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

2. 2 Intuition: L ocally D ecodable C ode LDC’s have applications in cryptography, complexity theory. Example - Public Key Cryptograohy: http://www.cs.ucla.edu/~rafail/PUBLIC/95.pdfhttp://www.cs.ucla.edu/~rafail/PUBLIC/95.pdf

3. 3 The rest of the presentation is organized as follows: 1) Formal definition for LDC. 2) Example LDC: Hadamard code. 3) Construction of LDC’s with fixed #queries and sub-exp. codeword length. 4) Construction of such binary LDC’s. Structure of Presentation

4. 4 Definition: L ocally D ecodable C ode

5. 5

6. 6 1) Formal definition for LDC. 2) Example LDC: Hadamard code. 3) Construction of LDC’s with fixed #queries and sub-exp. codeword length. 4) Construction of such binary LDC’s. Next Topic

7. 7 Example: Hadamard Code

8. 8

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

10. 10 Example: Hadamard Code - decoding

11. 11 Example: Hadamard Code - completeness

12. 12 Example: Hadamard Code - parameters

13. 13 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. Hadamard code has fixed num. queries and codeword length exponential in length of message. Next,we construct LDC’s with fixed num. queries and sub-exp. length – the main theme of this presentation.

14. 14 1) Formal definition for LDC. 2) Example LDC: Hadamard code. 3) Construction of LDC’s with fixed #queries and sub-exp. codeword length. 4) Construction of such binary LDC’s. Next Topic

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

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

17. 17 Definition 1: S-Matching Vectors

19. 19

20. 20 Example: S-Matching Vectors

21. 21 Definition 2: γ, group generator

22. 22

23. 23 Recap

24. 24 At last, the LDC !

25. 25

26. 26

27. 27 Decoding

28. 28 Definition: S-decoding Polynomial

29. 29 Example: S-decoding Polynomial

30. 30 The LDC, Decoding

31. 31

32. 32 Perfectly Smooth Decoder

33. 33 Perfectly Smooth Decoder, Cont.

34. 34 Success Probability of C

35. 35

36. 36 A (3,δ,3δ)-LDC

37. 37 Building {u i }, h, n

38. 38 Theorem, Grolmusz 2000

39. 39

40. 40 Grolmusz  {u i }

41. 41 Grolmusz  n, h

42. 42 1) Formal definition for LDC. 2) Example LDC: Hadamard code. 3) Construction of LDC’s with fixed #queries and sub-exp. codeword length. 4) Construction of such binary LDC’s. Next Topic

43. 43 Extension to Binary LDC’s

44. 44 Binary LDC - Encoding

45. 45

46. 46 Binary LDC ’ s - Decoding

47. 47

48. 48 Completeness of d i bin

49. 49 Smoothness of d i bin

50. 50 Smoothness of d i bin – Cont.

51. 51

52. 52

53. 53 LDC Parameters of C bin

54. 54

55. 55 We’ve presented: 1) What a locally decodable code (LDC) is. 2) The famous and popular Hadamard code. 3) How to construct LDC’s with fixed #queries and sub-exp. codeword length. 4) How to extend the construction to binary LDC’s. Summary