Kryptografie založená na kvadratických tělesech

Abstract

Imaginary quadratic fields were first suggested as a setting for public-key cryptography by Buchmann and Williams already in 1988 and more cryptographic schemes followed. Although the resulting protocols are currently not as efficient as those based on elliptic curves, they are comparable to schemes based on RSA and, moreover, their security is believed to be independent of other widely-used protocols including RSA, DSA and elliptic curve cryptography. This work gathers present results in the field of quadratic cryptography. It recapitulates the algebraic theory needed to work with the class group of imaginary quadratic fields. Then it investigates algorithms of class group operations, both asymptotically and practically effective. It also analyses feasible cryptographic schemes and attacks upon them. A library implementing described cryptographic schemes is a part of this work.

Nazev prace: Kryptografie zalozena na kvadratickych telesech Autor: Milan Straka Katedra (ustav): Katedra algebry Vedouci diplomove prace: RNDr. David Stanovsky, Ph.D. E-mail vedouciho: David.Stanovsky@mff.cuni.cz Abstrakt: Iraaginarni kvadraticka telesa byla navrzena pro pouziti v asyrnetricke kryptografii Buchmannem a Williamsern jiz v roce 1988 a od te doby vznikly i dalsi kryptograficke protokoly. I kdyz tyto protokolynejsou tak efektivni jako podobna schemata s eliptickyrni kfivkami, mohou konku- rovat schematum zalozenyrn na RSA, a navic je jejich bezpecnost pova- zovana za nezavislou na bezpecnosti beznych kryptosystemu jako RSA, DSA aEGG. Tato prace shrnuje dosavadni vysledky v oboru kvadraticke kryptografie. Jednak popisuje algebraickou teorii nutnou pro zavedeni tndove grupy imaginarnich kvadratickych teles a dale studuje algoritmy operaci v tri- dove grupe, jak asymptoticky, tak prakticky efektivni. Take rozebira vhodna kryptograficka schemata a utoky na ne. Soucasti teto prace je knihovna, ktera popsane protokoly efektivne im- plementuje. Klicova slova: tridova grupa imaginarniho kvadratickeho telesa, diskretni logaritmus, asymetricka kryptografie, sifrovaci a podpisove schema Title: Qudratic field based cryptography Author: Milan Straka Department: Department ofAlgebra Supervisor: RNDr. David...