The scattering of a single relativistic electron with few-cycle plane wave laser pulse with intensity of about is theoretically and numerically analyzed in the linear regime, and the radiated energy spectra of electron shows that zeptosecond X-ray pulses can be supported. The influences of the initial carrier-envelope phase offset of the incident few-cycle laser pulses are studied, and the results demonstrate that a single zeptosecond pulse can be produced from scattering by using a single-cycle laser pulse with fixed initial carrier-envelope phase offset . It is discovered that the influence of the initial carrier-envelope phase on the spectrum of the radiation is apparent for low and high frequency of the spectrum, but there is no influence of the central part of the spectrum.