On AVCs WITH Quadratic Constraints

1. On AVCs WITH Quadratic Constraints FarzinHaddadpour Joint work with MadhiJafariSiavoshani, MayankBakshiand SidharthJaggi Sharif University of Technology, Iran ISSL, EE Department Institute of Network Coding The Chinese University of Hong Kong 2013 ISIT July 7, 2013

2. Outline • Introduction • System Model • Relation with Prior Works • Main Result • Proof Steps • Conclusion

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

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

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

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

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

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

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

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

11. Achievability Proof • Based on this Criteria error probability is: for some and 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 • Using Lemma A1 and taking • we have for some and for some and for some and

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

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

15. Achievability Proof(Lemma2) 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: choosingwhere is a random vector over unit sphere and , then we can show that

16. Achievability Proof(Lemma3) 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

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

19. Thanks for Consideration Any Questions?