This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

A desirable property of one-way functions is that they be total, one-to-one, and onto—in other words, that they be permutations. We prove that one-way permutations exist exactly if PaUP-coUP: This provides the first characterization of the existence of one-way permutations based on a complexity-class separation and shows that their existence is equivalent to a number of previously studied complexitytheoretic hypotheses. We study permutations in the context of witness functions of nondeterministic Turing machines. A language is in PermUP if, relative to some unambiguous, nondeterministic, polynomial-time Turingmachine acceptingthe language, the function mappingeach stringto its unique witness is a permutation of the members of the language. We show that, under standard complexity-theoretic assumptions, PermUP is a strict subset of UP. We study SelfNP, the set of all languages such that, relative to some nondeterministic, polynomial-time Turing machine that accepts the language, the set of all witnesses of strings in the language is identical to the language itself. We show that SATASelfNP; and, under standard complexity-theoretic assumptions, SelfNPaNP:

Rabi and Sherman [RS97] presented novel digital signature and unauthenticated secret-key agreement protocols, developed by themselves and by Rivest and Sherman. These protocols use "strong," total, commutative (in the case of multi-party secret-key agreement), associative one-way functions as their key building blocks. Though Rabi and Sherman did prove that associative one-way functions exist if P 6= NP, they left as an open question whether any natural complexity-theoretic assumption is sufficient to ensure the existence of "strong," total, commutative, associative one-way functions. In this paper, we prove that if P 6= NP then "strong," total, commutative, associative one-way functions exist. Keywords: complexity-theoretic one-way functions, associativity. 1 Introduction and Preliminaries Rabi and Sherman [RS97] study associative one-way functions (AOWFs) and show that AOWFs exist exactly if P 6= NP. They also present the notion of strong AOWFs---AOWFs that are hard to invert even ...

This paper studies the power of three types of access to unambiguous computation: nonadaptive access, fault-tolerant access, and guarded access. (1) Though for NP it is known that nonadaptive access has exponentially terse adaptive simulations, we show that UP has no such relativizable simulations: there are worlds in which (k + 1)-truth-table access to UP is not subsumed by k-Turing access to UP. (2) Though fault-tolerant access (i.e., "l-helping" access) to NP is known to be no more powerful than NP itself, we give both structural and relativized evidence that fault tolerant access to UP suffices to recognize even sets beyond UP. Furthermore, we completely characterize, in terms of locally positive reductions, the sets that fault-tolerantly reduce to UP. (3) In guarded access, Grollmann and Selman's natural notion of access to unambiguous computation, a deterministic polynomial-time Turing machine asks questions to a nondeterministic polynomial-time Turing machine in such a way that the nondeterministic machine never accepts ambiguously. In contrast to guarded access, the standard notion of access to unambiguous computation is that of access to a set that is uniformly unambiguous--even for queries that it never will be asked by its questioner, it must be unambiguous. We show that these notions, though the same for nonadaptive reductions, differ for Turing and strong nondeterministic reductions.

A set A is m-reducible (or Karp-reducible) to B iff there is a polynomial-time computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1-equivalent iff each is m-reducible to the other by one-one reductions; ffl p-invertible equivalent iff each is m-reducible to the other by one-one, polynomial-time invertible reductions; and ffl p-isomorphic iff there is an m-reduction from one set to the other that is one-one, onto, and polynomial-time invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1-equivalent sets are p-isomorphic. (c) Every two p-invertible equivalent sets are p-isomorphic. 2 1. Overview If A is m-reducible to B, we usually interpret this to mean that A is computationally no more difficult than B, since a procedure for computing B is easily converted into a procedure for computing A of comparable complexity. In fact, this interpretation is supported by muc...

Rabi and Sherman [RS97] present a cryptographic paradigm based on associative, one-way functions that are strong (i.e., hard to invert even if one of their arguments is given) and total. Hemaspaandra and Rothe [HR99] proved that such powerful one-way functions exist exactly if (standard) one-way functions exist, thus showing that the associative one-way function approach is as plausible as previous approaches. In the present paper, we study the degree of ambiguity of one-way functions. Rabi and Sherman showed that no associative one-way function (over a universe having at least two elements) can be unambiguous (i.e., one-to-one). Nonetheless, we prove that if standard, unambiguous, one-way functions exist, then there exist strong, total, associative, one-way functions that are O(n)-to-one. This puts a reasonable upper bound on the ambiguity. Our other main results are: 1. P 6= FewP if and only if there exists an (n O(1) )-to-one, strong, total AOWF. 2. No O(1)-to-one total, associati...

Abstract. Recently, the property of unambiguity in alternating Turing machines has received considerable attention in the context of analyzing globally-unique games by Aida et al. [1] and in the design of efficient protocols involving globally-unique games by Crâsmaru et al. [7]. This paper investigates the power of unambiguity in alternating Turing machines in the following settings: 1. We construct a relativized world where unambiguity based hierarchies—AUPH, UPH, and UPH—are infinite. We construct another relativized world where UAP (unambiguous alternating polynomialtime) is not contained in the polynomial hierarchy. 2. We define the bounded-level unambiguous alternating solution class UAS(k), for every k ≥ 1, as the class of sets for which strings in the set are accepted unambiguously by some polynomial-time alternating Turing machine N with at most k alternations, while strings not in the set either are rejected or are accepted with ambiguity by N. We construct a relativized world where, for all k ≥ 1, UP≤k ⊂ UP≤k+1 and UAS(k) ⊂ UAS(k + 1). 3. Finally, we show that robustly k-level unambiguous polynomial-time alternating Turing machines accept languages that are computable in P Σp k ⊕A, for every oracle A. This generalizes a result of Hartmanis

Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist one-way functions (i.e., p-time computable, honest, p-time noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve the sufficient condition to P � = NP. More generally, in this paper we completely characterize which types of one-way functions stand or fall together with (plain) one-way functions—equivalently, stand or fall together with P � = NP. We look at the four attributes used in Rabi and Sherman’s seminal work on algebraic properties of one-way functions (see [RS97,RS93]) and subsequent papers—strongness (of noninvertibility), totality, commutativity, and associativity—and for each attribute, we allow it to be required to hold, required to fail, or “don’t care. ” In this categorization there are 3 4 = 81 potential types of one-way functions. We prove that each of these 81 feature-laden types stand or fall together with the existence of (plain) one-way functions. Key words: computational complexity, complexity-theoretic one-way functions, associativity, commutativity, strong noninvertibility.

We study circuit expressions of logarithmic and poly-logarithmic polynomial-time Kolmogorov complexity, focusing on their complexity-theoretic characterizations and learnability properties. They provide a nontrivial circuit-like characterization for a natural nonuniform complexity class that lacked it up to now. We show that circuit expressions of this kind can be learned with membership queries in polynomial time if and only if every NE-predicate is E-solvable. Thus they are learnable given that the learner is allowed the extra use of an oracle in NP. The precise way of accessing the oracle is shown to be optimal under relativization. We present a precise characterization of the subclass defined by Kolmogorov-easy circuit expressions that can be constructed from membership queries in polynomial time, with some consequences for the structure of reduction and equivalence classes of tally sets of very low density. Preliminary, sometimes weaker versions of the results in this paper were...

We investigate complexities of insertion operations on formal languages relatively to complexity classes. In this way, we introduce operations closely related to LOG(CFL) and NP . Our results relativize and give new characterizations of the ways to relativize nondeterministic space.