Luhn algorithm

From Wikipedia, the free encyclopedia

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.

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[edit] 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.

[edit] 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).

[edit] 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

[edit] Implementation

This PHP implementation returns true for valid numbers and false if it fails the Luhn Algorithm check. This, along with prefix and length check, is useful when verifying credit card numbers.

 function luhn_validate($number)
 {
    $number_array = array ();
    $k=1;
    for($i = strlen($number)-1; $i >= 0; $i--)
    {
    $number_array[$k]=$number[$i];
    ($k % 2) ? ($sum += $number_array[$k]) : ($sum += ($number_array[$k]*2 > 9) ? ($number_array[$k]*2 - 9) : ($number_array[$k]*2));
    $k++;
    }
   return $sum % 10 == 0;
 }

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;
 }

This is a Javascript implementation that takes a string as input.

 function CheckNumber(digits)
 {
   var sum = 0;
   var alt = false;
   var numvar = 0;
   for(var i = digits.length - 1; i >= 0; i--)
   {
     numvar = parseInt(digits.charAt(i));
     if(alt)
     {
       numvar *= 2;
       if(numvar > 9)
       {
         numvar -= 9;
       }
     }
     sum += numvar;
     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;
 }

The following is a Python function that takes a string of digits, and calculates a 'luhn check digit', so that the number (NNN) together with the luhn check digit (L) would pass the test (NNNL):

 # Return the "Luhn transform" of a digit.
 luhnify = lambda digit: sum(divmod(digit*2, 10))

 def luhn_checksum(stringOfDigits):
   """Returns a digit so that the Luhn checksum of the completed sequence is valid.
   """
   digits = [int(c) for c in stringOfDigits]
   oddDigits, evenDigits = digits[-2::-2], digits[-1::-2]
   total = sum(map(luhnify, oddDigits))+sum(evenDigits)
   return 10 - total % 10

[edit] Other implementations

[edit] References