Modular identification schemes: their error detecting effectiveness

In general, the check digit \(C\) of a number \(x_{1}x_{2}x_{3}...x_{n}\), in error detection systems using modular arithmetic, is the solution of the equation \[p_{1}x_{1}+p_{2}x_{2}+...+p_{n}x_{n}+C=0\;(\mbox{mod }k).\]

In the examples presented in this webpage, we have:

Identity Card (BI) and Fiscal Number (NIF) \[\{p_{1},p_{2},...,p_{8}\} \rightarrow \{9, 8, 7, ..., 2\}\] \(k=11\)
Barcode \[\{p_{1},p_{2},...,p_{8}\} \rightarrow \{1,3,1,3,1, ..., 3\}\] \(k= 10\)
Euro banknotes \[\{p_{L},p_{1},...,p_{10}\} \rightarrow \{1,1, ..., 1\}\] \(k= 9\)
Bank Account Number (NIB) \[\{p_{1},p_{2},...,p_{19}\} \rightarrow \{73, 17, 89, 38, 62, 45, 53, 15, 50, 5, 49, 34, 81, 76, 27, 90, 9, 30, 3\}\] \(k= 97\)

All identification systems of this kind satisfy the following properties:

1) The system detects a singular error \(x_{1}...x_{i}...x_{n} \rightarrow x_{1}...x_{j}...x_{n}\) if and only if \(\mbox{gcd}(p_{i},k)=1\);
2) The system detects a transposition error \(x_{1}...x_{i}...x_{j}...x_{n} \rightarrow x_{1}...x_{j}...x_{i}...x_{n}\) if and only if \(\mbox{gcd}(p_{i}-p_{j},k)=1\). (Proofs - only in Portuguese)

For more information, consult: [4] J. PICADO, A álgebra dos sistemas de identificação, Boletim da Sociedade Portuguesa de Matemática 44(2001) 39-73. (only in Portuguese)

The case of Visa cards is a little different, because the multiplication of \ (x_ {i} \) by its weight \ (p {i} \) is replaced by a diferent function weight. For more details see [4].

Another class of identifiocations systems is provided by the Verhoeff system, based on more advanced Group Theory.