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21
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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Cited by 188 (0 self)
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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 Two Faces of Lattices in Cryptology
, 2001
"... Lattices are regular arrangements of points in ndimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated LenstraLenstra Lov'asz lattice basis reduction algorithm twenty years ago, lattices have had surprising ..."
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Cited by 69 (16 self)
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Lattices are regular arrangements of points in ndimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated LenstraLenstra Lov'asz lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only negative, as lattices were used to break various cryptographic schemes. Paradoxically, several positive cryptographic applications of lattices have emerged in the past five years: there now exist publickey cryptosystems based on the hardness of lattice problems, and lattices play a crucial role in a few security proofs.
Lattice Reduction: a Toolbox for the Cryptanalyst
 Journal of Cryptology
, 1994
"... In recent years, methods based on lattice reduction have been used repeatedly for the cryptanalytic attack of various systems. Even if they do not rest on highly sophisticated theories, these methods may look a bit intricate to the practically oriented cryptographers, both from the mathematical ..."
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Cited by 55 (7 self)
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In recent years, methods based on lattice reduction have been used repeatedly for the cryptanalytic attack of various systems. Even if they do not rest on highly sophisticated theories, these methods may look a bit intricate to the practically oriented cryptographers, both from the mathematical and the algorithmic point of view. The aim of the present paper is to explain what can be achieved by lattice reduction algorithms, even without understanding of the actual mechanisms involved. Two examples are given, one of them being the attack devised by the second named author against Knuth's truncated linear congruential generator, which has been announced a few years ago and appears here for the first time in journal version.
The rise and fall of knapsack cryptosystems
 In Cryptology and Computational Number Theory
, 1990
"... ..."
Generalized compact knapsacks, cyclic lattices, and efficient oneway functions
 In STOC
, 2007
"... We investigate the averagecase complexity of a generalization of the compact knapsack problem to arbitrary rings: given m (random) ring elements a1,..., am ∈ R and a (random) target value b ∈ R, find coefficients x1,..., xm ∈ S (where S is an appropriately chosen subset of R) such that P ai · xi = ..."
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Cited by 47 (8 self)
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We investigate the averagecase complexity of a generalization of the compact knapsack problem to arbitrary rings: given m (random) ring elements a1,..., am ∈ R and a (random) target value b ∈ R, find coefficients x1,..., xm ∈ S (where S is an appropriately chosen subset of R) such that P ai · xi = b. We consider compact versions of the generalized knapsack where the set S is large and the number of weights m is small. Most variants of this problem considered in the past (e.g., when R = Z is the ring of the integers) can be easily solved in polynomial time even in the worst case. We propose a new choice of the ring R and subset S that yields generalized compact knapsacks that are seemingly very hard to solve on the average, even for very small values of m. Namely, we prove that for any unbounded function m = ω(1) with arbitrarily slow growth rate, solving our generalized compact knapsack problems on the average is at least as hard as the worstcase instance of various approximation problems over cyclic lattices. Specific worstcase lattice problems considered in this paper are the shortest independent vector problem SIVP and the guaranteed distance decoding problem GDD (a variant of the closest vector problem, CVP) for approximation factors n 1+ǫ almost linear in the dimension of the lattice. Our results yield very efficient and provably secure oneway functions (based on worstcase complexity assumptions) with key size and time complexity almost linear in the security parameter n. Previous constructions with similar security guarantees required quadratic key size and computation time. Our results can also be formulated as a connection between the worstcase and averagecase complexity of various lattice problems over cyclic and quasicyclic lattices.
Lattice Reduction in Cryptology: An Update
 Lect. Notes in Comp. Sci
, 2000
"... Lattices are regular arrangements of points in space, whose study appeared in the 19th century in both number theory and crystallography. ..."
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Cited by 36 (7 self)
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Lattices are regular arrangements of points in space, whose study appeared in the 19th century in both number theory and crystallography.
A Knapsack Type Public Key Cryptosystem Based On Arithmetic in Finite Fields
 IEEE Trans. Inform. Theory
, 1988
"... { A new knapsack type public key cryptosystem is introduced. The system is based on a novel application of arithmetic in nite elds, following a construction by Bose and Chowla. By appropriately choosing the parameters, one can control the density of the resulting knapsack, which is the ratio between ..."
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Cited by 35 (2 self)
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{ A new knapsack type public key cryptosystem is introduced. The system is based on a novel application of arithmetic in nite elds, following a construction by Bose and Chowla. By appropriately choosing the parameters, one can control the density of the resulting knapsack, which is the ratio between the number of elements in the knapsack and their size in bits. In particular, the density can be made high enough to foil \low density" attacks against our system. At the moment, no attacks capable of \breaking" this system in a reasonable amount of time are known. Research supported by NSF grant MCS{8006938. Part of this research was done while the rst author was visiting Bell Laboratories, Murray Hill, NJ. A preliminary version of this work was presented in Crypto 84 and has appeared in [8]. 1 1.
Cryptanalysis of the AjtaiDwork Cryptosystem
 Advances in Cryptology – Crypto ’98, LNCS 1462
, 1998
"... . Recently, Ajtai discovered a fascinating connection between the worstcase complexity and the averagecase complexity of some wellknown lattice problems. Later, Ajtai and Dwork proposed a cryptosystem inspired by Ajtai's work, provably secure if a particular lattice problem is difficult in the wor ..."
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Cited by 20 (2 self)
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. Recently, Ajtai discovered a fascinating connection between the worstcase complexity and the averagecase complexity of some wellknown lattice problems. Later, Ajtai and Dwork proposed a cryptosystem inspired by Ajtai's work, provably secure if a particular lattice problem is difficult in the worstcase. We present a heuristic attack (to recover the private key) against this celebrated cryptosystem. Experiments with this attack suggest that in order to be secure, implementations of the AjtaiDwork cryptosystem would require very large keys, making it impractical in a reallife environment. We also adopt a theoretical point of view: we show that there is a converse to the AjtaiDwork security result, by reducing the question of distinguishing encryptions of one from encryptions of zero to approximating some lattice problems. In particular, this settles the open question regarding the NPhardness of the AjtaiDwork cryptosystem: from a recent result of Goldreich and Goldwasser, our re...