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