(Astron. Astrophys.). In this paper it is shown that the Alfvén wave turbulence created behind a shock wave propagating along the ambiant magnetic field in a collision-less plasma, accelerates electrons of the high energy tail of the distribution up to mildly relativistic energies. The efficiency of the process depends upon the ratio of the acceleration time by the turbulence to the scattering time of the same turbulence. At high energies the accelerated particles have a power law spectrum, the steepness of which depends upon the turbulent energy.