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, IMEI numbers, National Provider Identifier numbers in US and Canadian Social Insurance Numbers. It was created by IBM scientist Hans Peter Luhn and described in U.S. Patent No. 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 specified in ISO/IEC 7812-1. 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.

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). It will detect 7 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) 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 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.

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 the check digit, which is the rightmost, and moving left, double the value of every second digit.
2. Sum the digits of the products (e.g., 10 = 1 + 0 = 1, 14 = 1 + 4 = 5) together with the undoubled digits from the original number.
3. If the total 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 "4992739871" that will have a check digit added, making it of the form 4992739871x:

Account number	4	9	9	2	7	3	9	8	7	1	x
Double every other	4	18	9	4	7	6	9	16	7	2	x
Sum of digits	4	9	9	4	7	6	9	7	7	2	=64

The check digit (x) is obtained by computing the sum of digits modulo 10 and then computing 10 less that value modulo 10. In layman's terms:

1. Compute the sum of the digits.
2. Divide by 10, and hold on to the remainder.
3. Compute 10 minus the remainder from the previous step.
4. The result is your check digit. If the answer is "10", you use 0.

This, makes the full account number read 49927398716.

The account number 49927398716 in turn is validated as follows:

1. Double every second digit, from the rightmost: (1×2) = 2, (8×2) = 16, (3×2) = 6, (2×2) = 4, (9×2) = 18
2. Sum all the individual digits (digits in parentheses are the products from Step 1): 6 + (2) + 7 + (1+6) + 9 + (6) + 7 + (4) + 9 + (1+8) + 4 = 70
3. Take the sum modulo 10: 70 mod 10 = 0; the account number is probably valid.

Implementation of standard Mod 10

Verification of the check digit

In Python:

def is_luhn_valid(cc):
    num = map(int, str(cc))
    return sum(num[::-2] + [sum(divmod(d * 2, 10)) for d in num[-2::-2]]) % 10 == 0

Calculation of the check digit

The above implementations check the validity of an input with a check digit. Calculating the check digit requires only a slight adaptation of the algorithm—namely:

1. Switch the odd / even multiplication.
2. If the (sum mod 10) == 0, then the check digit is 0
3. Else, the check digit = 10 - (sum mod 10)

In Python:

def calculate_luhn(cc):
    num = map(int, str(cc))
    check_digit = 10 - sum(num[-2::-2] + [sum(divmod(d * 2, 10)) for d in num[::-2]]) % 10
    return 0 if check_digit == 10 else check_digit

Other implementations

• Luhn implementations in Javascript
• Luhn generation and validation in php
• Validation of Luhn in PHP
• Implementation of the above php class regarding alphanumerics
• Luhn implementation in Ruby

See also

• Bank card number
• Luhn checking bank card numbers

References

• U.S. patent 2,950,048, Computer for Verifying Numbers, Hans P. Luhn, August 23, 1960.