Ulrich Vollmer

Perspectives for cryptographic long-term security

Cryptographic long-term security is needed, but difficult to achieve. Use flexible cryptographic tools, and have replacements ready.

Polynomial time quantum algorithm for the computation of the unit group of a number field

We present a quantum algorithm for the computation of the irrational period lattice of a function on Zn which is periodic in a relaxed sense. This algorithm is applied to compute the unit group of finite extensions of Q. Execution time for fixed field degree over Q is polynomial in the discriminant of the field. Our algorithms generalize and improve upon Hallgrens work [9] for the one-dimensional case corresponding to real-quadratic fields.

Binary Quadratic Forms

1. Quadratic forms and symmetric bilinear forms on vector spaces, and going from one to another, examples. Matrix of a form with respect to a basis, and how the matrix changes with respect to a different basis. 2. Discriminant of a form with respect to a fixed basis, change of discriminant of a form as the basis changes. Radical of the form, relationship between radical and dsicriminant. 3. Subspaces of a vector space with a bilinear form, orthogonal subspaces, and orthogonal complements 4. Witt’s Theorem (Theorem 3 on page number 31 of Serre’s book)

Asymptotically Fast Discrete Logarithms in Quadratic Number Fields

This article presents algorithms for computing discrete logarithms in class groups of quadratic number fields. In the case of imaginary quadratic fields, the algorithm is based on methods applied by Hafner and McCurley [HM89] to determine the structure of the class group of imaginary quadratic fields. In the case of real quadratic fields, the algorithm of Buchmann [Buc89] for computation of class group and regulator forms the basis. We employ the rigorous elliptic curve factorization algorithm of Pomerance [Pom87], and an algorithm for solving systems of linear Diophantine equations proposed and analysed by Mulders and Storjohann [MS99]. Under the assumption of the Generalized Riemann Hypothesis, we obtain for fields with discriminant d a rigorously proven time bound of \(L_{|d|} [\frac{1}{2}, \frac{3}{4}\sqrt{2}]\).

An Accelerated Buchmann Algorithm for Regulator Computation in Real Quadratic Fields

We present a probabilistic algorithm for computing the regulator R of a real quadratic order of discriminant � running in time L(1/2,3/�8+o(1)).

A note on the hermite basis computation of large integer matrices

A new algorithm is given and analyzed for the computation of the Hermite basis of a large integer matrix whose HNF has small essential part. The algorithm improves the results from [3] by dropping two key requirements on the matrix considered---sparsity and small kernel dimension---at the cost of relying on small determinant size.