  We consider a modified version of the BB84 quantum key distribution protocol
in which the angle between two different bases are less than $\pi/4$. We show
that the channel parameter estimate becomes the same as the original protocol
with sufficiently many transmitted qubits. On the other hand, the statistical
correlation between bits transmitted in one basis and those received in the
other basis becomes stronger as the angle between two bases becomes narrower.
If the angle is very small, the statistical correlation between bits
transmitted in one basis and those received in the other basis is as strong as
those received in the same basis as transmitting basis, which means that the
modified protocol can generate almost twice as long secret key as the original
protocol, provided that Alice and Bob choose two different bases with almost
the same probability. We also point out that the reverse reconciliation often
gives different amount of secret key to the direct reconciliation over Pauli
channels with our modified protocol.
