Gaussian wiretap channel with shared keys between transmitter and helpers

We study the secure degrees of freedom (d.o.f.) of helper-assisted Gaussian wiretap channel with shared key between the transmitter and the helper. Given that the rate of the key scales with power as γ/2 log SNR, we show that secure d.o.f. is min{1+γ / 2, 1}. The achievability proof combines real interference alignment with the artificial noise transmission technique. Using the shared key we sample common artificial noise symbols from a PAM constellation at the transmitter and the helper and transmit them in the null space of the legitimate receiver's channel. We further sample independent noise symbols, also from the same PAM constellation, at the transmitter and the helper and align these symbols at the legitimate receiver. The noise symbols together occupy sufficient dimensions to mask the message at the eavesdropper. A converse proof, which extends the technique of Xie and Ulukus to incorporate common randomness, establishes the optimality of secure d.o.f..We also extend the result to the case with M helpers with two different assumptions on the key sharing structure: the transmitter shares a common key or distinct keys with the M helpers. The exact secure d.o.f.s for these two cases, before they saturate to 1, are proved to be M+γ /M+1 and M+Mγ/M+1 respectively.