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
In this thesis three different topics in number theory are studied. The first part deals with exponential Diophantine equations in positive characteristic. In the second part various... Show moreIn this thesis three different topics in number theory are studied. The first part deals with exponential Diophantine equations in positive characteristic. In the second part various statistical properties of class groups are proven. In the final part we apply the circle method to count the number of representations of an odd integer n as the sum of three primes all with a fixed primitive root. Show less
We prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number... Show moreWe prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number field of discriminant D. Theorem A. The class group C(-4p) has an element of order 16 for one-fourth of prime numbers p of the form a^2+16c^4. Theorem B. The class group C(-8p) has an element of order 16 for one-eighth of prime numbers p = -1 mod 4. These are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. They prove an instance of the Cohen-Lenstra conjectures. The proofs of these theorems involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. In case of Theorem B, we prove a power-saving error term for a prime-counting function related to the 16-rank of C(-8p), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion. Show less
In this thesis we look at three counting problems connected to orders in number fields. First we study the probability that for a random polynomial f in Z[X] the ring Z[X]/f is the maximal order in... Show moreIn this thesis we look at three counting problems connected to orders in number fields. First we study the probability that for a random polynomial f in Z[X] the ring Z[X]/f is the maximal order in Q[X]/f. Connected to this is the probability that a random polynomial has a squarefree discriminant. The second counting problem counts the number of subrings within maximal orders. We know that the number of subrings of given index is finite. We determine bounds for the number of suborders in terms of the rank of the maximal order and the index of the suborder. Connected to this is a question from Manjul Bhargava on the number of suborders in quintic rings. The final problem deals with class groups. There are bounds known for the class number of a maximal order, and we use these bounds to bound the class number of general orders. Show less
