Verhoeff Calculator

How Verhoeff Calculator Works?

The Verhoeff Calculator works by passing each digit through a position-dependent permutation table and combining the results with the multiplication table of the dihedral group D5. The final value is inverted to give the check digit.

1. 1Reverse the payload so the rightmost digit is processed first.
2. 2Set the running check value c to 0.
3. 3Look up p = P[(i + 1) mod 8][digit] in the permutation table for each position i (starting at 0).
4. 4Update the running value with c = D[c][p] from the D5 multiplication table.
5. 5Take the inverse inv[c] after the last digit: that value is the check digit.
6. 6Validate a complete number by running the loop over all digits without the +1 shift: a valid number ends with c = 0.

Verhoeff Calculator Formula

The Verhoeff formula is Check Digit = inv( D(…D(D(0, P₁(a₁)), P₂(a₂))…, Pₙ(aₙ)) ), where aᵢ are the payload digits read from the right.

c₀ = 0
cᵢ = D[cᵢ₋₁][ P[i mod 8][aᵢ] ]   for i = 1…n (aᵢ read right to left)
Check Digit = inv[cₙ]

• D is the 10 × 10 multiplication table of the dihedral group D5.
• P is the permutation table: row k applies the base permutation (1 5 7 6 2 8 3 0 9 4) k times.
• inv is the inverse table [0, 4, 3, 2, 1, 5, 6, 7, 8, 9].
• aᵢ is the payload digit at position i from the right.

Position 1: digit 6 → P = 3, D[0][3] = 3
Position 2: digit 3 → P = 3, D[3][3] = 1
Position 3: digit 2 → P = 1, D[1][1] = 2
inv[2] = 3
Complete number = 2363

Verhoeff Calculator Conversion Table

The Verhoeff conversion table shows the inverse lookup for every final value of c and sample payloads converted to complete Verhoeff numbers.

What is Verhoeff Calculator?

A Verhoeff Calculator is a tool that computes or validates a decimal check digit using the Verhoeff algorithm. Dutch mathematician Jacobus Verhoeff published the algorithm in 1969.

The Verhoeff algorithm detects 100% of single-digit errors and 100% of adjacent transpositions. The method uses no modulus and no weights: three lookup tables do the work.

India's 12-digit Aadhaar number ends with a Verhoeff check digit, which makes Aadhaar the largest deployment of the algorithm with more than 1.3 billion numbers issued.

What are the uses of Verhoeff Calculator?

There are 4 main uses of the Verhoeff Calculator, including Aadhaar checks, identity number design, and error-rate studies.

• Aadhaar validation: Developers confirm the 12th digit of an Aadhaar number before submitting a KYC request.
• Identity number design: Agencies pick Verhoeff for new citizen or member numbers when full transposition detection is required.
• Data entry protection: Call centres protect order and policy numbers read aloud and typed by hand.
• Algorithm teaching: Teachers show group theory in practice through the D5 table.