Algorithms and Complexity Seminar
Monday, November 13, 2006, 4-5:15pm in 32-G575.
New Locally Decodable Codes and Private Information Retrieval
Schemes
Sergey Yekhanin (MIT CSAIL)
A q-query Locally Decodable Code (LDC) encodes an n-bit message x as an
N-bit codeword C(x), such that one can probabilistically recover any
bit x_i of the message by querying only q bits of the codeword C(x),
even after some constant fraction of codeword bits has been corrupted.
We give new constructions of three query LDCs of vastly shorter length
than that of previous constructions. Specifically, given any Mersenne
prime p=2^t-1, we design three query LDCs of length N=EXP(n^{1/t}), for
every $n$. Based on the largest known Mersenne prime, this translates
to a length of less than EXP(n^{10^{-7}}), compared to EXP(n^{1/2}) in
the previous constructions. It has often been conjectured that there
are infinitely many Mersenne primes. Under this conjecture, our
constructions yield three query locally decodable codes of length
N=EXP(n^{1/log log n}) for infinitely many n.
We also obtain analogous improvements for Private Information Retrieval
(PIR) schemes. We give 3-server PIR schemes with communication
complexity of O(n^{10^{-7}}) to access an n-bit database, compared to
the previous best scheme with complexity O(n^{1/5.25}). Assuming again
that there are infinitely many Mersenne primes, we get 3-server PIR
schemes of communication complexity n^{O(1/log log n)} for infinitely
many $n$.
Previous families of LDCs and PIR schemes were based on the properties
of low-degree multivariate polynomials over finite fields. Our
constructions are completely different and are obtained by constructing
a large number of vectors in a small dimensional vector space whose
inner products are restricted to lie in an algebraically nice set.
Host: Madhu Sudan
(The Algorithms
and Complexity Seminar series talks are usually held Mondays from
4-5:15pm in 32-G575.)