Results 1  10
of
20
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
Abstract

Cited by 30 (3 self)
 Add to MetaCart
This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete 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 NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Oneway permutations and selfwitnessing languages
 Journal of Computer and System Sciences
, 2003
"... A desirable property of oneway functions is that they be total, onetoone, and onto—in other words, that they be permutations. We prove that oneway permutations exist exactly if PaUPcoUP: This provides the first characterization of the existence of oneway permutations based on a complexityclas ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
A desirable property of oneway functions is that they be total, onetoone, and onto—in other words, that they be permutations. We prove that oneway permutations exist exactly if PaUPcoUP: This provides the first characterization of the existence of oneway permutations based on a complexityclass 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, polynomialtime Turingmachine acceptingthe language, the function mappingeach stringto its unique witness is a permutation of the members of the language. We show that, under standard complexitytheoretic assumptions, PermUP is a strict subset of UP. We study SelfNP, the set of all languages such that, relative to some nondeterministic, polynomialtime 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 complexitytheoretic assumptions, SelfNPaNP:
Creating Strong Total Commutative Associative OneWay Functions from Any OneWay Function
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1998
"... Rabi and Sherman [RS97] presented novel digital signature and unauthenticated secretkey agreement protocols, developed by themselves and by Rivest and Sherman. These protocols use "strong," total, commutative (in the case of multiparty secretkey agreement), associative oneway functions ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
Rabi and Sherman [RS97] presented novel digital signature and unauthenticated secretkey agreement protocols, developed by themselves and by Rivest and Sherman. These protocols use "strong," total, commutative (in the case of multiparty secretkey agreement), associative oneway functions as their key building blocks. Though Rabi and Sherman did prove that associative oneway functions exist if P 6= NP, they left as an open question whether any natural complexitytheoretic assumption is sufficient to ensure the existence of "strong," total, commutative, associative oneway functions. In this paper, we prove that if P 6= NP then "strong," total, commutative, associative oneway functions exist.
Promise Problems and Access to Unambiguous Computation
, 1992
"... This paper studies the power of three types of access to unambiguous computation: nonadaptive access, faulttolerant 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: ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
This paper studies the power of three types of access to unambiguous computation: nonadaptive access, faulttolerant 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)truthtable access to UP is not subsumed by kTuring access to UP. (2) Though faulttolerant access (i.e., "lhelping" 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 faulttolerantly reduce to UP. (3) In guarded access, Grollmann and Selman's natural notion of access to unambiguous computation, a deterministic polynomialtime Turing machine asks questions to a nondeterministic polynomialtime 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 unambiguouseven 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.
Every PolynomialTime 1Degree Collapses iff P = PSPACE
, 1996
"... A set A is mreducible (or Karpreducible) to B iff there is a polynomialtime computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1equivalent iff each is mreducible to the other by oneone reductions; ffl pinvertible equivalent iff each is mreducible to the othe ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
A set A is mreducible (or Karpreducible) to B iff there is a polynomialtime computable function f such that, for all x, x 2 A () f(x) 2 B. Two sets are: ffl 1equivalent iff each is mreducible to the other by oneone reductions; ffl pinvertible equivalent iff each is mreducible to the other by oneone, polynomialtime invertible reductions; and ffl pisomorphic iff there is an mreduction from one set to the other that is oneone, onto, and polynomialtime invertible. In this paper we show the following characterization. Theorem The following are equivalent: (a) P = PSPACE. (b) Every two 1equivalent sets are pisomorphic. (c) Every two pinvertible equivalent sets are pisomorphic. 2 1. Overview If A is mreducible 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...
Low ambiguity in strong, total, associative, oneway functions
, 2000
"... Rabi and Sherman [RS97] present a cryptographic paradigm based on associative, oneway 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 oneway functions exist exactly if (standard) oneway fun ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Rabi and Sherman [RS97] present a cryptographic paradigm based on associative, oneway 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 oneway functions exist exactly if (standard) oneway functions exist, thus showing that the associative oneway function approach is as plausible as previous approaches. In the present paper, we study the degree of ambiguity of oneway functions. Rabi and Sherman showed that no associative oneway function (over a universe having at least two elements) can be unambiguous (i.e., onetoone). Nonetheless, we prove that if standard, unambiguous, oneway functions exist, then there exist strong, total, associative, oneway functions that are O(n)toone. This puts a reasonable upper bound on the ambiguity. Our other main results are: 1. P = FewP if and only if there exists an (n O(1))toone, strong, total AOWF. 2. No O(1)toone total, associative functions exist in Σ ∗ × Σ ∗ → Σ ∗. 3. For every nondecreasing, unbounded, total, recursive function g: N → N, there is a g(n)toone, total, commutative, associative, recursive function in Σ ∗ × Σ ∗ → Σ ∗.
On the power of unambiguity in alternating machines
 In Proceedings of the 15th International Symposium on Fundamentals of Computation Theory
, 2004
"... Abstract. Recently, the property of unambiguity in alternating Turing machines has received considerable attention in the context of analyzing globallyunique games by Aida et al. [1] and in the design of efficient protocols involving globallyunique games by Crâsmaru et al. [7]. This paper investig ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. Recently, the property of unambiguity in alternating Turing machines has received considerable attention in the context of analyzing globallyunique games by Aida et al. [1] and in the design of efficient protocols involving globallyunique 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 boundedlevel 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 polynomialtime 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 klevel unambiguous polynomialtime alternating Turing machines accept languages that are computable in P Σp k ⊕A, for every oracle A. This generalizes a result of Hartmanis
Enforcing and defying associativity, commutativity, totality, and strong noninvertibility for oneway functions in complexity theory
 In ICTCS
, 2005
"... Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist oneway functions (i.e., ptime computable, honest, ptime noninvertible functions) that are total, commutative, and associative but not strongly noninvertible. In this paper we improve th ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Rabi and Sherman [RS97,RS93] proved that the hardness of factoring is a sufficient condition for there to exist oneway functions (i.e., ptime computable, honest, ptime 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 oneway functions stand or fall together with (plain) oneway 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 oneway 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 oneway functions. We prove that each of these 81 featureladen types stand or fall together with the existence of (plain) oneway functions. Key words: computational complexity, complexitytheoretic oneway functions, associativity, 1.1
Circuit Expressions of Low Kolmogorov Complexity
 In preparation
, 1999
"... We study circuit expressions of logarithmic and polylogarithmic polynomialtime Kolmogorov complexity, focusing on their complexitytheoretic characterizations and learnability properties. They provide a nontrivial circuitlike characterization for a natural nonuniform complexity class that lacked ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We study circuit expressions of logarithmic and polylogarithmic polynomialtime Kolmogorov complexity, focusing on their complexitytheoretic characterizations and learnability properties. They provide a nontrivial circuitlike 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 NEpredicate is Esolvable. 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 Kolmogoroveasy 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...
On the power of a unique quantum witness
"... In a celebrated paper, Valiant and Vazirani [29] raised the question of whether the difficulty of NPcomplete problems was due to the wide variation of the number of witnesses of their instances. They gave a strong negative answer by showing that distinguishing between instances having zero or one w ..."
Abstract
 Add to MetaCart
In a celebrated paper, Valiant and Vazirani [29] raised the question of whether the difficulty of NPcomplete problems was due to the wide variation of the number of witnesses of their instances. They gave a strong negative answer by showing that distinguishing between instances having zero or one witnesses is as hard as recognizing NP, under randomized reductions. We consider the same question in the quantum setting and investigate the possibility of reducing quantum witnesses in the context of the complexity class QMA, the quantum analogue of NP. The natural way to quantify the number of quantum witnesses is the dimension of the witness subspace W in some appropriate Hilbert space H. We present an efficient deterministic procedure that reduces any problem where the dimension d of W is bounded by a polynomial to a problem with a unique quantum witness. The main idea of our reduction is to consider the Alternating subspace of the tensor power H ⊗d. Indeed, the intersection of this subspace with W ⊗d is onedimensional, and therefore can play the role of the unique quantum witness.