Security of quantum key distribution with discrete rotational symmetry

We prove the unconditional security of quantum key distribution protocols using attenuated laser pulses with M different linear polarizations. When M=4, the proof covers the so-called SARG04 protocol [V.~Scarani et al., Phys. Rev.\ Lett. {\bf 92}, 057901 (2004)], which uses exactly the same quantum communication as the Bennett-Brassard 1984 protocol. For a channel with transmission $\eta$, we show that the key rate in SARG04 scales as $O(\eta^{3/2})$. When we increase the number of states to M=2k-1 or 2k, the key rate scaling improves as $O(\eta^{(k+1)/k})$.