We show that in the context of nonuniform complexity, nondeterministic logarithmic space bounded computation can be made unambiguous. An analogous result holds for the class of problems reducible to context-free languages. In terms of complexity classes, this can be stated as: NL/poly = UL/poly LogCFL/poly = UAuxPDA(log n; n O(1) )/poly

In this work a complete problem for an unambiguous logspace class is presented. This is surprising since unambiguity is a `promise' or `semantic' concept. These usually lead to classes apparently without complete problems. 1 Introduction One of the most central questions of complexity theory is to compare determinism with nondeterminism. Our inability to exhibit the precise relationship between these two features motivates the investigation of intermediate features such as symmetry or unambiguity. In this paper we will concentrate on the notion of unambiguity. Unfortunately, unambiguity of a device or of a language is in general an undecidable property. Unambiguous classes are not defined by a `syntactical' machine property but rather by a `semantical' restriction. A nasty consequence is the apparent lack of complete sets. In the case of time bounded computations there are relativizations of unambiguity which provably have no complete problem ([10]). For space bounded computations t...

Randomizing polynomials allow to represent a function f(x) by a low-degree randomized mapping ˆf(x, r) whose output distribution on an input x is a randomized encoding of f(x). It is known that any function f in uniform-⊕L/poly (and in particular in NC 1) can be efficiently represented by degree-3 randomizing polynomials. Such a degree-3 representation gives rise to an NC 0 4 representation, in which every bit of the output depends on only 4 bits of the input. In this paper, we study the relaxed notion of computationally private randomizing polynomials, where the output distribution of ˆ f(x, r) should only be computationally indistinguishable from a randomized encoding of f(x). We construct degree-3 randomizing polynomials of this type for every polynomial-time computable function, assuming the existence of a cryptographic pseudorandom generator (PRG) in uniform-⊕L/poly. (The latter assumption is implied by most standard intractability assumptions used in cryptography.) This result is obtained by combining a variant of Yao’s garbled circuit technique with previous “information-theoretic ” constructions of randomizing polynomials. We then present the following applications: • Relaxed assumptions for cryptography in NC 0. Assuming a PRG in uniform-⊕L/poly, the

We prove the first lower bounds for restricted read–once parity branching programs with unlimited parity nondeterminism where for each input the variables may be tested according to several orderings. Proving a superpolynomial lower bound for read–once parity branching programs is an important open problem. The following variant is well–motivated more general than variants for that lower bounds are known. Let k be a fixed integer. For each input a there are k orderings σ1(a),...,σk(a) of the variables such that for each computation path activated by a the bits are queried according to σi(a) forsomei, 1 ≤ i ≤ k. Weexamine a slightly more restricted variant, where the collections of allowed orderings have compact representations. Namely, sums of k graph–driven ⊕BP1s with polynomial size graph–orderings are under consideration. We prove lower bounds for the characteristic function of linear codes. We generalize a result by Savick´y and Sieling in [1] and recent results on graph–driven parity branching programs. Key words: read–once parity branching programs, lower bounds, computational complexity 1

We study the parallel time-complexity of basic cryptographic primitives such as one-way functions (OWFs) and pseudorandom generators (PRGs). Specifically, we study the possibility of computing instances of these primitives by NC 0 circuits, in which each output bit depends on a constant number of input bits. Despite previous efforts in this direction, there has been no significant theoretical evidence supporting this possibility, which was posed as an open question in several previous works. We essentially settle this question by providing overwhelming positive evidence for the possibility of cryptography in NC 0. Our main result is that every “moderately easy ” OWF (resp., PRG), say computable in NC 1, can be compiled into a corresponding OWF (resp., low-stretch PRG) in NC 0 4, i.e. whose output bits each depend on at most 4 input bits. The existence of OWF and PRG in NC 1 is a relatively mild assumption, implied by most number-theoretic or algebraic intractability assumptions commonly used in cryptography. Hence, the existence of OWF and PRG in NC 0 follows from a variety of standard assumptions. A similar compiler can also be obtained for other cryptographic primitives such as one-way permutations, encryption, commitment, and collision-resistant hashing. The above results leave a small gap between the possibility of cryptography in NC 0 4 and the known impossibility of implementing even OWF in NC 0 2. We partially close this gap by providing evidence for the existence of OWF in NC 0 3. Finally, our techniques can also be applied to obtain unconditionally provable constructions of non-cryptographic PRGs. In particular, we obtain ɛ-biased generators in NC 0 3, resolving an open question posed by Mossel et al. [25], as well as a PRG for logspace in NC 0. Our results make use of the machinery of randomizing polynomials [19], which was originally motivated by questions in the domain of information-theoretic secure multiparty computation.

Nondeterministic space bounded computation and its unambiguous version have been the focus of attention because of their signi cance in various contexts. In particular, nondeterministic logspace NL has been the the focus of much attention because NL contains many natural computational problems.

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Let G = (V, E) be an undirected graph with a nonnegative edge-weight function w. The routing cost of a spanning tree T of G is ∑ u,v∈V dT (u, v), where dT (u, v) denotes the weight of the simple u-v path in T. The Minimum Routing Cost Spanning Tree (MRCT) problem [WLB+ 00] asks for a spanning tree of G with the minimum routing cost. In this paper, we parallelize several previously proposed approximation algorithms for the MRCT problem and some of its variants. Let ɛ> 0 be an arbitrary constant. When the edge-weight function w is given in unary, we parallelize the (4/3 + ɛ)-approximation algorithm for the MRCT problem [WCT00b] by implementing it using an RN C circuit. There are other variants of the MRCT problem. In the Sum-Requirement Optimal Communication Spanning Tree (SROCT) problem [WCT00a], each vertex u is associated with a requirement r(u) ≥ 0. The objective is to find a spanning tree T of G minimizing ∑ u,v∈V (r(u) + r(v)) dT (u, v). When the edge-weight function w and the vertex-requirement function r are given in unary, we parallelize the 2-approximation algorithm for the SROCT problem [WCT00a] by realizing it using RN C circuits, with a slight degradation in the approximation ratio from 2 to 2 + o(1). In the weighted 2-MRCT problem [Wu02], we have additional inputs s1, s2 ∈ V and λ ≥ 1. The objective is to find a spanning tree T of G minimizing ∑ v∈V λ dT (s1, v) + dT (s2, v). When the edge-weight function w is given in unary, we parallelize the 2-approximation algorithm [Wu02] into RN C circuits, with a slight degradation in the approximation ratio from 2 to 2 + o(1). To the best of our knowledge, our results are the first parallelized approximation algorithms for the MRCT problem and its variants. 1