Introduction :- Specifications for credit card numbering have been drawn up by the International Standards Organization (ISO/IEC 7812-1:1993) and the American National Standards Institute (ANSI X4.13). These eminent organizations refuse to make their publications freely available on-line, and so the following information on the format of credit card numbers comes largely from an Internet Engineering Task Force (IETF) draft by Donald E. Eastlake 3rd, “ISO 7812/7816 Numbers and the Domain Name System (DNS)” (draft-eastlake-card-map-08, expires August 2001), available at the time of this writing at http://www.globecom.net/ietf/draft/draft-eastlake-card-map-08.html. I have not linked to this URL, because individual versions of IETF drafts are notoriously ephemeral.

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.

Major Industry Identifier

The first digit of your credit card number is the Major Industry Identifier (MII), which represents the category of entity which issued your credit card. Different MII digits represent the following issuer categories:

MII Digit Value Issuer Category
0 ISO/TC 68 and other industry assignments
1 Airlines
2 Airlines and other industry assignments
3 Travel and entertainment
4 Banking and financial
5 Banking and financial
6 Merchandizing and banking
7 Petroleum
8 Telecommunications and other industry assignments
9 National assignment

 For example, American Express, Diner’s Club, and Carte Blanche are in the travel and entertainment category, VISA, MasterCard, and Discover are in the banking and financial category, and SUN Oil and Exxon are in the petroleum category.

Issuer Identifier

The first 6 digits of your credit card number (including the initial MII digit) form the issuer identifier. This means that the total number of possible issuers is a million (10 raised to the sixth power, or 1,000,000).

Some of the better known issuer identifiers are listed in the following table:

Issuer Identifier Card Number Length
Diner’s Club/Carte Blanche 300xxx-305xxx,
36xxxx, 38xxxx
14
American Express 34xxxx, 37xxxx 15
VISA 4xxxxx 13, 16
MasterCard 51xxxx-55xxxx 16
Discover 6011xx 16

 If the MII digit is 9, then the next three digits of the issuer identifier are the 3-digit country codes defined in ISO 3166, and the remaining final two digits of the issuer identifier can be defined by the national standards body of the specified country in whatever way it wishes.

Account Number

Digits 7 to (n – 1) of your credit card number are your individual account identifier. The maximum length of a credit card number is 19 digits. Since the initial 6 digits of a credit card number are the issuer identifier, and the final digit is the check digit, this means that the maximum length of the account number field is 19 – 7, or 12 digits. Each issuer therefore has a trillion (10 raised to the 12th power, or 1,000,000,000,000) possible account numbers.

If we consider the large number of potential customers and usurious interest rates charged by issuers, there is obviously a lot of money to be made in the credit card industry. In more civilized ages, people believed that usury was a grievous offense contrary to nature or a mortal sin, not an acceptable business practice (Aristotle, Politics 1.10; St. Thomas Aquinas, De Malo 13.4; Dante, Inferno 11.94-111; etc.).

Check Digit

The final digit of your credit card number is a check digit, akin to a checksum. The algorithm used to arrive at the proper check digit is called the Luhn algorithm, after IBM scientist Hans Peter Luhn (1896-1964), who was awarded US Patent 2950048 (“Computer for Verifying Numbers”) for the technique in 1960.

For a card with an even number of digits, double every odd numbered digit and subtract 9 if the product is greater than 9. Add up all the even digits as well as the doubled-odd digits, and the result must be a multiple of 10 or it’s not a valid card. If the card has an odd number of digits, perform the same addition doubling the even numbered digits instead.

The bit about even and odd is a little confusing. The main point is that you don’t want to double the check digit, and this can easily be done by starting with the check digit, going backwards, and doubling every other digit

Examples

These examples are drawn from junk mail I received from credit card issuers in August 2001. Some of this junk mail contained glossy pictures of credit cards, and the sample numbers come directly from two of these pictures.

4408 0412 3456 7890

The first credit card offer showed a picture of a card with the number 4408 0412 3456 7890.

The Major Industry Identifier (MII) is 4 (banking and financial), the issuer identifier is 440804 (a VISA partner), the account number is 123456789, and the check digit is 0.

Let’s apply the Luhn check to 4408 0412 3456 7890. In the following table,

  • The top row is the original number.
  • In the second row, we multiply alternate digits by 2. Don’t multiply the check digit by 2.
  • In the third row, we force all digits to be less than 10, by subtracting 9 where necessary.
  • The bottom row contains the digits to be added together.
4 4 0 8 0 4 1 2 3 4 5 6 7 8 9 0
4 x 2 = 8 4 0 x 2 = 0 8 0 x 2 = 0 4 1 x 2 = 2 2 3 x 2 = 6 4 5 x 2 = 10 6 7 x 2 = 14 8 9 x 2 = 18 0
8 4 0 8 0 4 2 2 6 4 10 – 9 = 1 6 14 – 9 = 5 8 18 – 9 = 9 0
8 4 0 8 0 4 2 2 6 4 1 6 5 8 9 0

If we add all of the digits in the bottom row together, we get 67, which is not a multiple of 10, and therefore we conclude that the number 4408 0412 3456 7890 is an invalid credit card number.

By changing the check digit from 0 to 3, we arrive at the number 4408 0412 3456 7893, which does pass the Luhn check, since the sum of the digits in the bottom row would be 70, which is divisible by 10. 4408 0412 3456 7893 is, on the face of it, a valid credit card number.

4417 1234 5678 9112

The second credit card offer showed a picture of a card with the number 4417 1234 5678 9112.

The Major Industry Identifier (MII) is 4 (banking and financial), the issuer identifier is 441712 (a VISA partner), the account number is 345678911, and the check digit is 2.

Let’s apply the Luhn check to 4417 1234 5678 9112, as we did in the previous example.

4 4 1 7 1 2 3 4 5 6 7 8 9 1 1 2
4 x 2 = 8 4 1 x 2 = 2 7 1 x 2 = 2 2 3 x 2 = 6 4 5 x 2 = 10 6 7 x 2 = 14 8 9 x 2 = 18 1 1 x 2 = 2 2
8 4 2 7 2 2 6 4 10 – 9 = 1 6 14 – 9 = 5 8 18 – 9 = 9 1 2 2
8 4 2 7 2 2 6 4 1 6 5 8 9 1 2 2

 If we add all of the digits in the bottom row together, we get 69, which is not a multiple of 10, and therefore we conclude that the number 4417 1234 5678 9112 is an invalid credit card number.

By changing the check digit from 2 to 3, we arrive at the number 4417 1234 5678 9113, which does pass the Luhn check, since the sum of the digits in the bottom row would be 70, which is divisible by 10. 4417 1234 5678 9113 is, on the face of it, a valid credit card number.

Warning

These two credit card offers contained pictures with numbers which the Luhn check proved to be invalid. A change to their check digits made them ostensibly valid. But if I were you, I wouldn’t try to charge anything with them.

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.

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