# Luhn Algorithm

Last Updated :

21 Aug, 2024

Blog Author :

Edited by :

Ashish Kumar Srivastav

Reviewed by :

Dheeraj Vaidya, CFA, FRM

Table Of Contents

## What Is Luhn Algorithm?

Luhn Algorithm, also known as 'modulus 10' or 'mod 10' algorithm, is a mathematical formula for validating identification numbers for credit cards, IMEI, social security numbers, corporate identification numbers, etc. This checksum formula is widely utilized today, especially to facilitate online payments by detecting accidental input errors of identification numbers.

The Luhn algorithm formula was created by German Computer Scientist Hans Peter Luhn in the 1960s, after whom this formula derives its name. He worked at IBM as a researcher and created this validator using modern mathematical algorithms that enabled computers to ascertain the correct input of identification numbers quickly. However, it identifies incorrect number inputs and not malicious attacks.

##### Table of contents

- Luhn algorithm calculator is a tool that helps determine the correct sequence and input of identification numbers for card users, account numbers, corporate identity numbers, etc.
- The global financial electronic funds transfer companies like Visa and Master Card use the checksum formula to facilitate faster online payments and transactions. In addition, it eliminates redirecting the entire purchasing process due to the user's incorrect input of credit or debit card numbers.
- The algorithm generates a check value that verifies the correct input of the identification number and is mostly the same as the last digit of a credit card number.

### Luhn Algorithm Explained

The Luhn algorithm validator assists in checking and distinguishing the valid numbers from the mistyped or incorrect inputs. It is a method widely used by government and corporate entities to eliminate accidental errors by customers or users while using credit cards or identification numbers for online payments or transactions.

For instance, validating** **the Luhn algorithm for a credit card requires detecting incorrect card numbers at the point of sale terminal when a customer makes payment to any sellers online. Day-to-day transactions for individual buyers, businesses, and governments are assisted with this algorithm, eliminating the complexities of redirecting the entire process and saving time.

The algorithm is rather a complex one that determines a check key (also known as a checksum) to validate a particular number in the sequence or complete identification code. Thus, the computer systems are in sync with Luhn's algorithm, which automatically reads the incorrect inputs to make the user cautious. Moreover, it can determine even a single incorrect input from the whole number with the help of arithmetic calculations to check the resultant number with the expected result.

Today, all the leading companies that provide financial services, such as Master Card, Visa, and American Express, use the Luhn algorithm generator to verify the identification numbers and facilitate payments transfer. The checksum generated by the algorithm thus is suitable to eliminate any possibilities of human error that can redirect the entire purchase activity.

Similarly, certain brands that issue gift cards, such as Amazon or H&M, use the Luhn algorithm generator to generate the checksum for identification numbers.

### Formula

The Luhn algorithm calculator generates a checksum or checking key that is a single digit dependent determined by the sum of the sequence of numbers given. This check key is sometimes similar to the sequence's last digit for verification purposes.

The Luhn algorithm formula for verification of identification numbers includes the following steps,

**Step 1 -**The sequence of numbers starting from the first number on the right or the rightmost number is multiplied by two or doubled, and this follows for every second number from the first rightmost number.

**Step 2 -**If the double of a digit is equal to or larger than 10, then the sum of the resultant double digit numbers is taken,

**Step 3 -**Next step is to calculate the total of all the resultants that we have after multiplying by 2,

**Step 4 -**After realizing the sum of all digits, the check key or control digit 'x' equals = x = (10 – (x mod 10) mod 10). Thus, the value of 'x' is the checksum or check key according to Luhn Algorithm to verify a given sequence of numbers. Usually, the 'x' is the last digit of the identification number sequence that the algorithm determines.

### Examples of Luhn Algorithm

Let us now look at some of the calculations using the Luhn's algorithm formula mentioned above,

#### Example #1

Consider the following account number sequence '0974864982x.'

Let us determine the correct value of 'x' using Luhn's algorithm formula,

Sequence | 0 | 9 | 7 | 4 | 8 | 6 | 4 | 9 | 8 | 2 | x |

Step 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | x |

0stays | 18 | 7stays | 8 | 8stays | 12 | 4stays | 18 | 8stays | 4 | xstays | |

Step 2 | 1+8 | 1+2 | 1+8 | ||||||||

9 | 3 | 9 | |||||||||

Step 3 (sum of) | 0 | 9 | 7 | 8 | 8 | 3 | 4 | 9 | 8 | 4 | x |

**Step 3 –**Sum of all the values after multiplying = 0+9+7+8+8+3+4+9+8+4+x =**60**

**Step 4 -**Check digit is = (10 – (60 mod 10)) mod 10 = 5

- Thus, the value of x and the complete account number is '09728649825.'

- One can verify the check number '5' by calculating the sum of the complete account number = 0+9+7+2+8+6+4+9+8+2+5 =
**60**

Thus, if there is an error in putting the identification number and the resultant is not equal to 0 or multiples of 10, the system detects the error and cautions the user.

#### Example #2

The companies and networks issuing credit cards use a mathematical formula to curb data breaches, secure their networks, and ease user experience. Thus, the Luhn algorithm or Modulus 10 is one such tool to assist them in validating the identification numbers.

Credit cards are user-friendly for instant payments and thrive by providing faster resolution and error detection. In addition, banks and card issuing networks use these algorithms to encrypt and decrypt data quickly through software incorporated with Luhn algorithms.

Similarly, Visa – the global electronic funds transferring company, uses the __13th__ character or digit of the identification number, while other major networks use the last digit.

### Limitations

Albeit Luhn's algorithm formula is a useful tool to verify a vast set of identification numbers, this method also has certain limitations, such as,

- The algorithm limits the detection of certain sequences wherein a pair can exchange positions and still have the same checksum. For instance, a sequence of the numbers '02374598' and '82374590' have 0 and 8. However, if 0 is replaced by 8 and 8 is replaced by 0, the resultant checksum remains the same.
- Secondly, the algorithm fails to detect a double error, such as a sequence number 001133 and 001166 will have the same checksum '3' and can fail to detect a wrong input.
- Additionally, if 0 is present at the starting of the sequence or its absence in a sequence will not bring a change in Luhn's checksum. For instance, the number '000786' and only '786' will have the same checksum. However, the incorrect sequence of the numbers input may not be able to determine the error by a user if their checksum is similar.

### Frequently Asked Questions (FAQs)

**1.**

**What is the Luhn algorithm used for?**The Luhn algorithm is useful for accurate identification and verification of credits card, debit cards, social security, or corporate identity numbers. When a card user inputs their card number while purchasing online, Luhn's algorithm assists in cautioning them of accidental and incorrect inputs. Thus, card issuing companies incorporate this algorithm in their networks and operating systems.

**2.**

**How does Luhn algorithm work?**The algorithm calculator determines a check key or checksum to verify an identification number. Thus, the rightmost number is multiplied by two or doubled, which follows every alternate number in the sequence. The next step derives a sum of all these values equal to zero or a multiple of 10. Thus, adding the check number to the sequence must have a sum value of zero or a multiple of 10, according to Luhn's algorithm. Otherwise, the algorithm identifies an incorrect or accidental input.

**3.**

**Does Visa use Luhn algorithm?**Yes, Visa uses the Luhn algorithm to facilitate funds transfers across the globe via credit and debit card transactions. In addition, it helps the company gain user trust and enhances user experience by eliminating the long redirecting process due to accidental input of identification numbers. But this algorithm does not generate or verify CVV, expiry date, or magnetic stripes information.

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