1. Compute-and-Forward Can Buy Secrecy Cheap
Parisa Babaheidarian, Somayeh Salimi

2. What is the problem? Gaussian wiretap multiple-access channel
K users wish to communicate with a legitimate receiver
An external eaves-dropper listens to the channel
Channel noise is AWGN
Channel input power constraint per user is P

3. What have been done?
The problem is treated using two different approaches
Gaussian random codes [Tekin & Yener 2008]
Alignment approach
Lower bound on total secure degrees of freedom (s.d.o.f.) [ Bagherikaram et al. 2010]
Optimal secure degrees of freedom [Xie and Ulukus 2012, 2013]
what is s.d.of.?

4. Bounding the minimum distance in ML decoder
How good they perform?
At low SNR random codes do a good job, poor performance at moderate and high SNR regimes!
Alignment schemes : they are optimal at very high SNR regime
Don't have a trivial extension to the finite-SNR regime
Bounding the minimum distance in ML decoder

5. How to fill the gap?
We addressed the finite SNR regime using Compute- and-forward framework
Where did we get the idea?
[Ordentlich et al ] MAC, symmetric setting
[Ntranos et al. 2013] MAC, asymmetric setting

6. Overview of Compute-and-Forward
Messages are encoded into lattice codewords.
The receiver decodes K linearly independent equations of the codewords first and then through successive cancellation it finds the individual transmitted codewords.
The transmitter job:
Choose a proper
fine and coarse
lattice sets
The receiver job:
Choose a proper
equations coeff
and an order of
decoding

7. How the security constraint changes the game?
Using alignment and knowledge of eaves-dropper channel state, we limit the leaked information to the sum of the message codewords , i.e., give only one equation to the eaves-dropper.
Scaling the user codewords
Dither
Coarse lattice

8. Nested lattice structure
Each user has its own pair of fine and coarse lattice sets
Nesting ratio defines the number of
generated codewords.
In our scheme users generate their
codewords using different lattice sets
which satisfy the following nested structure.
[Erez an Zamir 2004]

9. Codebook Construction
Assume the equation rates at which the receiver can decode them with a vanishing error probability are given as
Assign the K computation rates to the K users in a proper order. Each user builds its codebook with the rate as
Fine Lattice
Voronoi region

10. Are we done?
Adding an outer layer which consists of B i.i.d. copies of the inner lattice codewords
Is it necessary?
Only for technical reasons to bound the leakage rate using Packing Lemma!
Since the average leakage over all B blocks goes to zero, at least one copy of the inner code exists with a vanishing leakage rate.
1(n)
Dithering
Scaling
N=n×B
B(n)

11. Any other trick? Bounding the nearest neighbor quantization value
A combination of Sphere packing and Lemma 11 in [Erez and Zamir 2004]
Approximate the uniform distribution with a Gaussian distribution and apply the large deviation theory

12. Achievable result grows with log(P)
A lower bound on the secure sum rate
Then
grows with log(P)

13. OK, what we have bought?
Achievable rates and upper bounds for k=3

14. A special case A reversely degraded channel
Gaussian random codes zero secure sum rate
How about the proposed algorithm?

15. Asymptotic analysis Asymptotic behavior of the achievable scheme
Can we reach to the optimal s.d.o.f?
[Xie and Ulukus 2012]
Yes, an adjustment in the construction of the inner codebook is needed (extension to the presented work)

16. Summary
we proposed a security scheme built on the asymmetric compute-and-forward framework, which works at any finite SNR.
The achievable secure sum rate presented in our scheme scales with log(SNR) and therefore, it signficantly outperforms the existing random coding result for the most SNR regimes.
Our presented scheme also achieves a total s.d.o.f. of
This result can be further improved to reach the optimal secure degrees of freedom at high SNR regime.

17. Acknowledgment
The authors would like to thank Bobak Nazer and Prakash Ishwar for their helpful comments and discussions.
Also, the authors would like to thank the anonymous reviewers for helping us improving the paper.

18. Thank you for your attention!
The authors appreciate any feedback or question to be addressed to