=12 so you would add 1+2 to your total (not 12).
5. always include the Visa or M/C/ prefix.



3. How To Generate *Valid* Credit Card Numbers

What do the credit card numbers mean and how are they generated? I need to start with a disclaimer: Do
not use any credit card numbers, except your own, to buy things off internet. It’s wrong and it’s illegal. The
purpose of this post is *not* to create fraudulent workable card numbers. It is to explain the math and the
science behind those numbers that most of us see day in and day out; and hence this post should be
viewed from a purely academic perspective.

3.1. Typical credit card anatomy

Before we understand how credit card numbers are generated, here is a brief explanation of
what a typical credit card number means.

 Out of the 16 numbers on a typical credit card, the set of first 6 digits is known as the
issuer identifier number (read this for details), and the last digit is known as the “check
digit” which is generated in such a way as to satisfy a certain condition (the Luhn or Mod
10 check). “Luhn check” is explained later in this post. The term sounds intimidating, but
it’s really a very simple (and elegant) concept.
 Taking away the 6 identifier digits and 1 check digit leaves us with 9 digits in the middle
that form the “account number”.
 Now, there are 10 possible numbers (from 0 to 9) that can be arranged in these 9
places. This gives rise to 109 combinations, that is, 1 billion possible account numbers
(per issuer identifier).
 With each account number, there is always an unique check digit associated (for a given
issuer identifier and an account number, there cannot be more than one correct check
digit)
 Amex issues credit cards with15 digits. The account numbers in this case are 8 digit
long.



3.2. What is the “Luhn” or “Mod 10″ check?

In 1954, Hans Luhn of IBM proposed an algorithm to be used as a validity criterion for a given
set of numbers. Almost all credit card numbers are generated following this validity
criterion…also called as the Luhn check or the Mod 10 check. It goes without saying that the
Luhn check is also used to verify a given existing card number. If a credit card number does not
satisfy this check, it is not a valid number. For a 16 digit credit card number, the Luhn check can
be described as follows:

1. Starting with the check digit, double the value of every second digit (never double the
check digit). For example, in a 16 digit credit card number, double the 15th, 13th, 11th,
9th…digits (digits in odd places). In all, you will need to double eight digits.
2. If doubling of a number results in a two digit number, add up the digits to get a single
digit number. This will result in eight single digit numbers.
3. Now, replace the digits in the odd places (in the original credit card number) with these
new single digit numbers to get a new 16 digit number.
4. Add up all the digits in this new number. If the final total is perfectly divisible by 10, then
the credit card number is valid (Luhn check is satisfied), else it is invalid.

When credit card numbers are generated, the same steps are followed with one minor change.
First, the issuer identifier and account numbers are assigned (issuer numbers are fixed for a
given financial institution, whereas the account numbers are randomly allocated - I think). Then,
the check digit is assumed to be some variable, say X. After this, the above steps are followed,
and during the last step, X is chosen in such a way that it satisfies the Luhn check.

This part is a bit confusing and takes some time to understand. However, don’t get stuck
here…continue reading through the examples below and you will figure out what this is all
about.

3.3. Credit card numbers valid or invalid?

Have you ever wondered if those numbers on the fake plastic or cardboard credit cards that
come with the “preapproved” offers are real or imaginary? If they are not valid, how do you know
it?…Just apply the Luhn check and all the those fake credit cards will invariably fail.Here is an
example of a VISA credit card (look at the expiry date - 01/09 ..it’s still valid ! )



Note that the credit card number starts with “4″…so it is indeed a VISA issued credit card (VISA
cards start with “4″ and MasterCard/Maestro cards start with “5″). Now, let us apply the Luhn
algorithm to this card. To make it easier on you guys, I have created a schematic of the steps
towards the Luhn check (below) for this card number 4552 7204 1234 5678:

 In this case, when we sum up the total, it comes to 61 which is not perfectly divisible by
10, and hence this credit card number is invalid.
 If such a credit card number is ever generated, the value of the check digit would be
adjusted in such a way as to satisfy the Luhn condition. In this case, the only value of the
check digit, that will create a valid credit card number, is 7. Choosing 7 as the check digit
will bring the total to 60 (which is perfectly divisible by 10) and the Luhn condition will be
satisfied. So the valid credit card number will be 4552 7204 1234 5677.

Let’s try another example, this time with a MasterCard.

Again, performing the Luhn check on this credit card number, we have:



 The total comes to 65 which is not perfectly divisible by 10. Hence this credit card
number is invalid.
 In this case, a valid credit card number will result only if the check digit is 8. This will
bring the total to 70 which is perfectly divisible by 10. So the valid credit card number will
be 5490 1234 5678 9128.

3.4. Closing remarks

If I still have your attention, here are some additional thoughts. In the context of this post, by the
term “valid”, I mean “mathematically valid”. A mathematically valid credit card does not mean a
“working” credit card. The Luhn formula validates only the credit card number; it does not
validate the expiry date and/or card security code (CVV, CVC). Plus, as discussed before, the 9
digit account number will yield 1 billion combinations; so the chances of getting a working credit
card number are very remote. It should also be noted that, this validation is usually employed at
the transaction end; which means that numbers that do not satisfy the Luhn check are not
forwarded to the card issuer and the transaction is terminated. If you have a fake credit card
which satisfies the Luhn check, it will go through at the transaction end, but the card issuer will
most likely catch the mischief. So don’t go about trying to use these numbers to buy stuff.

Just to be clear on this, I don’t expect comments like these (check out the source of this
comment):

hey. im hearing good things about your site! i need some money to jump start my poker career.
Probably about 40-100$ would do. i dont have a credit card to use and it pisses me off because
i know i could beat the majority of the people online. please help

If you intend to post such comments, at least be extremely funny.

So you think you can separate out valid and invalid account numbers now? Here are a couple of
trial numbers for you:

 5491 9469 1544 4923 - Valid or invalid? If invalid, what should have been the correct
check digit to make it valid?
 4539 9920 4349 1562 - Valid or invalid? If invalid, what should have been the correct
check digit to make it valid?

Sudoku fans will quickly figure out multiple valid combinations of the above numbers. If you
don’t want to do the math, here are some ready made valid (”test”) credit card numbers from
Paypal.By the way, the Luhn check is also valid for debit card numbers.I am still in the learning
phase with this topic and trying to further understand how people use (or misuse (?)) such
information. If you have some insight in this matter, please feel free to share it with us.If you



liked what you read above, go ahead and subscribe to this blog to get more updates. It’s easy -
just click on one of the buttons below and get the feed.

3.5. Resources and References

There is a vast amount of literature on the Luhn algorithm and a quick Google search will
enlighten you on how popular this topic is. If you don’t want to read all that, here are links to
some interesting reading.

 Card Identification Features (pdf file) - extremely useful, must read.
 Credit Card Numbers and Credit Card Generator @ Graham King.
 Anatomy of Credit Card Numbers.
 Credit Card Number - Wikipedia.



4. Actionscript Credit Card Validation - Luhn Check
By no means did I write this, if you do a search for Luhn Check in google you will come with a million
results. However, most of them are Java implementations, JavaScript, ASP, JSP, but never ActionScript.
Below is a version I converted from Javascript into Actionscript 2.0. This Luhn Check validates the users
credit card numbers and makes sure A) there are enough for the given card, and B) matches the number
pattern of a given card. This example only checks Visa, MasterCard, American Express, and Diners Club.

BY NO MEANS IS THIS THE ONLY CHECK YOU NEED TO DO TO PROCESS AN
ORDER.

You must use a valid payment gateway that does proper server side credit card validation. The
purpose of this script is to reduce traffic to the server by weeding out the invalid credit card
numbers. If the client side tests are successful the application then has to send the Credit Card
information to the server for proper Credit Card Validation and order processing.

lass com.scottgmorgan.utils.CreditCardValidation {

/**
* checks for datein the format MM/YY or MM/YYYY against the current date
* @param month a number representing the month to be validated against.
* @param year a number representing the year to be validated against.
*/
public function isValidExpDate(month:Number,year:Number):Boolean {
var result:Boolean = true;
var now:Date = new Date();
var nowMonth:Number = now.getMonth() + 1;
var nowYear:Number = now.getFullYear();
if ((nowYear > year) || ((nowYear == year ) && (nowMonth > month))){
result = false;
}

return result;
}

/**
* checks for valid credit card format using the Luhn check and known digit
* about various cards
* @ccType a string indicating the entered credit card name
* @ccNum a string indicating the credit card number being used.
*/
public function
isValidCreditCardNumber(ccType:String,ccNum:String):Boolean {
if (ccNum.length>0){
if (isNaN(ccNum)){
result = false;
}
if (result){
if (!luhnCheck(ccNum) || !validateCCNum(ccType,ccNum)){
result = false;
}
}



}
return result;
}

private function luhnCheck(str:String){
var sum:Number = 0;
var mul:Number = 1;
var strLen:Number = str.length;
for (var i:Number = 0; i < strLen; i++){
var digit:String = str.substring(strLen-i-1,strLen-i);
var tproduct:Number = parseInt(digit ,10)*mul;
if (tproduct >= 10){
sum += (tproduct % 10) + 1;
} else {
sum += tproduct;
}

if (mul == 1){
mul++;
} else {
mul--;
}
}

if ((sum % 10) != 0){
result = false;
}

return result;
}

private function validateCCNum(cardType:String,cardNum:String):Boolean{
var result:Boolean = false;
cardType = cardType.toUpperCase();
var cardLen:Number = cardNum.length;
var firstdig:String = cardNum.substring(0,1);
var seconddig:String = cardNum.substring(1,2);
var first4digs:String = cardNum.substring(0,4);

switch (cardType){
case "VISA":
result = ((cardLen == 16) || (cardLen == 13)) && (firstdig ==
"4");
break;
case "AMEX":
var validNums = "47";
result = (cardLen == 15) && (firstdig == "3") &&
(validNums.indexOf(seconddig)>=0);
break;
case "MASTERCARD":
break;
case "DISCOVER":
result = (cardLen == 16) && (first4digs == "6011");



break;
case "DINERS":
break;
}
return result;
}
}

And here is the sample code that instantiates this bad boy:

var ccValidator = new CreditCardValidation();
var valid = ccValidator.isValidCreditCardNumber('visa', '4111111111111111');

trace('card is valid: ' + valid);

Pretty straightforward. I have hardcoded the values in the method call above but in the real world
this would be puuled from a textfield or a combo box for the card type.

As a side note, if you ever want to get around the validation for Visa simply enter
„4111111111111111′ (1 “4″, and 15 “1‟s”). Unfortunatly, this number will not get you past the
real creditcard validation that occurs on the server or at the payment gateway.


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