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The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself 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 will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
New Publickey Cryptosystem Using Braid Groups
 Advances in cryptology—CRYPTO 2000 (Santa Barbara, CA), 166–183, Lecture Notes in Comput. Sci. 1880
, 2000
"... Abstract. The braid groups are infinite noncommutative groups naturally arising from geometric braids. The aim of this article is twofold. One is to show that the braid groups can serve as a good source to enrich cryptography. The feature that makes the braid groups useful to cryptography includes ..."
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Cited by 98 (4 self)
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Abstract. The braid groups are infinite noncommutative groups naturally arising from geometric braids. The aim of this article is twofold. One is to show that the braid groups can serve as a good source to enrich cryptography. The feature that makes the braid groups useful to cryptography includes the followings: (i) The word problem is solved via a fast algorithm which computes the canonical form which can be efficiently manipulated by computers. (ii) The group operations can be performed efficiently. (iii) The braid groups have many mathematically hard problems that can be utilized to design cryptographic primitives. The other is to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in the braid groups. The efficiency of our systems is demonstrated by their speed and information rate. The security of our systems is based on topological, combinatorial and grouptheoretical problems that are intractible according to our current mathematical knowledge. The foundation of our systems is quite different from widely used cryptosystems based on number theory, but there are some similarities in design. Key words: public key cryptosystem, braid group, conjugacy problem, key exchange, hard problem, noncommutative group, oneway function, public key infrastructure 1
The rise and fall of knapsack cryptosystems
 In Cryptology and Computational Number Theory
, 1990
"... ..."
The Foundations of Modern Cryptography
, 1998
"... In our opinion, the Foundations of Cryptography are the paradigms, approaches and techniques used to conceptualize, define and provide solutions to natural cryptographic problems. In this essay, we survey some of these paradigms, approaches and techniques as well as some of the fundamental result ..."
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In our opinion, the Foundations of Cryptography are the paradigms, approaches and techniques used to conceptualize, define and provide solutions to natural cryptographic problems. In this essay, we survey some of these paradigms, approaches and techniques as well as some of the fundamental results obtained using them. Special effort is made in attempt to dissolve common misconceptions regarding these paradigms and results. c flCopyright 1998 by Oded Goldreich. Permission to make copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that new copies bear this notice and the full citation on the first page. Abstracting with credit is permitted. A preliminary version of this essay has appeared in the proceedings of Crypto97 (Springer's Lecture Notes in Computer Science, Vol. 1294). 0 Contents 1 Introduction 2 I Basic Tools 6 2 Central Paradigms 6 2.1 Computati...
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 12 (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.
Enhancing privacy through negative representations of data
, 2004
"... The paper introduces the concept of a negative database, in which a set of records DB is represented by its complement set. That is, all the records not in DB are represented, and DB itself is not explicitly stored. After introducing the concept, several results are given regarding the feasibility o ..."
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Cited by 12 (9 self)
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The paper introduces the concept of a negative database, in which a set of records DB is represented by its complement set. That is, all the records not in DB are represented, and DB itself is not explicitly stored. After introducing the concept, several results are given regarding the feasibility of such a scheme and its potential for enhancing privacy. It is shown that a database consisting of n, lbit records can be represented negatively using only O(ln) records. It is also shown that membership queries for DB can be processed against the negative representation in time no worse than linear in its size and that reconstructing the database DB represented by a negative database NDB given as input is an NPhard problem when time complexity is measured as a function of the size of NDB.
Associative oneway functions: A new paradigm for secretkey agreement and digital signatures
, 1993
"... Abstract We propose associative oneway functions as a new cryptographic paradigm for exchanging secret keys and for signing digital documents. First, we precisely define these functions and establish some of their basic properties. Next, generalizing a theorem of Selman, we constructively prove tha ..."
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Abstract We propose associative oneway functions as a new cryptographic paradigm for exchanging secret keys and for signing digital documents. First, we precisely define these functions and establish some of their basic properties. Next, generalizing a theorem of Selman, we constructively prove that they exist if and only if P 6 = NP. In addition, we exhibit an implementation based on integer multiplication. We present a novel protocol that enables two parties to agree on a secret key, and we discuss the security of this protocol. Finally, we generalize our protocol to enable two or more parties to agree on a secret key, and we present a similar protocol for signing documents.
Hidden Field Equations HFE and Isomorphisms of Polynomials IP: two new Families of Asymmetric Algorithms
, 1996
"... In #11# T. Matsumoto and H. Imai described a new asymmetric algorithm based on multivariate polynomials of degree twoover a #nite #eld. Then in #14# this algorithm was broken. The aim of this paper is to show that despite this result it is probably possible to use multivariate polynomials of degree ..."
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Cited by 7 (0 self)
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In #11# T. Matsumoto and H. Imai described a new asymmetric algorithm based on multivariate polynomials of degree twoover a #nite #eld. Then in #14# this algorithm was broken. The aim of this paper is to show that despite this result it is probably possible to use multivariate polynomials of degree two in carefully designed algorithms for asymmetric cryptography. In this paper we will give some examples of suchschemes. All the examples that we will give, belong to two large family of schemes: HFE and IP. With HFE we will be able to do encryption, signatures or authentication in an asymmetric way. Moreover HFE #with properly chosen parameters# resist to all known attacks and can be used in order to givevery short asymmetric signatures or very short encrypted messages #of length 128 bits or 64 bits for example#. IP can be used for asymmetric authentications or signatures. IP authentications are zero knowledge. Note 1 : Another title for this paper could be #How to repair MatsumotoImai algorithm with the same kind of public polynomials". Note 2 : This paper is the extended version of the paper with the same title published at Eurocrypt '96. 1
The Security of the Gabidulin Public Key Cryptosystem
, 1996
"... The Gabidulin Public Key Cryptosystem (PKC), like the well known McEliece PKC, is based on error correcting codes, and was introduced as an alternative to the McEliece system with the claim that much smaller codes could be used, resulting in a more practical system. ..."
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The Gabidulin Public Key Cryptosystem (PKC), like the well known McEliece PKC, is based on error correcting codes, and was introduced as an alternative to the McEliece system with the claim that much smaller codes could be used, resulting in a more practical system.