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Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers

Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers. An Inequality. . Error Correcting Codes. n bit message. Decoder processes the (corrupted) codeword.

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Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers

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  1. Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers

  2. An Inequality

  3. Error Correcting Codes n bit message Decoder processes the (corrupted) codeword In classical error correcting codes decoder needs to process the whole (corrupted) codeword to recover even a single bit of the original message! Adversarial noise N bit codeword

  4. Locally Decodable Codes Codes with sub-linear decoding complexity! Definition: A code C encoding n bits to N bits is called k-LDC if given a (linearly) corrupted codeword one can recover any particular bit of the message (w.h.p.) by reading only k randomly chosen bits. n bit message Decoder reads only k bits Adversarial noise N bit codeword

  5. Locally Decodable Codes • Example: There is a 2-query LDC of length Exp(n). • Major question: What is the length of optimal k-query LDCs? • Applications: • Cryptography (private information retrieval). • Worst-case to average case reductions. • Fault tolerant computation. • Data transmission / storage.

  6. LDCs: progress in bounds Primes • 2-query: Tight bound - Exp(n) [KdW]. • 3-query: Lower bound: - Ω(n2 / log log n) [W]. Upper bounds: - Exp(n1/2) [BIK]. (Polynomial interpolation.) - Exp(n1/t), where 2t-1 is prime [Y]. (Point removal method.) • Exp(n1/32,582,657) - unconditionally. • Exp(no(1)) - if there exist infinitely many Mersenne primes. • Goal: Obtain constant-query LDCs of length Exp(no(1)) unconditionally. Mersenne primes

  7. This work We undertake an in-depth study of the point removal method of [Y] to answer two questions: • Are Mersenne primes essential to the method? • Has the method been pushed to its limit?

  8. Heart of the point removal method • Definition: A set S  Fq is t - combinatorially nice if …. • Definition: A set S  Fq is k - algebraically nice if …. • Theorem: If for some Fq there exists S  Fq such that: • - S is t-combinatorially nice and • - S is k-algebraically nice; • then there exist k-query LDCs of length Exp(n1/t). • Lemma: Let p=2t-1 be a Mersenne prime; then S = {1,2,4,…,2t-1} in Fp is t-combinatorially nice and 3-algebraically nice.

  9. Are Mersenne primes essential? Primes Answer: No. Mersenne numbers with large prime factors are good enough! Theorem: Let  > 0. If P(2t-1) > (2t-1) = p; then {1,2,…,2t-1}  Fp is t-comb. nice and k()-algebr. nice; thus exist k() – query LDCs of length Exp(n1/t). Notation: P(m) = the largest prime factor of m. Large prime factors of Mersenne numbers Mersenne primes

  10. Has the method been pushed to its limit? Answer: Yes. Unless we progress on some old number theory questions. Primes that are somewhat large factors of Mersenne numbers are necessary! Theorem: If for infinitely many t there is an Fq and S  Fq that is k-algebraically nice and t-combinatorially nice; then infinitely often:P(2t-1) > (t/2)1+1/(k-2). The largest function f(t) for that P(2t-1) > f(t) unconditionally infinitely often is: f(t) = t log2t / log log t. [Stewart]

  11. LDCs and factors of Mersenne numbers P(2t-1) > t log2 t / log log t Known P(2t-1) >(t/2)1+1/(k-2) Necessary P(2t-1) >(2t-1) Sufficient P(2t-1)= 2t-1 Goal: Obtain constant-query codes of subexponential length.

  12. About the proof • Mersenne numbers with large prime factors yield nice subsets. • Nice subsets of finite fields yield Mersenne numbers with somewhat large prime factors. (We will see a piece of the second proof.)

  13. Nice subsets to large factors of Mersenne numbers Claim: 3-algebraically nice subsets of prime fields yield large prime factors of Mersenne numbers. Theorem: Suppose S  Fp is 3-algebraically nice; then - p | 2t-1; - p > 0.75 t2.

  14. Proof: two steps • S  Fp is 3-algebraically nice; then there exist 1 2 3 in Cp such that: 1 + 2 + 3 = 0. • There exist 1 2 3 in Cp such that: 1 + 2 + 3 = 0; then p | 2t-1 and p > 0.25 t2. Notation: Cp - the set of p-th roots of unity in F2. (We will go over the second step.)

  15. Proof of the second step - I Lemma: There exist 1 2 3 in Cp such that: 1 + 2 + 3 = 0; then p | 2t-1 and p > 0.25 t2. Proof: • Let t be the smallest such that Cp  F2 . • p | 2t-1; • Elements of Cp\{1} are proper elements of F2 i.e., for  in Cp\{1}, and f(x) in F2[x], deg f < t: f() = 0. F2 t t Cp t

  16. Proof of the second step - II Proof (continued): • Let i denote elements of Cp. • 1 + 2 + 3 = 0; yields 4 = 1 + 5. • 4= 2-1.1 ; 5= 2-1.3 • Fix  in Cp such that (1+ ) is in Cp. • Consider the set Z={a (1 + )b| a,b in [0 ,…, t/2-1]}. • a (1 + )bc (1 + )d else we would have: f() = 0, where deg f < t. Thus, |Z| = (t/2)2 and hence p > (t/2)2 .

  17. Conclusions: • Summary: Further progress on upper bounds for LDCs via point removal method is tied to progress on lower bounds for prime factors of Mersenne numbers. • Hopes: • Progress in number theory problems. • Broader generalizations of the method. (finite rings?)

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