MV
M.M.J. Vons
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Since identification numbers play an important role in many everyday systems, even minor typing errors can have significant consequences. This thesis investigates the error detection capabilities of several widely used check digit systems, with a particular focus on modular arithmetic-based algorithms. The study begins by introducing the mathematical background of modular arithmetic and describing different types of common data entry errors.
Next, three real-world systems are analysed in detail. In particular, credit card numbers using the Luhn algorithm, International Bank Account Numbers (IBANs) using the MOD97-10 algorithm, and GTIN-13 product codes using an alternating-weight modulo 10 method. For each system, the structure of the number format and the corresponding check digit algorithm are formally defined and explained. A mathematical analysis is then carried out to determine which types of errors, including single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors can be detected. This mathematical analysis includes both formal proofs and a computational experiment.
The final part of the thesis introduces a generalised version of the GTIN-13 check digit algorithm, in which the fixed weight 3 is replaced by an arbitrary integer p. The resulting modulo 10 algorithm with alternating weight p is then analysed in detail. Necessary and sufficient conditions for error detection are derived in terms of greatest common divisors involving p, p-1, p+1, and the modulus 10. This generalisation illustrates how the value of the weighting parameter p affects the error detection properties of the system.
Overall, the results show that check digit algorithms can detect many common types of input errors such as single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors. Even though the algorithms have some limitations, they are still relatively effective in their ability to detect errors. ...
Next, three real-world systems are analysed in detail. In particular, credit card numbers using the Luhn algorithm, International Bank Account Numbers (IBANs) using the MOD97-10 algorithm, and GTIN-13 product codes using an alternating-weight modulo 10 method. For each system, the structure of the number format and the corresponding check digit algorithm are formally defined and explained. A mathematical analysis is then carried out to determine which types of errors, including single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors can be detected. This mathematical analysis includes both formal proofs and a computational experiment.
The final part of the thesis introduces a generalised version of the GTIN-13 check digit algorithm, in which the fixed weight 3 is replaced by an arbitrary integer p. The resulting modulo 10 algorithm with alternating weight p is then analysed in detail. Necessary and sufficient conditions for error detection are derived in terms of greatest common divisors involving p, p-1, p+1, and the modulus 10. This generalisation illustrates how the value of the weighting parameter p affects the error detection properties of the system.
Overall, the results show that check digit algorithms can detect many common types of input errors such as single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors. Even though the algorithms have some limitations, they are still relatively effective in their ability to detect errors. ...
Since identification numbers play an important role in many everyday systems, even minor typing errors can have significant consequences. This thesis investigates the error detection capabilities of several widely used check digit systems, with a particular focus on modular arithmetic-based algorithms. The study begins by introducing the mathematical background of modular arithmetic and describing different types of common data entry errors.
Next, three real-world systems are analysed in detail. In particular, credit card numbers using the Luhn algorithm, International Bank Account Numbers (IBANs) using the MOD97-10 algorithm, and GTIN-13 product codes using an alternating-weight modulo 10 method. For each system, the structure of the number format and the corresponding check digit algorithm are formally defined and explained. A mathematical analysis is then carried out to determine which types of errors, including single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors can be detected. This mathematical analysis includes both formal proofs and a computational experiment.
The final part of the thesis introduces a generalised version of the GTIN-13 check digit algorithm, in which the fixed weight 3 is replaced by an arbitrary integer p. The resulting modulo 10 algorithm with alternating weight p is then analysed in detail. Necessary and sufficient conditions for error detection are derived in terms of greatest common divisors involving p, p-1, p+1, and the modulus 10. This generalisation illustrates how the value of the weighting parameter p affects the error detection properties of the system.
Overall, the results show that check digit algorithms can detect many common types of input errors such as single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors. Even though the algorithms have some limitations, they are still relatively effective in their ability to detect errors.
Next, three real-world systems are analysed in detail. In particular, credit card numbers using the Luhn algorithm, International Bank Account Numbers (IBANs) using the MOD97-10 algorithm, and GTIN-13 product codes using an alternating-weight modulo 10 method. For each system, the structure of the number format and the corresponding check digit algorithm are formally defined and explained. A mathematical analysis is then carried out to determine which types of errors, including single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors can be detected. This mathematical analysis includes both formal proofs and a computational experiment.
The final part of the thesis introduces a generalised version of the GTIN-13 check digit algorithm, in which the fixed weight 3 is replaced by an arbitrary integer p. The resulting modulo 10 algorithm with alternating weight p is then analysed in detail. Necessary and sufficient conditions for error detection are derived in terms of greatest common divisors involving p, p-1, p+1, and the modulus 10. This generalisation illustrates how the value of the weighting parameter p affects the error detection properties of the system.
Overall, the results show that check digit algorithms can detect many common types of input errors such as single-digit errors, transposition errors, twin errors, jump-transposition errors, and jump-twin errors. Even though the algorithms have some limitations, they are still relatively effective in their ability to detect errors.