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Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers Kiran Kedlaya Sergey Yekhanin MIT Microsoft Research An Inequality Error Correcting Codes n bit message 0 1 Decoder processes the (corrupted) codeword … 0 1 Adversarial noise N bit codeword 0 010 … 011 0 1 1 0 … 0 0 1 In classical error correcting codes decoder needs to process the whole (corrupted) codeword to recover even a single bit of the original message! Locally Decodable Codes Codes with sub-linear decoding complexity! n bit message 0 1 Decoder reads only k bits … 0 1 Adversarial noise N bit codeword 0 010 … 011 0 1 1 0 … 0 0 1 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. 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. LDCs: progress in bounds • 2-query: Tight bound - Exp(n) [KdW]. Primes • 3-query: Mersenne primes 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. 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? 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. Are Mersenne primes essential? Primes Answer: No. Mersenne numbers with large prime factors are good enough! Large prime factors of Mersenne numbers Mersenne primes 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. 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 kalgebraically 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 log2 t / log log t. [Stewart] LDCs and factors of Mersenne numbers Known P(2t-1) > t log2 t / log log t 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. 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.) 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. 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.) 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. F2t Proof: • Let t be the smallest such that Cp F2 t. Cp • p | 2t-1; • Elements of Cp \ {1} are proper elements of F2 t i.e., for in Cp \ {1}, and f(x) in F2[x], deg f < t: f() = 0. 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 . 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?)