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An observation on associative oneway functions in complexity theory
 Information Processing Letters
, 1997
"... Abstract We introduce the notion of associative oneway functions and prove that they exist if and only if P 6 = NP. As evidence of their utility, we present two novel protocols that apply strong forms of these functions to achieve secret key agreement and digital signatures. ..."
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Cited by 11 (0 self)
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Abstract We introduce the notion of associative oneway functions and prove that they exist if and only if P 6 = NP. As evidence of their utility, we present two novel protocols that apply strong forms of these functions to achieve secret key agreement and digital signatures.
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 ..."
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Cited by 7 (4 self)
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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.
If P != NP then Some Strongly Noninvertible Functions are Invertible
 IN PROCEEDINGS OF THE 13TH INTERNATIONAL SYMPOSIUM ON FUNDAMENTALS OF COMPUTATION THEORY
, 2000
"... Rabi, Rivest, and Sherman alter the standard notion of noninvertibility to a new notion they call strong noninvertibility, and show  via explicit cryptographic protocols for secretkey agreement ([RS93,RS97] attribute this to Rivest and Sherman) and digital signatures [RS93,RS97]  that strong ..."
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Cited by 4 (2 self)
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Rabi, Rivest, and Sherman alter the standard notion of noninvertibility to a new notion they call strong noninvertibility, and show  via explicit cryptographic protocols for secretkey agreement ([RS93,RS97] attribute this to Rivest and Sherman) and digital signatures [RS93,RS97]  that strongly noninvertible functions would be very useful components in protocol design. Their denition of strong noninvertibility has a small twist (\respecting the argument given") that is needed to ensure cryptographic usefulness. In this paper, we show that this small twist has a large, unexpected consequence: Unless P = NP, some strongly noninvertible functions are invertible.
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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Cited by 2 (2 self)
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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
A new cryptosystem based on hidden order groups. Cryptology ePrint Archive, Report 2006/178, 2006. Available at http://eprint.iacr.org/2006/178.pdf
"... Let G1 be a cyclic multiplicative group of order n. It is known that the DiffieHellman problem is random selfreducible in G1 with respect to a fixed generator g if φ(n) is known. That is, given g, g x ∈ G1 and having oracle access to a ‘DiffieHellman Problem ’ solver with fixed generator g, it is ..."
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Let G1 be a cyclic multiplicative group of order n. It is known that the DiffieHellman problem is random selfreducible in G1 with respect to a fixed generator g if φ(n) is known. That is, given g, g x ∈ G1 and having oracle access to a ‘DiffieHellman Problem ’ solver with fixed generator g, it is possible to compute g 1/x ∈ G1 in polynomial time (see theorem 3.2). On the other hand, it is not known if such a reduction exists when φ(n) is unknown (see conjuncture 3.1). We exploit this “gap” to construct a cryptosystem based on hidden order groups and present a practical implementation of a novel cryptographic primitive called an Oracle Strong Associative OneWay Function (OSAOWF). OSAOWFs have applications in multiparty protocols. We demonstrate this by presenting a key agreement protocol for dynamic adhoc groups. 1
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
Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms
"... Abstract This is a collection of one page abstracts of recent results of interest to the Complexity community. The purpose of this document is to spread this information, not to judge the truth or interest of the results therein. 1 ..."
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Abstract This is a collection of one page abstracts of recent results of interest to the Complexity community. The purpose of this document is to spread this information, not to judge the truth or interest of the results therein. 1
Creating Strong, Total, Commutative, Associative . . .
, 1999
"... Rabi and Sherman 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 buildi ..."
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Rabi and Sherman 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. Although Rabi and Sherman did prove that associative oneway functions exist if P != 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 != NP then strong, total, commutative, associative oneway functions exist.