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Generation of attosecond pulses of incoherent X-ray and gamma radiation through probe laser pulse Thomson scattering off electron bunches formed by an accelerating laser from plasma layers is investigated. When a non-adiabatic laser pulse of relativistic amplitude falls normally at a plasma film of subcritical density, a simultaneous longitudinal (relative to the axis of the laser beam) displacement of all electrons can occur under the action of the longitudinal component of the Lorentz force. With a sufficiently large field amplitude, this force accelerates electrons to relativistic velocities. As a result, a relativistic electron bunch with a diameter of the order of several microns with a thickness of several nanometers can be formed. Using numerical 2D simulations, it is shown that for relatively moderate dimensionless laser amplitudes of the order of 20 (laser power of about 200 TW), relativistic electron bunches with a minimum thickness of about ten nanometers and an electron energy of hundreds of MeV are formed. The characteristics of the Thomson scattering of a probe laser pulse at such electron bunches are investigated. It is shown that the energy of scattered quanta can achieve several MeVs with the brightness of radiation at the level of the best sources currently available.