  Quantum Cryptography uses the counter-intuitive properties of Quantum
Mechanics for performing cryptographic tasks in a secure and reliable way. The
Quantum Key Distribution (QKD) protocol BB84 has been proven secure against
several important types of attacks: collective attacks and 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 a classical
information-theoretical analysis and proves a fully composable security. 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.
