Most of current public-key cryptography is considered insecure against attacks from sufficiently powerful quantum computers. Post-quantum cryptography studies methods to secure information... Show moreMost of current public-key cryptography is considered insecure against attacks from sufficiently powerful quantum computers. Post-quantum cryptography studies methods to secure information resistant against such attacks. One proposal is isogeny-based cryptography, which bases its security on computational hardness assumptions related to maps between elliptic curves. We analyze the security of isogeny-based cryptographic schemes, in particular those based on class group actions. We find special cases in which the underlying computational hardness assumptions can be broken, sometimes even by classical computers. Furthermore, we study a method, known as radical isogenies, to accelerate the execution of isogeny-based protocols. Finally, we introduce multivariate generalizations of Hilbert class polynomials, which may yield computational benefits compared to their univariate counterparts. Show less
The main topic of this PhD thesis is the Arakelov ray class group of a number field, an algebraic object that contains both the ideal class group structure and the unit group structure. The main... Show moreThe main topic of this PhD thesis is the Arakelov ray class group of a number field, an algebraic object that contains both the ideal class group structure and the unit group structure. The main result consists of the fact that certain specific random walks on the Arakelov ray class group result in a target point that is uniformly distributed on this group, under the assumption of an extended version of the Riemann Hypothesis. Almost all other results of this work are consequences of this fact. Show less
