## Introduction

The Verhoeff algorithm, created in 1969 by Dutch mathematician Jacobus Verhoeff, is a checksum technique used to detect errors in numerical data entry, specifically those resulting from digit transpositions. It is a straightforward but efficient technique that is frequently used in a variety of sectors, including data processing, healthcare, and finance, to guarantee the accuracy of numerical data.

## How Does the Verhoeff Algorithm Work?

The Verhoeff algorithm is based on permutations and the mathematical dihedral group—a collection of symmetries that maintains an object's shape. To put it simply, it generates a checksum digit for a given numeric input by using a series of permutation tables and a multiplication operation.

**Setup**

**Name permutation tables:**The Verhoeff algorithm makes use of permutation tables to list all possible digit permutations. For error detection, these tables are essential.**Define a multiplication table:**The results of multiplying digit pairs are shown in this table.

**Input Digit Array**:

- Create an array of digits from the provided numeric input.

**Permutation**

- In accordance with the permutation tables, apply permutations to the digit array. This stage guarantees the detection of mistakes like transpositions.

**Checksum Calculation**

- Multiply the permuted digits by the multiplication table's corresponding digits.
- Add up all of these multiplications' results.
- The final checksum digit is obtained by finding the digit that, when added to the sum, makes it divisible by 10.

**Verification**

- Add the checksum digit to the array of original numbers.
- Recalculate the checksum using the same procedure to confirm that the input is accurate. The input is regarded to be valid if the recalculated checksum matches with the added digit.

### Example

Verhoeff algorithm with a simple example

**Input**: 1234

**1. Setup**

Permutation table

```
[1 2 3 4 5 6 7 8 9 0]
[0 4 3 2 1 5 6 7 8 9]
[0 1 5 7 6 2 8 9 3 4]
[0 1 2 8 3 4 9 5 6 7]
[0 1 2 3 9 7 4 8 5 6]
```

Multiplication table

```
× | 0 1 2 3 4 5 6 7 8 9
--|--------------------
0 | 0 0 0 0 0 0 0 0 0 0
1 | 0 1 2 3 4 5 6 7 8 9
2 | 0 2 4 6 8 10 12 14 16 18
3 | 0 3 6 9 12 15 18 21 24 27
4 | 0 4 8 12 16 20 24 28 32 36
5 | 0 5 10 15 20 25 30 35 40 45
6 | 0 6 12 18 24 30 36 42 48 54
7 | 0 7 14 21 28 35 42 49 56 63
8 | 0 8 16 24 32 40 48 56 64 72
9 | 0 9 18 27 36 45 54 63 72 81
```

**2. Input Digit Array**: [1, 2, 3, 4]

**3. Permutation**

- Using the permutation table, apply the permutations to the digit array:

a) Original: [1, 2, 3, 4]

b) Permuted: [1, 3, 2, 4]

**4. Checksum Calculation**

- Multiply the permuted digits with the corresponding digits from the multiplication table:

a)Multiplications: (1×1) + (3×2) + (2×3) + (4×4) = 1 + 6 + 6 + 16 = 29 - Find the checksum digit: 29 + 1 = 30 (divisible by 10)

**5. Verification**:

- Append the checksum digit (0) to the original digit array: [1, 2, 3, 4, 0]
- Recalculate the checksum using the same process:

a) Permutation: [1, 3, 2, 4, 0]

b)Checksum Calculation: (1×1) + (3×2) + (2×3) + (4×4) + (0×5) = 1 + 6 + 6 + 16 + 0 = 29 - Since the recalculated checksum (29) matches the appended digit (0), the input (1234) is considered valid.

## Conclusion

An accurate way to identify mistakes in numerical data entering, especially digit transpositions, is by the use of the Verhoeff algorithm. Utilizing a multiplication operation and permutation tables, it produces a checksum digit that can be used to verify the accuracy of the input. Data correctness is critical in many applications, and this instrument's simplicity, efficacy, and ease of deployment make it a useful tool.