  This article is a supplement to our recent one about the analysis of the
noise properties in the Kirchhoff-Law-Johnson-Noise (KLJN) secure key exchange
system [Gingl and Mingesz, PLOS ONE 9 (2014) e96109,
doi:10.1371/journal.pone.0096109]. Here we use purely mathematical statistical
derivations to prove that only normal distribution with special scaling can
guarantee security. Our results are in agreement with earlier physical
assumptions [Kish, Phys. Lett. A 352 (2006) 178-182, doi:
10.1016/j.physleta.2005.11.062]. Furthermore, we have carried out numerical
simulations to show that the communication is clearly unsecure for improper
selection of the noise properties. Protection against attacks using time and
correlation analysis is not considered in this paper.
