Locally Decodable Codes of fixed number of queries and Sub-exponential Length

1. Locally Decodable Codesof fixed number of queries and Sub-exponential Length Article By Klim Efremenko Presented by Inon Peled 30 November 2008

2. Intuition: LocallyDecodableCode • LDC’s have applications in cryptography, complexity theory. Example - Public Key Cryptograohy: http://www.cs.ucla.edu/~rafail/PUBLIC/95.pdf

3. Structure of Presentation • The rest of the presentation is organized as follows: • Formal definition for LDC. • Example LDC: Hadamard code. • Construction of LDC’s with fixed #queries and sub-exp. codeword length. • Construction of such binary LDC’s. 3

4. Definition: LocallyDecodableCode

5. Definition: LocallyDecodableCode

6. Next Topic • Formal definition for LDC. • Example LDC: Hadamard code. • Construction of LDC’s with fixed #queries and sub-exp. codeword length. • Construction of such binary LDC’s. 6

7. Example: Hadamard Code 7

8. Example: Hadamard Code 8

9. Because in every non-zero codeword, exactly half of the letters are 1. Example: Hadamard Code 9

10. Example: Hadamard Code - decoding 10

11. Example: Hadamard Code - completeness 11

12. Example: Hadamard Code - parameters 12

13. Hadamard code has fixed num. queries and codeword length exponential in length of message. • Next ,we construct LDC’s with fixed num. queries andsub-exp. length – the main theme of this presentation. Non-adaptive, Linear • • The Hadamard code, as well as every code • that we will present later is: • • Non-adaptive: makes all queries at once. • • And so cannot adapt its queries one after another. • • Linear: a linear transformation.

14. Next Topic • Formal definition for LDC. • Example LDC: Hadamard code. • Construction of LDC’s with fixed #queries and sub-exp. codeword length. • Construction of such binary LDC’s. 14

15. Stages of Constructing our LDC The construction of our LDC begins with fixing a constant m, such that: 15

16. To continue the construction, we must first introducea couple of definitions: Stages of Constructing our LDC The construction of our LDC begins with fixing a constant m, such that: 16

17. Definition 1: S-Matching Vectors

18. Definition 1: S-Matching Vectors

19. Definition 1: S-Matching Vectors 19

20. Example: S-Matching Vectors 20

21. Definition 2: γ, group generator

22. Definition 2: γ, group generator 22

23. Recap

24. At last, the LDC ! 24

25. At last, the LDC ! 25

26. At last, the LDC ! 26

27. Decoding 27

28. Definition: S-decoding Polynomial

29. Example: S-decoding Polynomial 29

30. TheLDC,Decoding

31. TheLDC,Decoding 31

32. Perfectly Smooth Decoder

33. Perfectly Smooth Decoder, Cont.

34. Success Probability of C

35. Success Probability of C 35

36. A (3,δ,3δ)-LDC 36

37. Building {ui}, h, n

38. Theorem, Grolmusz 2000

39. Theorem, Grolmusz 2000 39

40. Grolmusz  {ui}

41. Grolmusz  n, h 41

42. Next Topic • Formal definition for LDC. • Example LDC: Hadamard code. • Construction of LDC’s with fixed #queries and sub-exp. codeword length. • Construction of such binary LDC’s. 42

43. Extension to Binary LDC’s

44. Binary LDC - Encoding

45. Binary LDC - Encoding 45

46. Binary LDC’s - Decoding

47. Binary LDC’s - Decoding 47

48. Completeness of dibin

49. Smoothness of dibin 49

50. Smoothness of dibin – Cont. 50