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.

A systematic and feasible method for computing nuclear contributions to electrical properties of polyatomic molecules

An analytic method to evaluate nuclear contributions to electrical properties of polyatomic molecules is presented. Such contributions control changes induced by an electric field on equilibrium geometry (nuclear relaxation contribution) and vibrational motion (vibrational contribution) of a molecular system. Expressions to compute the nuclear contributions have been derived from a power series expansion of the potential energy. These contributions to the electrical properties are given in terms of energy derivatives with respect to normal coordinates, electric field intensity or both. Only one calculation of such derivatives at the field-free equilibrium geometry is required. To show the useful efficiency of the analytical evaluation of electrical properties (the so-called AEEP method), results for calculations on water and pyridine at the SCF/TZ2P and the MP2/TZ2P levels of theory are reported. The results obtained are compared with previous theoretical calculations and with experimental values

Systematic study of the static electrical properties of the CO molecule: influence of the basis set size and correlation energy

The influence of the basis set size and the correlation energy in the static electrical properties of the CO molecule is assessed. In particular, we have studied both the nuclear relaxation and the vibrational contributions to the static molecular electrical properties, the vibrational Stark effect (VSE) and the vibrational intensity effect (VIE). From a mathematical point of view, when a static and uniform electric field is applied to a molecule, the energy of this system can be expressed in terms of a double power series with respect to the bond length and to the field strength. From the power series expansion of the potential energy, field-dependent expressions for the equilibrium geometry, for the potential energy and for the force constant are obtained. The nuclear relaxation and vibrational contributions to the molecular electrical properties are analyzed in terms of the derivatives of the electronic molecular properties. In general, the results presented show that accurate inclusion of the correlation energy and large basis sets are needed to calculate the molecular electrical properties and their derivatives with respect to either nuclear displacements or/and field strength. With respect to experimental data, the calculated power series coefficients are overestimated by the SCF, CISD, and QCISD methods. On the contrary, perturbation methods (MP2 and MP4) tend to underestimate them. In average and using the 6-311 + G(3df) basis set and for the CO molecule, the nuclear relaxation and the vibrational contributions to the molecular electrical properties amount to 11.7%, 3.3%, and 69.7% of the purely electronic μ, α, and β values, respectively

Elastic scattering of fast electrons and positrons by atoms

Elastic scattering of relativistic electrons and positrons by atoms is considered in the framework of the static field approximation. The scattering field is expressed as a sum of Yukawa terms to allow the use of various approximations. Accurate phase shifts have been computed by combining Bühring¿s power-series method with the WKB and Born approximations. This combined procedure allows the evaluation of differential cross sections for kinetic energies up to several tens of MeV. Numerical results are used to analyze the validity of several approximate methods, namely the first- and second-order Born approximations and the screened Mott formula, which are frequently adopted as the basis of multiple scattering theories and Monte Carlo simulations of electron and positron transport.

Compositional Time Series: An Application

The composition of the labour force is an important economic factor for a country.Often the changes in proportions of different groups are of interest.I this paper we study a monthly compositional time series from the Swedish LabourForce Survey from 1994 to 2005. Three models are studied: the ILR-transformed series,the ILR-transformation of the compositional differenced series of order 1, and the ILRtransformationof the compositional differenced series of order 12. For each of thethree models a VAR-model is fitted based on the data 1994-2003. We predict the timeseries 15 steps ahead and calculate 95 % prediction regions. The predictions of thethree models are compared with actual values using MAD and MSE and the predictionregions are compared graphically in a ternary time series plot.We conclude that the first, and simplest, model possesses the best predictive power ofthe three models

Lag-one autocorrelation in short series: Estimation and hypothesis testing

In the first part of the study, nine estimators of the first-order autoregressive parameter are reviewed and a new estimator is proposed. The relationships and discrepancies between the estimators are discussed in order to achieve a clear differentiation. In the second part of the study, the precision in the estimation of autocorrelation is studied. The performance of the ten lag-one autocorrelation estimators is compared in terms of Mean Square Error (combining bias and variance) using data series generated by Monte Carlo simulation. The results show that there is not a single optimal estimator for all conditions, suggesting that the estimator ought to be chosen according to sample size and to the information available of the possible direction of the serial dependence. Additionally, the probability of labelling an actually existing autocorrelation as statistically significant is explored using Monte Carlo sampling. The power estimates obtained are quite similar among the tests associated with the different estimators. These estimates evidence the small probability of detecting autocorrelation in series with less than 20 measurement times.