Mathematical methods in solutions of the problems from the Third International Students' Olympiad in Cryptography

The mathematical problems and their solutions of the Third International Students' Olympiad in Cryptography NSUCRYPTO'2016 are presented. We consider mathematical problems related to the construction of algebraic immune vectorial Boolean functions and big Fermat numbers, problems about secrete sharing schemes and pseudorandom binary sequences, biometric cryptosystems and the blockchain technology, etc. Two open problems in mathematical cryptography are also discussed and a solution for one of them proposed by a participant during the Olympiad is described. It was the first time in the Olympiad history

Approximate approach of research and assessment of crack resistance of cylindrical housings

In this work, we propose a procedure that allows us to quickly, without involving finite-element packages at the execution stage, evaluate the state of postulated cracks in cylindrical structural elements with internal anticorrosion cladding under pressure and unsteady temperature loading. The procedure contains three components. The first component of the procedure is based on the analytical solution of the unsteady thermal conductivity problem. The second component of the procedure is also analytical relations specifying circumferential and longitudinal stresses in a two-layer cylinder as a function of internal pressure and temperature distribution along the depth of the wall. In the third stage of the procedure, the coefficients of stress intensity along the crack front are determined by the method of influence functions using the known distribution of stresses. To implement this part of the procedure, the influence functions must first be calculated by the finite element method. In this work the example of calculating the functions of influence for longitudinal semi-elliptical cracks of 1/8 wall thickness depth and with the ratio of the principal half-diameters 0.3 and 0.7 is presented. As an example of the implementation of the procedure, comparative calculations were carried out and the elastic-plastic calculation of a cylindrical shell containing longitudinal cracks was given. Stress intensity coefficients along the part of the front located in the base metal were calculated. The results of analytical calculations were conservative until plastic deformations developed in the cladding. At significant plastic strains, the values of the stress intensity coefficients calculated by the proposed procedure, on the contrary, are significantly inferior. This demonstrates once again the fact that calculations in the framework of linear fracture mechanics are not always conservative in relation to calculations in the elastic-plastic formulation. The paper gives an explanation of the reason for the obtained effect

A statistical study of magnetic flux emergence in solar active regions prior to strongest flares

Using the data on magnetic field maps and continuum intensity for Solar Cycles 23 and 24, we explored 100 active regions (ARs) that produced M5.0 or stronger flares. We focus on the presence/absence of the emergence of magnetic flux in these ARs 2-3 days before the strong flare onset. We found that 29 ARs in the sample emerged monotonously amidst quiet-Sun. A major emergence of a new magnetic flux within pre-existing AR yielding the formation of a complex flare-productive configuration was observed in another 24 cases. For 30 ARs, an insignificant (in terms of the total magnetic flux of pre-existing AR) emergence of a new magnetic flux within the pre-existing magnetic configuration was observed; for some of them the emergence resulted in a formation of a configuration with a small $\delta$-sunspot. 11 out of 100 ARs exhibited no signatures of magnetic flux emergence during the entire interval of observation. In 6 cases the emergence was in progress when the AR appeared on the Eastern limb, so that the classification and timing of emergence were not possible. We conclude that the recent flux emergence is not a necessary and/or sufficient condition for strong flaring of an AR. The flux emergence rate of analyzed here flare-productive ARs was compared with that of flare-quiet ARs analyzed in our previous studies. We revealed that the flare-productive ARs tend to display faster emergence than the flare-quiet ones do.Comment: 18 pages, 8 figure

Statistical analysis of the total magnetic flux decay rate in solar active regions

We used line-of-sight magnetograms acquired by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory to derive the decay rate of total unsigned magnetic flux for 910 ephemeral and active regions (ARs) observed between 2010 and 2017. We found that: i) most of the ARs obey the power law dependence between the peak magnetic flux and the magnetic flux decay rate, $DR$, so that $DR\sim \Phi^{0.70}$; ii) larger ARs lose smaller fraction of their magnetic flux per unit of time than the smaller ARs; iii) there exists a cluster of ARs exhibiting significantly lower decay rate than it would follow from the power law and all of them are unipolar sunspots with total fluxes in the narrow range of $(2 - 8) \times 10^{21}$ Mx; iv) a comparison with our previous results shows that the emergence rate is always higher than the decay rate. The emergence rate follows a power law with a shallower slope than the slope of the decay-rate power law. The results allowed us to suggest that not only the maximum total magnetic flux determines the character of the decaying regime of the AR, some of the ARs end up as a slowly decaying unipolar sunspot; there should be certain physical mechanisms to stabilize such a sunspot