Autor/-in:

Vlachos Constantin

Some Applications of Algebra in Coding Theory and Cryptography

Betreuer/-in:
Niklas Gassner

Schule:
Freie Katholische Schule Zürich

Fach: Mathematik

The ideas for all my experiments originated from attending inspiring lectures and talking with interesting people. I am incredibly grateful for all the opportunities I've had to learn and look forward to many more. The stating and proving of our own theorem on the error-detectability of the IBAN code taught me to be open to approaching things from a different, unorthodox angle. Although the theorem concerns itself with a coding theory matter it was proven using techniques from number theory, a different area of mathematics. This is why I strongly believe learning about scientific fields or areas of fields different from ones own can provide new perspectives and ideas.

Abstract

In this project we reverse-engineered the linear code known as the ISBN (international standrad book number) code. We searched for a generator matrix and parity-check matrix experimentally and used them to deduce certain properties of the ISBN code.

Next, we reverse-engineered the non-linear IBAN (international bank account number) code. We came up with a decoding scheme utilizing p-adic numbers and used this very scheme to prove our theorem that the IBAN code is a single error detecting code.

Finally, we also constructed a hash-function. In order to do so it was necessary to also construct a pseudo-random generator, which was then combined with the pseudo-random function by Goldreich, Goldwasser and Micali, to get the desired result. We did not prove this or find experimental evidence to support a claim of first or second pre-image resistance, we did however test some known attacks.

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