  We consider the Bennett-Brassard cryptographic scheme, which uses two
conjugate quantum bases. An eavesdropper who attempts to obtain information on
qubits sent in one of the bases causes a disturbance to qubits sent in the
other basis. We derive an upper bound to the accessible information in one
basis, for a given error rate in the conjugate basis. Independently fixing the
error rate in the conjugate bases, we show that both bounds can be attained
simultaneously by an optimal eavesdropping probe, consisting of two qubits. The
qubits' interaction and their subsequent measurement are described explicitly.
These results are combined to give an expression for the optimal information an
eavesdropper can obtain for a given average disturbance when her interaction
and measurements are performed signal by signal. Finally, the relation between
quantum cryptography and violations of Bell's inequalities is discussed.
