On optimal language compression for sets in PSPACE/poly

We show that if DTIME[2^O(n)] is not included in DSPACE[2^o(n)], then, for every set B in PSPACE/poly, all strings x in B of length n can be represented by a string compressed(x) of length at most log(|B^{=n}|)+O(log n), such that a polynomial-time algorithm, given compressed(x), can distinguish x from all the other strings in B^{=n}. Modulo the O(log n) additive term, this achieves the information-theoretic optimum for string compression. We also observe that optimal compression is not possible for sets more complex than PSPACE/poly because for any time-constructible superpolynomial function t, there is a set A computable in space t(n) such that at least one string x of length n requires compressed(x) to be of length 2 log(|A^=n|).Comment: submitted to Theory of Computing System

On Pseudorandom Encodings

We initiate a study of pseudorandom encodings: efficiently computable and decodable encoding functions that map messages from a given distribution to a random-looking distribution. For instance, every distribution that can be perfectly and efficiently compressed admits such a pseudorandom encoding. Pseudorandom encodings are motivated by a variety of cryptographic applications, including password-authenticated key exchange, “honey encryption” and steganography. The main question we ask is whether every efficiently samplable distribution admits a pseudorandom encoding. Under different cryptographic assumptions, we obtain positive and negative answers for different flavors of pseudorandom encodings, and relate this question to problems in other areas of cryptography. In particular, by establishing a two-way relation between pseudorandom encoding schemes and efficient invertible sampling algorithms, we reveal a connection between adaptively secure multiparty computation for randomized functionalities and questions in the domain of steganography

Compression of samplable sources

We study the compression of polynomially samplable sources. In particular, we give efficient prefix-free compression and decompression algorithms for three classes of such sources (whose support is a subset of ¤¦¥¨§�©��� �). 1. We show how to compress � sources samplable by logspace machines to expected lengt

RESOURCE BOUNDED SYMMETRY OF INFORMATION REVISITED

Abstract. The information contained in a string x about a string y is the difference between the Kolmogorov complexity of y and the conditional Kolmogorov complexity of y given x, i.e., I(x: y) = C(y) − C(y | x). The Kolmogorov–Levin Theorem says that I(x: y) is symmetric up to a small additive term. We investigate if this property also holds for several versions of polynomial time bounded Kolmogorov complexity. We study symmetry of information for some variants of distinguishing complexity CD where CD(x) is the length of a shortest program which accepts x and only x. We show relativized worlds where symmetry of information does not hold in a strong way for deterministic and nondeterministic polynomial time distinguishing complexities CD poly and CND poly. On the other hand, for nondeterministic polynomial time distinguishing complexity with randomness, CAMD poly, we show that symmetry of information holds for most pairs of strings in any set in NP. Our techniques extend work of Buhrman et al. (CCC 2004) on language compression by AM algorithms, and have the following application to the compression of samplable sources, introduced in Trevisan et al. (CCC 2004): any element x in the support of a polynomial time samplable source X can be given a description of size − log Pr[X = x] + O(log 3 n), from which x can be recovered by an AM algorithm