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The main purpose of this bachelor thesis is the description of the attack on the Niederreiter cryptosystem based on GRS codes. This attack was published by Sidelnikov and Shestakov in 1992. In the beginning the problem of group action, which is used in the attack, is introduced. A short preface into the coding theory follows, GRS codes are described and McEliece and Niederreiter cryptosystems are introduced, both as representatives of post-quantum cryptography. The following part of the thesis is dedicated to the attack itself. It is showed how one uses the group action, the process of the attack is also described in detail and its computing complexity is mentioned. Everything is illustrated by examples. Powered by TCPDF (www.tcpdf.org)

Title: Goppa codes and their applications Author: Bc. Jaroslav Kotil Department: Department of algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc. Abstract: In this diploma paper we introduce Goppa codes, describe their para- metres and inclusion in Alternant codes, which are residual Generalized Reed- Solomon codes, and Algebraic-geometry codes. Aftewards we demonstrate deco- ding of Goppa codes and introduce Wild Goppa codes. We also describe post- quantum cryptography member: McEliece cryptosystem for which no effective attacks with quantum computers are known. We outline a usage of this crypto- system with Goppa codes and describe the security of the cryptosystem together with possible attacks of which the most effective ones are based on information- set decoding. Keywords: Goppa codes, Generalized Reed-Solomon codes, Algebraic-geometry codes, Post-quantum cryptography, McEliece cryptosystem 1