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.
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