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Engineering of attosecond relativistic electron beams from thin gas jet targets with superintense ultra-short laser pulses is considered. For perpendicular incidence of the relativistic laser pulse at the gas jet, a longitudinal component of the Lorentz force causes a shift of the plasma electrons in the direction of propagation of the laser pulse and can accelerate them to relativistic velocities. If the pulse amplitude is large enough, and the thickness of the gas jet is reasonable, then full evacuation of the electrons from plasma near the laser beam axis takes place. These electrons can form a relativistic electron bunch, which diameter is determined by the diameter of the laser pulse and length is considerably smaller than the laser wavelength. Such a structure can be called a relativistic electron mirror. The threshold thickness of the gas jet is determined by the amplitude and duration of the laser pulse and the initial density of electrons in the gas jet. Requirements on the front duration of the laser pulse for the case of the gas targets are significantly decreased in comparison with the nanofilm targets; besides, the use of the gas targets facilitates creation of installations with high pulse repetition rate. All these features are very important for experiment. Formation of relativistic electron mirrors from gas jet targets was studied with two-dimensional particle-in-cell simulations. Achievable characteristics of the mirrors were found for different transverse distribution of the gas jet density as well as for the laser pulses of different duration. It was shown that, in all considered cases, it is possible to choose the system parameters (pulse amplitude, density and thickness of the gas jet) in such a way that a single relativistic electron mirror can be formed. Thomson backscattering of a counter-streaming probe laser pulse off such a relativistic electron mirror was also simulated and the parameters of generated coherent ultrashort x-ray pulses were studied.