  The Quantum Key Distribution (QKD) protocol BB84 has been proven secure
against several important types of attacks: the collective attacks and the
joint attacks. Here we analyze the security of a modified BB84 protocol, for
which information is sent only in the z basis while testing is done in both the
z and the x bases, against collective attacks. The proof follows the framework
of a previous paper (Boyer, Gelles, and Mor, 2009), but it avoids the classical
information-theoretical analysis that caused problems with composability. We
show that this modified BB84 protocol is as secure against collective attacks
as the original BB84 protocol, and that it requires more bits for testing.
