In this paper we highlight the benefits of using genus-2 curves in public-key cryptography. Compared to the standardized genus-1 curves, or elliptic curves, arithmetic on genus-2 curves is typically more involved but allows us to work with moduli of half the size. We give a taxonomy of the best known techniques to realize genus-2 based cryptography, which includes fast formulas on the Kummer su...

Given a curve C of genus 2 defined over a field k of characteristic different from 2, with Jacobian variety J , we show that the two-coverings corresponding to elements of a large subgroup of H ` Gal(k/k), J [2](k) ́ (containing the Selmer group when k is a global field) can be embedded as intersection of 72 quadrics in P k , just as the Jacobian J itself. Moreover, we actually give explicit equ...

Instance complexity was introduced by Orponen, Ko, Schh oning, and Watanabe 14] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity, they introduced the notion of p-hard instances, and conjectured that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Fortnow and Kummer 6] proved ...

In our original article we outlined the type of somato-psychosomatic feedback cycle occurring in panic attacks in chronic obstructive pulmonary disease (COPD) that F. Kummer has further elaborated in his correspondence. In fact, this type of feedback cycle is described in the most widely accepted theory of panic attacks and panic disorder in the physically healthy, the cognitive (or ‘‘catastrop...