Rigorously Analyzed Algorithmsfor the Discrete Logarithm Problemin Quadratic Number FieldsVom Fachbereich Informatikder Technischen Universitat¨ DarmstadtgenehmigteDissertationzur Erreichung des akademischen GradesDoctor rerum naturalium (Dr. rer.nat.)vonDipl.-Math. Ulrich Vollmeraus BerlinReferenten: Prof. Dr. Johannes Buchmann (TU Darmstadt)Prof.dr. Peter Stevenhagen (Universiteit Leiden, NL)Tag der Einreichung: 15. September 2003Tag der mundlic¨ hen Prufung:¨ 28. Oktober 2003Darmstadt, 2003Hochschulkennziffer: D17To my parentsandTo all who were denied an academic educationby political arbitrarinessAcknowledgementsAt this point I would like to take the opportunity to thank at least some ofthose people whose support enabled me to do the research which led to thisthesis.First of all I would like to thank my thesis advisor, Prof. Dr. JohannesBuchmann, for proposing the topic, for his advice and support and theinvestmentoftimefordiscussionsthatprovidedideasandimpetusespeciallyat the important initial phase of my thesis research. He has proved to be awonderful guide into the world of academia.I would like to thank Prof.dr. Stevenhagen for evaluating and carefullyreading this thesis on short notice, and the detailed comments for improve-ment he made.