The Luhn algorithm or Luhn formula, also known as the "modulus
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
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A checksum or hash sum is a fixed-size datum computed from an arbitrary block of digital data for the purpose of detecting accidental errors that may have been introduced during its transmission or storage. The integrity of the data can be checked at any later time by recomputing the checksum and...
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A social insurance number is a number issued in Canada to administer various government programs. The SIN was created in 1964 to serve as a client account number in the administration of the Canada Pension Plan and Canada's varied employment insurance programs. In 1967, Revenue Canada started...
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Works are in the public domain if the intellectual property rights have expired, if the intellectual property rights are forfeited, or if they are not covered by intellectual property rights at all...
A cryptographic hash function is a deterministic procedure that takes an arbitrary block of data and returns a fixed-size bit string, the hash value, such that an accidental or intentional change to the data will change the hash value...
; 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
Verhoeff algorithm
The Verhoeff algorithm, a checksum formula for error detection first published in 1969, was developed by Dutch mathematician Jacobus Verhoeff . Like the more widely known Luhn algorithm, it works with strings of decimal digits of any length...
The Luhn mod N algorithm is an extension to the Luhn algorithm that allows it to work with sequences of non-numeric characters...
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:
Counting from the check digit, which is the rightmost, and moving left, double the value of every second digit.
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.
In the mathematical community, the word modulo is often used informally. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
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" that will have a check digit added, making it of the form 7992739871x:
Account number
7
9
9
2
7
3
9
8
7
1
x
Double every other
7
18
9
4
7
6
9
16
7
2
x
Sum of digits
7
9
9
4
7
6
9
7
7
2
=67
The check digit (x) is obtained by computing the sum of digits then computing 9 times that value modulo 10 (so that's: (67 * 9 mod 10)). In layman's terms:
Compute the sum of the digits (67).
Multiply by 9 (603).
Take the last digit (3).
The result is your 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
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 checkdigit.
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
:
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:
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
:
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
A bank card number is the primary account number found on credit cards and bank cards. It has a certain amount of internal structure and shares a common numbering scheme. Credit card numbers are a special case of ISO/IEC 7812 bank card numbers....
