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Goppa codes and their applications
Kotil, Jaroslav ; Drápal, Aleš (advisor) ; Šťovíček, Jan (referee)
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
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Algebraicko-geometrické kódy a Gröbnerovy báze
Heglasová, Veronika ; Šťovíček, Jan (advisor) ; Drápal, Aleš (referee)
In this master thesis we introduce algebraic geometry codes (AG codes). Be- sides basic definitions, properties and attributes of AG codes and algebraic ge- ometry we show how to encode AG codes that has nontrivial Abelian group of permutation automorphisms and how to decode one-point AG codes. We also present Hermitian codes, which are example of one-point AG codes with nontriv- ial Abelian group of permutation automorphisms. We demonstrate the method for encoding and the method for decoding on specific Hermitian code. 1
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Goppa codes and their applications
Kotil, Jaroslav ; Drápal, Aleš (advisor) ; Šťovíček, Jan (referee)
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
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