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

We show how to do decoy-state quantum key distribution efficiently with large random errors in the intensity control. We present a theorem for efficiently calculating the lower bound of single-photon counts with many undetermined parameters. In the calculation of the single-photon counts of our protocol, the linear terms of the intensity fluctuation disappear and only the quadratic terms take effect. Given that the intensity fluctuation is upper bounded by ±5%, ±10%, and ±15%, the verified lower bound of the percentage of untagged bits from our protocol is as large as 99.7%, 99.0%, and 97.9% of that from an ideal protocol where the light intensity is exactly controlled.