Identification numbers with check digit algorithms 
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.
