What is the ISIN Checksum Algorithm?

The ISIN Checksum Algorithm is called the Luhn algorithm with "Modulus 10 Double Add Double" technique.

Based on 2 examples given on wikipedia.com, the ISIN Checksum Algorithm can be described in pseudo code as follows:

algorithm Isin-Check-Digit is
  input: isin, a 11-character ISIN.

  isin_num := ""
  for 0 ≤ i < 11 do
    c := the ith character of isin
    if c is a digit then
      isin_num := append c to the end 
    else if c is a letter then
      p := ordinal position of c in the alphabet (A=1, B=2...)
      v := p + 9
      isin_num := append v to the end 
    end if 
  repeat

  sum := 0
  n: = number digites in isin_num
  for 0 ≤ i < n do
    d := pop out the last digit of isin_num
    if i is even then
      d := d × 2
    end if
    sum := sum + int(v div 10) + v mod 10
  repeat

  return (10 - (sum mod 10)) mod 10

Here is a Python version of the above pseudo code published on stackoverflow.com.

import string
isin = 'US4581401001'

def digit_sum(n):
    return (n // 10) + (n % 10)

alphabet = {letter: value for (value, letter) in
    enumerate(''.join(str(n) for n in range(10)) + string.ascii_uppercase)}

isin_to_digits = ''.join(str(d) for d in (alphabet[v] for v in isin[:-1]))
isin_sum = 0
for (i, c) in enumerate(reversed(isin_to_digits), 1):
    if i % 2 == 1:
        isin_sum += digit_sum(2*int(c))
    else:
        isin_sum += int(c)

checksum_digit = abs(- isin_sum % 10)

Here is a PHP version of the above pseudo code modified from the "isin" PHP library by David Marland on GitHub.

function isin_checksum($input) {
    $characters = str_split($input);
    // convert all characters to numbers (ints)
    foreach ($characters as $i => $char) {
        // cast to int, by using intval at base 36 we also convert letters to numbers
        $characters[$i] = intval($char, 36);
    }

    // pull out the checkDigit
    $checkDigit = array_pop($characters);

    // put the string back together
    $number = implode('', $characters);

    // performs the Luhn algorithm
    // to obtain a check digit

    // first split up the string
    $numbers = str_split($number);

    // calculate the positional value.
    // when there is an even number of digits the second group will be multiplied, so p starts on 0
    // when there is an odd number of digits the first group will be multiplied, so p starts on 1
    $p = count($numbers) % 2;
    // run through each number
    foreach ($numbers as $i => $num) {
        $num = (int) $num;
        // every positional number needs to be multiplied by 2
        if ($p % 2) {
            $num = $num*2;
            // if the result was more than 9
            // add the individual digits
            $num = array_sum(str_split($num));
        }
        $numbers[$i] = $num;
        $p++;
    }

    // get the total value of all the digits
    $sum = array_sum($numbers);

    // get the remainder when dividing by 10
    $mod = $sum % 10;

    // subtract from 10
    $rem = 10 - $mod;

    // mod from 10 to catch if the result was 0
    $digit = $rem % 10;

    return $digit;
}

⇒ Online Resources on CUSIP

⇐ Converting CUSIP to ISIN

⇑ Introduction on CUSIP

⇑⇑ CUSIP - North American Financial Security ID