On some class number relations of algebraic tori.

Algebraic tori in cryptography

We give a mathematical interpretation in terms of algebraic tori of Lucas-based cryptosystems, XTR, and the conjectural generalizations in [2]. We show that the varieties underlying these systems are quotients of algebraic tori by actions of products of symmetric groups. Further, we use these varieties to disprove conjectures from [2].

Derandomizing Some Number-theoretic and Algebraic Algorithms

Motivic Invariants of Algebraic Tori

We prove a trace formula and a global form of Denef and Loeser’s motivic monodromy conjecture for tamely ramified algebraic tori over a discretely valued field. If the torus has purely additive reduction, the trace formula gives a cohomological interpretation for the number of components of the Néron model.

Essential Dimension of Algebraic Tori

The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group. We obtain similar formulas for the essential p-dimension of a broad class of groups, which includes all algebraic tori.

On the Computation of the Class Number of an Algebraic Number Field

It is shown how the analytic class number formula can be used to produce an algorithm which efficiently computes the class number h of an algebraic number field F. The method assumes the truth of the Generalized Riemann Hypothesis in order to estimate the residue of the Dedekind zeta function of F at s = 1 sufficiently well that h can be determined unambiguously. Given the regulator R of F and ...