A key-exchange system based on imaginary quadratic fields

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Artur Travesa i Grau[en] The aim of this project is to give an overview of the field of mathematical cryptography through the lenses of asymmetric protocols based on the Discrete Logarithm Problem over imaginary quadratic fields. The mathematical foundation is illustrated with the study of quadratic orders and their class groups, which are the relevant algebraic infrastructure for a Diffie-Hellman-type protocol known as Buchmann-Willams cryptosystem. The relationship between quadratic orders and binary quadratic forms is exploited to develop and explain the computational aspect of cryptography, providing convenient ways of machine computation. The connection between ideals in the maximal and non-maximal orders is the key to developing computationally-efficient cryptographic protocols over quadratic fields. In that sense, the Hühnlein-Jacobson and the Paulus-Takagi cryptosystems are introduced. Finally, the security component of the protocols is analyzed by discussing the Discrete Logarithm Problem and measures to obtain conjectural security