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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^[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.

Contents 1 Strengths and weaknesses 2 Informal explanation 3 Mod 10+5 Variant 4 Implementation of standard Mod 10 5 Other implementations 6 References

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 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.

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 together with the undoubled digits from the original number.
3. If the total ends in 0 (put another way, if the total modulo 10 is equal to 0), then the number is valid according to the Luhn formula; else it is not valid.

As an illustration, if the account number is 49927398716, it will be 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 valid.

Mod 10+5 Variant

Some credit cards use the "Mod 10 plus 5" variant to extend the space of valid card numbers.^{[citation needed]} In this variant, if the sum ends in 0 or 5, the number is considered valid.

Implementation of standard Mod 10

Python variant:

def is_mod10(cc):
    dub, tot = 0, 0
    for i in range(len(cc) - 1, -1, -1):
        for c in str((dub + 1) * int(cc[i])):
            tot += int(c)
        dub = (dub + 1) % 2
    return (tot % 10) == 0

Java variant:

  public static boolean isValidCC(String num) {
 
    final int[][] sumTable = {{0,1,2,3,4,5,6,7,8,9},{0,2,4,6,8,1,3,5,7,9}};
    int sum = 0, flip = 0;
 
    for (int i = num.length() - 1; i >= 0; i--, flip++)
      sum += sumTable[flip & 0x1][num.charAt(i) - '0'];
    return sum % 10 == 0;
  }

c variant:

int LuhnMod10 ( char * cardNumber , int size )
{
 	static int table [ 10 ] = { { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , { 0 , 2 , 4 , 6 , 8 , 1 , 3 , 5 , 7 , 9 } } ;

 	for ( int i = size - 1 , odd = 0 , sum = 0 ; i >= 0 ; i -- )
 		if ( isdigit ( cardNumber [ i ] ) )
 			sum += table [ ( odd = 1 - odd ) ] [ cardNumber [i] - '0' ] ;
 	sum %= 10 ;
 	return ( sum ? 10 - sum : 0 ) ; /* return the check digit */
}

這裡有一個表格，概述了主要的信用卡，您可能要驗證。

Here is a table outlining the major credit cards that you might want to validate.

卡型別CARD TYPE	字首Prefix	長度Length	校驗碼演算法Check digit algorithm
萬事達卡 MASTERCARD	51-55 51-55	16 16	10 mod 10
簽證 VISA	4 4	13，16 13, 16	10 mod 10
美國證券交易所 AMEX	34 34 37 37	15 15	10 mod 10
大來卡/ Diners Club/ 全權委託 Carte Blanche	300-305 300-305 36 36 38 38	14 14	10 mod 10
探索 Discover	6011 6011	16 16	10 mod 10
途中 enRoute	2014年 2014 2149 2149	15 15	任何 any
聯合協調 JCB	3 3	16 16	10 mod 10
聯合協調 JCB	2131 2131 1800 1800	15 15	10 mod 10