Non-Malleable Extractors with Short Seeds and Applications to Privacy Amplification

1. Non-Malleable Extractors with Short Seeds and Applications to Privacy Amplification Gil Cohen Weizmann Institute Joint work with Ran Razand Gil Segev

2. Randomness Seeded-Extractors

3. Randomness Seeded-Extractors truly random bits (seed) bits from an imperfect source of randomness Ext almost-truly random bits

4. Randomness Seeded-Extractors If all points has probability , the source is called an-source. truly random bits (seed) bits from an imperfect source of randomness Ext almost-truly random bits For simplicity, think of it as “uniform hidden bits”.

5. Randomness Seeded-Extractors Measured in statistical distance. truly random bits (seed) bits from an imperfect source of randomness Ext almost-truly random bits

6. Strong Seeded-Extractors For any -source and independent is called strongif

7. Parameters Given • Maximize • Minimize Non-Constructive and Optimal [Sips88], [RTS00] truly random bits (seed) bits from an -source Ext almost-truly random bits Almost matching explicit constructions (…,[LRVW03],[GUV07] ,[DW08],[DKSS09]).

8. Defined by[DW09] Non-Malleable Extractors

9. Non-Malleable Extractors with no fixed point.

10. ANotNon-Malleable Extractor Expanders are low-degree undirected graphs that “look random”. Are known to induce extractors. Nodes Labeled neighbors (think of 1-16).

11. ANotNon-Malleable Extractor

12. ANotNon-Malleable Extractor

13. ANotNon-Malleable Extractor

14. ANotNon-Malleable Extractor

15. ANotNon-Malleable Extractor

16. ANotNon-Malleable Extractor

17. ANotNon-Malleable Extractor

18. ANotNon-Malleable Extractor

19. ANotNon-Malleable Extractor

20. ANotNon-Malleable Extractor

21. Non-Constructive[DW09] • Seed length • Output length Compared with strong extractors • Seed length • Output length

22. Non-Constructive[DW09] • Seed length • Output length Compared with strong extractors • Seed length • Output length

23. Non-Constructive[DW09] • Seed length • Output length Compared with strong extractors • Seed length • Output length

24. The Explicit Construction of [DLWZ11] • Conditional efficiency

25. The Explicit Construction of[DLWZ11] • Conditional efficiency

26. The Explicit Construction of[DLWZ11] • Conditional efficiency

27. The Explicit Construction of[DLWZ11] • Conditional efficiency

28. Main Result

29. Main Result • Unconditionally efficient

30. Main Result • Unconditionally efficient

31. Main Result • Unconditionally efficient

32. Main Result • Unconditionally efficient

33. Main Result • Unconditionally efficient

34. Privacy Amplification - passive adversary - [BBR88], [Mau92],[BBCM95]

35. is an -source Alice Bob =? Computationally unbounded! Eve

36. Interesting Measures • Entropy loss • Communication complexity • Number of rounds Alice Bob =? Eve

37. Interesting Measures • Entropy loss • Communication complexity • Number of rounds Alice Bob from Eve’s point of view =? Eve

38. Interesting Measures • Entropy loss • Communication complexity • Number of rounds Alice Bob Number of communicated bits =? Eve

39. Interesting Measures • Entropy loss • Communication complexity • Number of rounds Alice Bob =? Eve

40. Strong Extractors to the Rescue

41. Strong Extractors to the Rescue • Entropy loss • Communication complexity • Number of rounds s

42. Privacy Amplification -active adversary - [Mau97],[MM97],[Wol98],[MW03],[RW03],[DKRS06],[DW09],[KR09],[CKOR10]

43. Privacy Amplification Protocol Active Adversary • Correctness • Privacy • Authenticity

44. Privacy Amplification Protocol Active Adversary • Correctness • Privacy • Authenticity If both parties are honest then they agree.

45. Privacy Amplification Protocol Active Adversary • Correctness • Privacy • Authenticity For any Eve, from Eve’s view.

46. Privacy Amplification Protocol Active Adversary • Correctness • Privacy • Authenticity For any Eve, .

47. If

48. Privacy Amplification Protocols [DS02, DW09] 1 round requires [RW03] gave rounds for [CKOR10] gave rounds for

49. 2 Rounds Privacy Amplification Protocols ()

50. The Extractor of [Raz05]