Encryption methods using formal power series rings

Recently there has been a great deal of work on noncommutative algebraic cryptography. This involves the use of noncommutative algebraic objects as the platforms for encryption systems. Most of this work, such as the Anshel-Anshel-Goldfeld scheme, the Ko-Lee scheme and the Baumslag-Fine-Xu Modular group scheme use nonabelian groups as the basic algebraic object. Some of these encryption methods have been successful and some have been broken. It has been suggested that at this point further pure group theoretic research, with an eye towards cryptographic applications, is necessary.In the present study we attempt to extend the class of noncommutative algebraic objects to be used in cryptography. In particular we explore several different methods to use a formal power series ring R && x1; :::; xn && in noncommuting variables x1; :::; xn as a base to develop cryptosystems. Although R can be any ring we have in mind formal power series rings over the rationals Q. We use in particular a result of Magnus that a finitely generated free group F has a faithful representation in a quotient of the formal power series ring in noncommuting variables.

Wave-vector dependence of spin and density multipole excitations in quantum dots

We have employed time-dependent local-spin density-functional theory to analyze the multipole spin and charge density excitations in GaAs-AlxGa1-xAs quantum dots. The on-plane transferred momentum degree of freedom has been taken into account, and the wave-vector dependence of the excitations is discussed. In agreement with previous experiments, we have found that the energies of these modes do not depend on the transferred wave vector, although their intensities do. Comparison with a recent resonant Raman scattering experiment [C. Schüller et al., Phys. Rev. Lett. 80, 2673 (1998)] is made. This allows us to identify the angular momentum of several of the observed modes as well as to reproduce their energies

Ion-beam induced oxidation of GaAs and AlGaAs

The oxidation of GaAs and AlxGa1−xAs targets by oxygen irradiation has been studied in detail. It was found that the oxidation process is characterized by the strong preferential oxidation of Al as compared to Ga, and of Ga as compared to As. This experimental observation, which has been accurately quantified by using x‐ray photoelectron spectroscopy, is connected to the different heats of formation of the corresponding oxides. The oxide grown by ion beam oxidation shows a strong depletion in As and relatively low oxidation of As as well. The depletion can be associated with the preferential sputtering of the As oxide in respect to other compounds whereas the low oxidation is due to the low heat of formation. In contrast Al is rapidly and fully oxidized, turning the outermost layer of the altered layer to a single Al2O3 overlayer, as observed by transmission electron microscopy. The radiation enhanced diffusion of oxygen and aluminum in the altered layer explains the large thickness of these altered layers and the formation of Al oxides on top of the layers. For the case of ion‐beam oxidation of GaAs a simulation program has been developed which describes adequately the various growth mechanisms experimentally observed

Integrability and non-integrability of periodic non-autonomous Lyness recurrences (revised and enlarged version)

This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an+xn+1)=xn, where fang is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k 2 f1; 2; 3; 6g the behavior of the sequence fxng is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some di erent features.