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We present a general theorem for the efficient verification of the lowerbound of single-photon transmittance. We show how to do decoy-state quantum keydistribution efficiently with large random errors in the intensity control. Inour protocol, the linear terms of fluctuation disappear and only the quadraticterms take effect. We then show the unconditional security of decoy-statemethod with whatever error pattern in intensities of decoy pulses and signalpulses provided that the intensity of each decoy pulse is less than $\mu$ andthe intensity of each signal pulse is larger than $\mu'$

Decoy-state quantum key distribution with large random errors of light intensity