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The npcompleteness column: Finding needles in haystacks
 ACM Transactions on Algorithms
, 2007
"... Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 197 ..."
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Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at
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 ..."
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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
Querymonotonic Turing Reductions
"... ... A for which any set that Turing reduces to A will also reduceto A via both queryincreasing and querydecreasing Turing reductions. In particular, this holds for all tight paddable sets, where a set is said to be tight paddable exactly if it is paddable via a function whose output length is bou ..."
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... A for which any set that Turing reduces to A will also reduceto A via both queryincreasing and querydecreasing Turing reductions. In particular, this holds for all tight paddable sets, where a set is said to be tight paddable exactly if it is paddable via a function whose output length is bounded tightlyboth from above and from below in the length of the input. We prove that many natural NPcomplete problems such as satisfiability, clique, and vertex cover aretight paddable.
Quantum Computation and Information Project,
, 2004
"... We first give a full characterization of averagecase quantum oneway permutations. Our characterization is an extension of the characterization of worstcase quantum oneway permutations (or, a partial characterization of averagecase quantum oneway permutations) by Kashefi, Nishimura and Vedral. ..."
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We first give a full characterization of averagecase quantum oneway permutations. Our characterization is an extension of the characterization of worstcase quantum oneway permutations (or, a partial characterization of averagecase quantum oneway permutations) by Kashefi, Nishimura and Vedral. As in the previous results, our characterization is also written in terms of reflection operator and pseudo identity. To prove the full characterization of averagecase quantum oneway permutations, we incorporate their basic ideas with the universal hashing technique and modify the reduction between inverting averagecase quantum oneway permutation and another problem appeared in the characterization of worstcase quantum oneway permutations. In a sense, our characterization says that the hardness of inverting quantum oneway permutations comes from the hardness to efficiently implement some reflection operators. 1
Remark on a “NonBreakable Data Encryption” Scheme by
"... We break a cryptosystem by Kish and Sethuraman and show that the authentication problem of their protocol can be fixed. We prove that finding an instance of this cryptosystem, which meets the design criteria, would show that P � = NP. Keywords: Classical information; encryption; public key cryptogra ..."
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We break a cryptosystem by Kish and Sethuraman and show that the authentication problem of their protocol can be fixed. We prove that finding an instance of this cryptosystem, which meets the design criteria, would show that P � = NP. Keywords: Classical information; encryption; public key cryptography; Kish/Sethuraman cipher
Remark on a “NonBreakable Data Encryption” Scheme by
"... We break a cryptosystem by Kish and Sethuraman and show that the authentication problem of their protocol can be fixed. We prove that finding an instance of this cryptosystem, which meets the design criteria, would show that P � = NP. Keywords: Classical information; encryption; public key cryptogra ..."
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We break a cryptosystem by Kish and Sethuraman and show that the authentication problem of their protocol can be fixed. We prove that finding an instance of this cryptosystem, which meets the design criteria, would show that P � = NP. Keywords: Classical information; encryption; public key cryptography; Kish/Sethuraman cipher
Quantum Cryptography: A Survey
, 2005
"... We survey some results in quantum cryptography. After a brief introduction to classical cryptography, we provide the quantummechanical background needed to present some fundamental protocols from quantum cryptography. In particular, we review quantum key distribution via the BB84 protocol and its s ..."
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We survey some results in quantum cryptography. After a brief introduction to classical cryptography, we provide the quantummechanical background needed to present some fundamental protocols from quantum cryptography. In particular, we review quantum key distribution via the BB84 protocol and its security proof, as well as the related quantum bit commitment protocol and its proof of insecurity.
Abstract
, 2005
"... We prove that Psel, the class of all Pselective sets, is EXPimmune, but is not EXP/1immune. That is, we prove that some infinite Pselective set has no infinite EXPtime subset, but we also prove that every infinite Pselective set has some infinite subset in EXP/1. Informally put, the immunity ..."
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We prove that Psel, the class of all Pselective sets, is EXPimmune, but is not EXP/1immune. That is, we prove that some infinite Pselective set has no infinite EXPtime subset, but we also prove that every infinite Pselective set has some infinite subset in EXP/1. Informally put, the immunity of Psel is so fragile that it is pierced by a single bit of information. The above claims follow from broader results that we obtain about the immunity of the Pselective sets. In particular, we prove that for every recursive function f, Psel is DTIME(f)immune. Yet we also prove that Psel is not Π p 2 /1immune. 1