Damm Algorithm (Checksum)

Damm's algorithm is a check digit / checksum calculation system designed to detect input errors in decimal numeric identifiers (serial numbers, administrative codes, etc.). Designed to be simple to implement by hand or programmatically, this algorithm detects all single substitution errors, all adjacent digit transpositions, as well as some multiple errors.

The calculation relies on the use of a unique substitution table, a quasigroup of order 10, known as a Damm table.

The calculation starts with c = 0. For each digit of the number, read from left to right, update c = table[c][digit].

To verify a number, append the check digit, then process all digits: the final result must be 0.

The check digit is 4, so the number with checksum is 1234. To verify, process 1234 and the final c must be 0.

The algorithm guarantees detection of the following types of errors:

— All single substitution errors (replacing one digit with another, e.g.: $ 1234 \rightarrow 1239 $).

— All adjacent transpositions (swapping two adjacent digits, e.g.: $ 1234 \rightarrow 1324 $).

— Some multiple errors.

However, some complex permutations are not detected, such as circular permutations (e.g.: $ 1234 \rightarrow 4123 $) or some non-adjacent transpositions.

The time complexity is linear, $ O(n) $, where $ n $ is the number of digits in the sequence, since each digit is examined exactly once.

The space complexity is constant, $ O(1) $: the Damm table has a fixed size ($ 10 \times 10 $), which requires minimal memory space independent of the length of the data processed.

The Damm table is built from a quasigroup of order 10 with specific algebraic properties that guarantee detection of transposition and substitution errors. More precisely, the main diagonal of the table contains only zeros ($ N(x, x) = 0 $), and the mathematical structure satisfies strict antisymmetry conditions that prevent collisions when adjacent digits are swapped.

In the Damm table, $ \text{table}[0][0] = 0 $. Since the accumulator starts at $ c = 0 $, any zero at the start of the sequence leaves $ c $ unchanged at $ 0 $. Thus, 123, 0123 and 00123 produce exactly the same check digit ($ 4 $).

Yes, the method extends to any base $ n $ provided that a weakly totally antisymmetric quasigroup of order $ n $ exists. H. Michael Damm proved the existence of such quasigroups for all orders $ n \neq 2 $ and $ n \neq 6 $ (possible use in base $ 16 $ hexadecimal, base $ 32 $ or base $ 36 $ alphanumeric, but excluding base $ 2 $ binary and base $ 6 $).

The algorithm was introduced in 2004 by German mathematician H. Michael Damm in his doctoral thesis entitled Totally Antisymmetric Quasigroups.