## Structure of Adhaar Number

The Aadhaar number typically consists of 12 digits. The first 11 digits are randomly generated, while the last digit is a checksum generated using the Verhoeff algorithm, ensuring accuracy in data entry and validation.

## Aadhaar Number Validator

The

**Verhoeff algorithm**is a checksum formula used to detect errors in data entry or transmission. It adds a digit to a number (typically at the end) to create a checksum, which is then used to validate the number. This algorithm is particularly effective in detecting single-digit errors, transpositions, and most other errors that might occur during data input or transmission.Let's say we have an Aadhaar number: 1234 5678 9012.

To calculate the checksum using the Verhoeff algorithm:

1. Reverse the Aadhaar number: 2109 8765 4321.

2. Create a multiplication table using a pre-defined set of permutations and then multiply corresponding digits in the Aadhaar number with digits in the first row of the table, summing up the results.

3. Repeat step 2 until all digits are multiplied and summed.

4. The checksum is the digit in the table corresponding to the sum obtained in step 3.

For example, using a pre-defined Verhoeff multiplication table:

0 1 2 3 4 5 6 7 8 9

---------------------

0|0 1 2 3 4 5 6 7 8 9

1|1 2 3 4 0 6 7 8 9 5

2|2 3 4 0 1 7 8 9 5 6

3|3 4 0 1 2 8 9 5 6 7

4|4 0 1 2 3 9 5 6 7 8

5|5 9 8 7 6 0 4 3 2 1

6|6 5 9 8 7 1 0 4 3 2

7|7 6 5 9 8 2 1 0 4 3

8|8 7 6 5 9 3 2 1 0 4

9|9 8 7 6 5 4 3 2 1 0

Let's calculate the checksum for Aadhaar number 1234 5678 9012:

1. Reverse the Aadhaar number: 2109 8765 4321.

2. Multiply corresponding digits and sum them:

(2*0) + (1*4) + (0*3) + (9*2) + (8*1) + (7*0) + (6*9) + (5*8) + (4*7) + (3*6) + (2*5) + (1*4) = 236

3. The checksum digit is in the table at row 2, column 3, so the checksum is 3.

So, the complete Aadhaar number with the Verhoeff checksum is 1234 5678 9012 3.

**Another example**

Let's take the Aadhaar number 9876 5432 1098 as an example.

1. Reverse the Aadhaar number:

Reverse the digits to get 8901 2345 6789.

2. Create a multiplication table:

This table is predefined and used for multiplication.

3. Multiply and Sum:

Multiply each digit of the reversed Aadhaar number with the corresponding digit from the first row of the multiplication table, then sum up all these products.

(8*0) + (9*1) + (0*2) + (1*3) + (2*4) + (3*5) + (4*6) + (5*7) + (6*8) + (7*9) + (8*0) + (9*1) = 240

4. Locate the checksum digit: The sum obtained corresponds to a specific digit in the table. This digit is the checksum.

In the table, the digit at row 2, column 4 is 4.

5. Append the checksum to the Aadhaar number: Add this checksum digit to the end of the Aadhaar number.

So, the complete Aadhaar number with the Verhoeff checksum is 9876 5432 1098 4.

The checksum generated using the

**Verhoeff algorithm**is unique for a given input. This uniqueness ensures that the checksum can reliably detect errors in the Aadhaar number during data entry or transmission. If there's even a single-digit error in the Aadhaar number, the resulting checksum will almost always be different, allowing for easy error detection.### Verhoeff Algorithm in more details and its use and importance

The Verhoeff algorithm is a checksum formula designed to detect errors in data entry or transmission, particularly in numeric data. Here's a more detailed explanation of the algorithm and its importance:

**How Verhoeff Algorithm Works:**

**1. Permutation Table:**The algorithm uses a predefined permutation table, typically a 10x10 matrix, with unique permutations of the digits 0 through 9.

**2. Data Reversal:**The input data (such as an Aadhaar number) is reversed to create a reversed sequence of digits.

**3. Multiplication and Summation:**Each digit of the reversed data is multiplied with a corresponding digit from the first row of the permutation table. The products are then summed up.

**4. Checksum Determination:**The sum obtained in step 3 corresponds to a specific digit in the permutation table. This digit is the checksum.

**5. Checksum Appending:**The checksum digit is appended to the end of the original data.

### Importance and Use:

**1. Error Detection:**The primary purpose of the Verhoeff algorithm is error detection. It can detect a wide range of errors, including single-digit errors, transpositions of adjacent digits, and most other errors that might occur during data input or transmission.

**2. Robustness:**The algorithm is robust and reliable. It provides a high level of assurance that errors will be detected, even in large datasets or complex numeric data.

**3. Data Integrity:**Verhoeff checksums help ensure the integrity of numeric data, particularly in critical systems where accuracy is paramount, such as Aadhaar systems where accurate identification is essential for various services.

**4. Efficiency:**Despite its effectiveness, the Verhoeff algorithm is computationally efficient and easy to implement. It does not require complex calculations or extensive computational resources.

**5. Wide Adoption:**The Verhoeff algorithm has been widely adopted in various applications requiring error detection in numeric data, including Aadhaar numbers, credit card numbers, and other identification or verification systems.

Overall, the Verhoeff algorithm plays a crucial role in ensuring the accuracy and integrity of numeric data, making it a valuable tool in various domains where data accuracy is critical.

### Uses of Verhoeff Algorithm

The Verhoeff algorithm is used in various other contexts beyond Aadhaar. Some of the common applications include:

**1. Credit Card Numbers:**Many credit card numbers contain a checksum digit generated using algorithms like Verhoeff to detect errors in the card number during transactions.

**2. IMEI Numbers:**International Mobile Equipment Identity (IMEI) numbers, which uniquely identify mobile devices, may include a checksum digit generated using the Verhoeff algorithm for error detection.

**3. ISBN Numbers:**International Standard Book Numbers (ISBN) assigned to books often incorporate a checksum digit calculated using checksum algorithms like Verhoeff to ensure the accuracy of the ISBN.

**4. VIN Numbers:**Vehicle Identification Numbers (VIN) on automobiles may include a checksum digit generated using the Verhoeff algorithm to detect errors in the VIN during manufacturing, registration, or maintenance processes.

**5. National Identification Numbers**: Similar to Aadhaar numbers, national identification numbers in other countries may utilize checksum digits generated using algorithms like Verhoeff to ensure the accuracy and integrity of the identification data.

**6. Barcode Numbers:**Some types of barcodes, such as EAN-13 barcodes used for retail products, include a checksum digit computed using algorithms like Verhoeff to verify the integrity of the barcode data.

In these and other applications, the Verhoeff algorithm provides a reliable method for detecting errors in numeric data, thereby enhancing data integrity and accuracy.

Last 4 or 6 digit of credit and debit card in India is unique?

In India, the last four or six digits of credit and debit card numbers are typically not guaranteed to be unique. These digits are known as the "cardholder account number" or "primary account number" (PAN). While the first few digits of a card number often represent the issuer and the type of card, the last four or six digits (the PAN) usually identify the individual account within the issuer's system.

However, these digits are not necessarily unique across all cardholders or even within the same issuer. Many card issuers recycle PANs, especially if an account is closed or if a card is reissued due to loss or expiration. As a result, different cardholders may have cards with the same last four or six digits.

The uniqueness of a card number typically lies in the combination of the issuer identification number (IIN) and the PAN. The combination of these digits, along with other components like the card expiration date and CVV, ensures the uniqueness and security of each card transaction.

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