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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
Combinatorially Based Cryptography for Children (and Adults)
, 2000
"... In this paper we show how certain notions of modern cryptography can be presented to youngsters using combinatorial constructions. Among the topics discussed are the use of Boolean circuits for bit commitment protocols and hash functions, and the construction of a public key message transmission sys ..."
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In this paper we show how certain notions of modern cryptography can be presented to youngsters using combinatorial constructions. Among the topics discussed are the use of Boolean circuits for bit commitment protocols and hash functions, and the construction of a public key message transmission system using perfect codes in a graph. We also discuss how efforts such as this in popularizing mathematics for children are related to mathematics education reform.
A Status Report on the P versus NP Question
"... We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation. ..."
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We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.
PostQuantum Signatures
, 2004
"... Digital signatures have become a key technology for making the Internet and other IT infrastructures secure. But in 1994 Peter Shor showed that quantum computers can break all digital signature schemes that are used today and in 2001 Chuang and his coworkers implemented Shor’s algorithm for the firs ..."
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Digital signatures have become a key technology for making the Internet and other IT infrastructures secure. But in 1994 Peter Shor showed that quantum computers can break all digital signature schemes that are used today and in 2001 Chuang and his coworkers implemented Shor’s algorithm for the first time on a 7qubit NMR quantum computer. This paper studies the question: What kind of digital signature algorithms are still secure in the age of quantum computers? 1 1
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.
Towards Foundations Of Cryptography: Investigation Of Perfect Secrecy
, 1996
"... In the spirit of Shannon's theory of secrecy systems we analyse several possible natural definitons of the notion of perfect secrecy; these definitions are based on arguments taken from probability theory, information theory, the theory of computational complexity, and the theory of programsiz ..."
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In the spirit of Shannon's theory of secrecy systems we analyse several possible natural definitons of the notion of perfect secrecy; these definitions are based on arguments taken from probability theory, information theory, the theory of computational complexity, and the theory of programsize complexity or algorithmic information. It turns out that none of these definitions models the intuitive notion of perfect secrecy completely: Some fail because a cryptographic system with weak keys can be proven to achieve perfect secrecy in their framework; others fail, because a system which, intuitively, achieves perfect secrecy cannot be proven to do so in their framework. To present this analysis we develop a general formal framework in which to express and measure secrecy aspects of information transmission systems. Our analysis leads to a clarification of the intuition which any definition of the notion of perfect secrecy should capture and the conjecture, that such a definition may be i...
Cryptography from tensor problems
, 2012
"... We describe a new proposal for a trapdoor oneway function. The new proposal belongs to the “multivariate quadratic” family but the trapdoor is different from existing methods, and is simpler. Known quantum algorithms do not appear to help an adversary attack this trapdoor. (Beyond the asymptotic ..."
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We describe a new proposal for a trapdoor oneway function. The new proposal belongs to the “multivariate quadratic” family but the trapdoor is different from existing methods, and is simpler. Known quantum algorithms do not appear to help an adversary attack this trapdoor. (Beyond the asymptotic squarerootspeedup which applies to all oracle search problems.)