Generalized beam-splitting attack in quantum cryptography with dim coherent states

As it is very difficult to prepare a good approximation of one-photon states, practical quantum cryptography uses highly attenuated laser pulses which can well be represented by coherent states with average photon number below one photon. In such a case more than one photon may appear in some pulses. Thus an eavesdropper has a chance to split the signal and gain some information on the key without disturbing the transmission in a substantial way. In this paper we derive the number of bits an eavesdropper can gain by this sort of attack and the question of Limits on the average number of photons in a pulse is discussed. It is assumed an eavesdropper may have detectors with 100% efficiency, she can store the 'extracted' qubits, she can non-destructively measure the number of photons, and also she is able to perform 'cascade' beam splitting resulting in extraction of just one photon. Besides, it is assumed that she can replace a lossy communication line by a lossless one. (C) 1999 Published by Elsevier Science B.V. All rights reserved.