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On the Theory of Average Case Complexity
 Journal of Computer and System Sciences
, 1997
"... This paper takes the next step in developing the theory of average case complexity initiated by Leonid A Levin. Previous works [Levin 84, Gurevich 87, Venkatesan and Levin 88] have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. ..."
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Cited by 124 (6 self)
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This paper takes the next step in developing the theory of average case complexity initiated by Leonid A Levin. Previous works [Levin 84, Gurevich 87, Venkatesan and Levin 88] have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. Our results include: ffl the equivalence of search and decision problems in the context of average case complexity; ffl an initial analysis of the structure of distributionalNP (i.e. NP problems coupled with "simple distributions") under reductions which preserve average polynomialtime; ffl a proof that if all of distributionalNP is in average polynomialtime then nondeterministic exponentialtime equals deterministic exponential time (i.e., a collapse in the worst case hierarchy); ffl definitions and basic theorems regarding other complexity classes such as average logspace. An exposition of the basic definitions suggested by Levin and suggestions for some alternative definitions ...
A personal view of averagecase complexity
 in 10th IEEE annual conference on structure in complexity theory, IEEE computer society press. Washington DC
, 1995
"... The structural theory of averagecase complexity, introduced by Levin, gives a formal setting for discussing the types of inputs for which a problem is dicult. This is vital to understanding both when a seemingly dicult (e.g. NPcomplete) problem is actually easy on almost all instances, and to d ..."
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Cited by 92 (0 self)
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The structural theory of averagecase complexity, introduced by Levin, gives a formal setting for discussing the types of inputs for which a problem is dicult. This is vital to understanding both when a seemingly dicult (e.g. NPcomplete) problem is actually easy on almost all instances, and to determining which problems might be suitable for applications requiring hard problems, such as cryptography. This paper attempts to summarize the state of knowledge in this area, including some \folklore " results that have not explicitly appeared in print. We also try to standardize and unify denitions. Finally, we indicate what we feel are interesting research directions. We hope that this paper will motivate more research in this area and provide an introduction to the area for people new to it.
Average Case Completeness
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1991
"... We explain and advance Levin's theory of average case completeness. In particular, we exhibit examples of problems complete in the average case and prove a limitation on the power of deterministic reductions. ..."
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Cited by 79 (2 self)
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We explain and advance Levin's theory of average case completeness. In particular, we exhibit examples of problems complete in the average case and prove a limitation on the power of deterministic reductions.
Hiding Cliques for Cryptographic Security
 Des. Codes Cryptogr
, 1998
"... We demonstrate how a well studied combinatorial optimization problem may be introduced as a new cryptographic function. The problem in question is that of finding a "large" clique in a random graph. While the largest clique in a random graph is very likely to be of size about 2 log 2 n, it ..."
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Cited by 41 (0 self)
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We demonstrate how a well studied combinatorial optimization problem may be introduced as a new cryptographic function. The problem in question is that of finding a "large" clique in a random graph. While the largest clique in a random graph is very likely to be of size about 2 log 2 n, it is widely conjectured that no polynomialtime algorithm exists which finds a clique of size (1 + ffl) log 2 n with significant probability for any constant ffl ? 0. We present a very simple method of exploiting this conjecture by "hiding" large cliques in random graphs. In particular, we show that if the conjecture is true, then when a large clique  of size, say, (1+2ffl) log 2 n  is randomly inserted ("hidden") in a random graph, finding a clique of size (1 + ffl) log 2 n remains hard. Our result suggests several cryptographic applications, such as a simple oneway function. 1 Introduction Many hard graph problems involve finding a subgraph of an input graph G = (V; E) with a certain propert...
Simple Strategies for Large ZeroSum Games with Applications to Complexity Theory
 In Proceedings of ACM Symposium on Theory of Computing. 734–740
, 1994
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Averagecase computational complexity theory
 Complexity Theory Retrospective II
, 1997
"... ABSTRACT Being NPcomplete has been widely interpreted as being computationally intractable. But NPcompleteness is a worstcase concept. Some NPcomplete problems are \easy on average", but some may not be. How is one to know whether an NPcomplete problem is \di cult on average"? ..."
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Cited by 31 (2 self)
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ABSTRACT Being NPcomplete has been widely interpreted as being computationally intractable. But NPcompleteness is a worstcase concept. Some NPcomplete problems are \easy on average&quot;, but some may not be. How is one to know whether an NPcomplete problem is \di cult on average&quot;? The theory of averagecase computational complexity, initiated by Levin about ten years ago, is devoted to studying this problem. This paper is an attempt to provide an overview of the main ideas and results in this important new subarea of complexity theory. 1
OneWay Functions, Hard on Average Problems, and Statistical ZeroKnowledge Proofs (Extended Abstract)
 IN PROCEEDINGS OF THE 6TH ANNUAL STRUCTURE IN COMPLEXITY THEORY CONFERENCE
, 1991
"... In this paper, we study connections among oneway functions, hard on the average problems, and statistical zeroknowledge proofs. In particular, we show how these three notions are related and how the third notion can be better characterized, assuming the first one. ..."
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Cited by 31 (9 self)
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In this paper, we study connections among oneway functions, hard on the average problems, and statistical zeroknowledge proofs. In particular, we show how these three notions are related and how the third notion can be better characterized, assuming the first one.
The tale of oneway functions
 PROBLEMS OF INFORMATION TRANSMISSION
, 2003
"... The existence of oneway functions (owf) is arguably the most important problem in computer theory. The article discusses and refines a number of concepts relevant to this problem. For instance, it gives the first combinatorial complete owf, i.e., a function which is oneway if any function is. The ..."
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Cited by 26 (0 self)
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The existence of oneway functions (owf) is arguably the most important problem in computer theory. The article discusses and refines a number of concepts relevant to this problem. For instance, it gives the first combinatorial complete owf, i.e., a function which is oneway if any function is. There are surprisingly many subtleties in basic definitions. Some of these subtleties are discussed or hinted at in the literature and some are overlooked. Here, a unified approach is attempted.
Matrix Transformation is Complete for the Average Case
 SIAM JOURNAL ON COMPUTING
, 1995
"... In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the a ..."
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Cited by 21 (1 self)
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In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the average case reduction theory is still at an early stage, and only a few average case complete problems are known. We present the first algebraic problem complete for the average case under a natural probability distribution. The problem is this: Given a unimodular matrix X of integers, a set S of linear transformations of such unimodular matrices and a natural number n, decide if there is a product of n (not necessarily different) members of S that takes X to the identity matrix.
On the time complexity of 2tag systems and small universal Turing machines
 In In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006
, 2006
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