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

2 Intuition: L ocally D ecodable C ode LDC’s have applications in cryptography, complexity theory. Example - Public Key Cryptograohy:

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 Definition: L ocally D ecodable C ode

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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 Example: Hadamard Code

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

10 Example: Hadamard Code - decoding

11 Example: Hadamard Code - completeness

12 Example: Hadamard Code - parameters

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

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 Definition 1: S-Matching Vectors

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20 Example: S-Matching Vectors

21 Definition 2: γ, group generator

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23 Recap

24 At last, the LDC !

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27 Decoding

28 Definition: S-decoding Polynomial

29 Example: S-decoding Polynomial

30 The LDC, Decoding

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32 Perfectly Smooth Decoder

33 Perfectly Smooth Decoder, Cont.

34 Success Probability of C

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36 A (3,δ,3δ)-LDC

37 Building {u i }, h, n

38 Theorem, Grolmusz 2000

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40 Grolmusz  {u i }

41 Grolmusz  n, h

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 Extension to Binary LDC’s

44 Binary LDC - Encoding

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46 Binary LDC ’ s - Decoding

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48 Completeness of d i bin

49 Smoothness of d i bin

50 Smoothness of d i bin – Cont.

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53 LDC Parameters of C bin

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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