1. ON AVCS WITH QUADRATIC CONSTRAINTS Farzin Haddadpour Joint work with Madhi Jafari Siavoshani, Mayank Bakshi and Sidharth Jaggi Sharif University of Technology, Iran ISSL, EE Department 2013 ISIT July 7, 2013 Institute of Network Coding The Chinese University of Hong Kong

2. Outline 1/18 Introduction System Model Relation with Prior Works Main Result Proof Steps Conclusion

3. Introduction 2/18 Goal: decode message Goal: interrupt Alice’s information of their movement Goal: transmit reliably How can I interrupt this transmission ? Alice Willie Bob

4. System Model 3/18 Enc Dec Power Constraints: : i.i.d. Gaussian Vector

5. Prior Works 4/18 [Hughes and Narayan 1988] Enc Dec Jammer Shared common randomness Message Aware Jammer Capacity Rate:

6. Prior Works 5/18 [Csizar and Narayan 1991] Capacity Rate: if otherwise Enc Dec Jammer

7. Our Model 6/18 Enc Dec Jammer Enc Dec Jammer Shared common randomness Message Aware Jammer

8. Our Model 7/18 Enc Dec Jammer Private randomization Message- aware Jamming Stochastic encoding Public code Message-aware jamming Oblivious adversary

9. Main Result 8/18 Enc Dec Jammer Private randomization Message- aware Jamming if otherwise Theorem (Capacity Rate):

10. Achievability Proof 9/18 Codebook : Error No Error Intuition : Because of our error probability we take average over colored row otherwise Csizar’s approach which has averaging over whole codewords Note: Decoder uses ML decoding if for if no such exists

11. Achievability Proof 10/18 Based on this Criteria error probability is: for some and Lemma1: fix vector then for every and uniformly distributed over for large if

12. Achievability Proof (Lemma1) Proof of Lemma 1 : Lemma A1 [Csizar and Narayan 1991] : Let be arbitrary r.v.’s and be arbitrary function with then the condition a,s, implies 1.Using Lemma A1 and taking we have 11/18 for some and

13. Achievability Proof (Lemma1) 2. So it remains to bound Where (a) follows by. 12/18

14. Achievability Proof (Lemma1) 13/18 Then terms (1) and (2) can be upper bounded using this Lemma. Lemma [Csizar and Narayan 1991]: u is a fix vector and U is distributed uniformly over sphere and for have

15. Achievability Proof (Lemma2) 14/18 Lemma 2(Quantizing Adversarial Vector): for a fixed vector, sufficient small and for every there exists a fixed codebook with rate which also does well for every. Proof of Lemma 2: choosing where is a random vector over unit sphere and, then we can show that

16. Achievability Proof (Lemma3) 15/18 Lemma 3(Codebook Existence): For every and enough large, there exist a fixed codebook with rate such that for every vector, and every transmitted message : Proof of Lemma 3: It’s enough to show that But using Lemma 2 we don’t need to check for every but only for that covers, therefore we can write Union bound

17. Achievability Proof (Lemma3) Consider this figure for upper bounding the Cardinality of 16/18

18. Conclusion 17/18 Such as Discrete Scenarios Using Stochastic Encoder won’t Improve Capacity Region

19. THANKS FOR CONSIDERATION Any Questions?