Security of two quantum cryptography protocols using the same four qubit states

The first quantum cryptography protocol, proposed by Bennett and Brassard in 1984 (BB84), has been widely studied in the last years. This protocol uses four states (more precisely, two complementary bases) for the encoding of the classical bit. Recently, it has been noticed that by using the same four states, but a different encoding of information, one can define a new protocol which is more robust in practical implementations, specifically when attenuated laser pulses are used instead of single-photon sources [V. Scarani et al., Phys. Rev. Lett. {\bf 92}, 057901 (2004); referred to as SARG04]. We present a detailed study of SARG04 in two different regimes. In the first part, we consider an implementation with a single-photon source: we derive bounds on the error rate $Q$ for security against all possible attacks by the eavesdropper. The lower and the upper bound obtained for SARG04 (Q14.9% respectively) are close to those obtained for BB84 (Q14.6% respectively). In the second part, we consider the realistic source consisting of an attenuated laser and improve on previous analysis by allowing Alice to optimize the mean number of photons as a function of the distance. SARG04 is found to perform better than BB84, both in secret key rate and in maximal achievable distance, for a wide class of Eve's attacks