Succinct Representation of Codes with Applications to Testing

Motivated by questions in property testing, we search for linear error-correcting codes that have the "single local orbit" property: i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every "sparse" binary code whose coordinates are indexed by elements of F_{2^n} for prime n, and whose symmetry group includes the group of non-singular affine transformations of F_{2^n} has the single local orbit property. (A code is said to be "sparse" if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known. Our result gives the first short (O(n)-bit, as opposed to the natural exp(n)-bit) description of a low-weight basis for BCH codes. The interest in the "single local orbit" property comes from the recent result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that have the single local orbit property under the affine symmetry group are locally testable. When combined with our main result, this shows that all sparse affine-invariant codes over the coordinates F_{2^n} for prime n are locally testable. If, in addition to n being prime, if 2^n-1 is also prime (i.e., 2^n-1 is a Mersenne prime), then we get that every sparse cyclic code also has the single local orbit. In particular this implies that BCH codes of Mersenne prime length are generated by a single low-weight codeword and its cyclic shifts

Combinatorial Construction of Locally Testable Codes

An error correcting code is said to be locally testable if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a constant number of symbols of the string. While the best known construction of LTCs by Ben-Sasson and Sudan (STOC 2005) and Dinur (J. ACM 54(3)) achieves very e cient parameters, it relies heavily on algebraic tools and on PCP machinery. In this work we present a new and arguably simpler construction of LTCs that is purely combinatorial, does not rely on PCP machinery and matches the parameters of the best known construction. However, unlike the latter construction, our construction is not entirely explicit

Sparse Random Linear Codes are Locally Decodable and Testable

We show that random sparse binary linear codes are locally testable and locally decodable (under any linear encoding) with constant queries (with probability tending to one). By sparse, we mean that the code should have only polynomially many codewords. Our results are the first to show that local decodability and testability can be found in random, unstructured, codes. Previously known locally decodable or testable codes were either classical algebraic codes, or new ones constructed very carefully. We obtain our results by extending the techniques of Kaufman and Litsyn [11] who used the MacWilliams Identities to show that “almost-orthogonal ” binary codes are locally testable. Their definition of almost orthogonality expected codewords to disagree in n 2 ± O(√n) coordi-nates in codes of block length n. The only families of codes known to have this property were the dual-BCH codes. We extend their techniques, and simplify them in the process, to include codes of distance at least n 2 − O(n1−γ) for any γ> 0, provided the number of codewords is O(nt) for some constant t. Thus our results derive the local testability of linear codes from the classical coding theory parameters, namely the rate and the distance of the codes. More significantly, we show that this technique can also be used to prove the “self-correctability” of sparse codes of sufficiently large distance. This allows us to show that random linear codes under linear encoding functions are locally decodable. This ought to be surprising in that the definition of a code doesn’t specify the encoding function used! Our results effectively say that any linear function of the bits of the codeword can be locally decoded in this case

Batch PIR and Labeled PSI with Oblivious Ciphertext Compression

In this paper, we study two problems: oblivious compression and decompression of ciphertexts. In oblivious compression, a server holds a set of ciphertexts with a subset of encryptions of zeroes whose positions are only known to the client. The goal is for the server to effectively compress the ciphertexts obliviously, while preserving the non-zero plaintexts and without learning the plaintext values. For oblivious decompression, the client, instead, succinctly encodes a sequence of plaintexts such that the server may decode encryptions of all plaintexts value, but the zeroes may be replaced with arbitrary values. We present solutions to both problems that construct lossless compressions only 5% more than the optimal minimum using only additive homomorphism. The crux of both algorithms involve embedding ciphertexts as random linear systems that are efficiently solvable. Using our compression schemes, we obtain state-of-the-art schemes for batch private information retrieval (PIR) where a client wishes to privately retrieve multiple entries from a server-held database in one query. We show that our compression schemes may be used to reduce communication by up to 30% for batch PIR in both the single- and two-server settings. Additionally, we study labeled private set intersection (PSI) in the unbalanced setting where one party\u27s set is significantly smaller than the other party\u27s set and each entry has associated data. By utilizing our novel compression algorithm, we present a protocol with 65-88% reduction in communication with comparable computation compared to prior works