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Coherent ultrashort x-ray pulses can be generated through Thomson backscattering of a probe laser pulse off a relativistic electron mirror [1]. In this case, parameters of x-ray pulse, such as amplitude, frequency, envelope, phase, etc., can be controlled easily. In this presentation, we consider problems, which should be solved for experimental realization of this idea: shaping of the laser pulse front, generation of relativistic electron mirror and back reflection of a probe laser pulse off such a mirror. An idea for synchronous acceleration of electrons from a nanofilm with a superintense nonadiabatic laser pulse was considered in [2]. For a nonadiabatic laser pulse of relativistic amplitude incident normally at the nanofilm, all electrons can be expelled simultaneously out of the nanofilm in the longitudinal direction (along the laser beam axis) due to the action of the longitudinal component of the Lorentz force. As a result, a relativistic electron mirror can be formed - an electron bunch with diameter of several micrometers and thickness of several nanometers or less. Accelerating laser pulse should be relativistic with amplitude exceeding some threshold, which is determined by material and thickness of the nanofilm, besides the front of the laser pulse should be sharp enough. We consider laser pulse front shaping using plasma layers with thickness of about laser wavelength or more. Such layers allow fully suppressing the part of the laser pulse with small amplitude, and the transmitted pulse maximal amplitude can be about that of the incoming laser pulse. Then, we studied numerically characteristics for relativistic electron mirrors generated with shaped pulses. It was shown that, for a nanofilm with a thickness of several nanometers, a lifetime of relativistic electron mirror can be tens of femtoseconds with good homogeneity and with momenta spread of no more than 1-2 %. At last, the reflections of the counter-propagating probe laser pulses off generated relativistic electron mirrors were investigated. It was shown that a single coherent X-ray pulse with duration of less than 100 as, wavelength of ten nanometers or less and power of several hundreds of gigawatts can be generated. Characteristics for x-ray pulses were investigated such as dependence of the reflected pulse parameters on amplitude and duration of the probe pulse, nanofilm thickness and electron density, etc. It was shown that if the relativistic electron mirror stays in the field of accelerating pulse during interaction with the probe pulse, a reflection coefficient of the mirror and a frequency up-shift coefficient for the probe pulse do not depend on the amplitude of the probe pulse up to that of the accelerating pulse.