Abstract. The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. This problem is NP-hard over any field. When the number of equations m is the same as the number of unknowns n the best known algorithms are exhaustive search for small fields, and a Gröbner base algorithm for large fields. Gröbner base algorithms have large exponential complexity and cannot solve in practice systems with n ≥ 15. Kipnis and Shamir [9] have recently introduced a new algorithm called "relinearization". The exact complexity of this algorithm is not known, but for sufficiently overdefined systems it was expected to run in polynomial time. In this paper we analyze the theoretical and practical aspects of relinearization. We ran a large number of experiments for various values of n and m, and analysed which systems of equations were actually solvable. We show that many of the equations generated by relinearization are linearly dependent, and thus relinearization is less efficient that one could expect. We then develop an improved algorithm called XL which is both simpler and more powerful than relinearization. For all 0 < ≤ 1/2, and m ≥ n 2 , XL and relinearization are expected to run in polynomial time of approximately n O(1/ √ ) . Moreover, we provide strong evidence that relinearization and XL can solve randomly generated systems of polynomial equations in subexponential time when m exceeds n by a number that increases slowly with n.

Drift, slope of sample, and indeterminate sensitivity of piezoceramics are considered as the origin of the linear scanning tunneling microscope (STM) image distortions. A special algorithm of STM scanning and reconstruction of image of unknown atomic structures is suggested and tested. This algorithm allows us to measure three components of the drift velocity and two angles, characterizing the average slope of scanning surface. On the one hand, using this algorithm, one can perform the STM calibration by a known surface structure (for example, the highly oriented pyrolytic graphite surface) even in the presence of a drift. This enables us to determine all the three piezoceramics constants of STM piezoscanner and the deviation of the real scanner axes X and Y from orthogonality. On the other hand, using such a calibrated STM and the described algorithm, it is possible to obtain the real STM image and make measurements for unknown surfaces with atomic resolution without the distortions mentioned above. As it has been proved, the error of interatomic distance measurements does not exceed 5% (∼0.1 Å) for any direction along the surface, even in the presence of a drift up to 0.4 Å/s and the sample slope angle up to 10°.

PACS. 78.47.+p -Time-resolved optical spectroscopies and other ultrafast optical measurements in condensed matter. PACS. 74.72.-h -High-Tc compounds. PACS. 74.76.Bz -High-Tc films.Abstract. -Using a femtosecond pump-probe optical technique, we measured the time-resolved change in reflectivity of three single-layer cuprate samples, including Bi2Sr2−xLaxCuO6+z, La2CuO4+y and La2−xSrxCuO4 grown on SrTiO3. We analyze the data in terms of a nonthermal electron relaxation model, and obtain quantitative agreement between the model and the data. The initial electron-electron thermalization process is much longer in the superconducting samples as opposed to the overdoped nonsuperconducting sample, suggesting a bottleneck in electron thermalization process due to the presence of the superconducting gap.c EDP Sciences