Verhoeff Algorithm

The Verhoeff algorithm, a checksum formula for error detection first published in 1969, was developed by Dutch mathematician Jacobus Verhoeff (born 1927). Like the more widely known Luhn algorithm, it works with strings of decimal digits of any length. It detects all single-digit errors and all transposition errors involving two adjacent digits.

Verhoeff devised his algorithm using the properties of D₅ (the dihedral group of order 10) — a non-commutative system of operations on ten elements, corresponding to the results of rotating or reflecting (flipping) a regular pentagon. In practice, however, the scheme would normally be implemented using precomputed lookup tables.