  Recently, we have shown the advantages of two-way quantum communications in
continuous variable quantum cryptography. Thanks to this new approach, two
honest users can achieve a non-trivial security enhancement as long as the
Gaussian interactions of an eavesdropper are independent and identical. In this
work, we consider asymmetric strategies where the Gaussian interactions can be
different and classically correlated. For several attacks of this kind, we
prove that the enhancement of security still holds when the two-way protocols
are used in direct reconciliation.
