  This paper introduces and demonstrates four new statistical attacks against
the Kirchhoff-Law-Johnson-Noise (KLJN) secure key exchange scheme. The attacks
utilize compromised random number generators at Alice's/Bob's site(s). The case
of partial correlations between Alice's/Bob's and Eve's probing noises is
explored, that is, Eve's knowledge of Alice's and Bob's noises is limited but
not zero. We explore the bilateral situation where Eve has partial knowledge of
Alice's and Bob's random number generators. It is shown that in this situation
Eve can crack the secure key bit by taking the highest cross-correlation
between her probing noises and the measured voltage noise in the wire. She can
also crack the secure key bit by taking the highest cross-correlation between
her noise voltages and her evaluation of Alice's/Bob's noise voltages. We then
explore the unilateral situation in which Eve has partial knowledge of only
Alice's random number generator thus only those noises (of Alice and Eve) are
correlated. In this situation Eve can still crack the secure key bit, but for
sufficiently low error probability, she needs to use the whole bit exchange
period for the attack. The security of the KLJN key exchange scheme, similarly
to other protocols, necessitates that the random number generator outputs are
truly random for Eve.
