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Closest Point Search in Lattices
 IEEE TRANS. INFORM. THEORY
, 2000
"... In this semitutorial paper, a comprehensive survey of closestpoint search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closestpoint search algorithm, ba ..."
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Cited by 197 (1 self)
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In this semitutorial paper, a comprehensive survey of closestpoint search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closestpoint search algorithm, based on the SchnorrEuchner variation of the Pohst method, is implemented. Given an arbitrary point x 2 R m and a generator matrix for a lattice , the algorithm computes the point of that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan algorithm and an experimental comparison with the Pohst algorithm and its variants, such as the recent ViterboBoutros decoder. The improvement increases with the dimension of the lattice. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, compu...
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 189 (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
On the spheredecoding algorithm I. Expected complexity
 IEEE Trans. Sig. Proc
, 2005
"... Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The ..."
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Cited by 76 (5 self)
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Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NPhard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worstcase complexity of the integer leastsquares problem, we study its expected complexity, averaged over the noise and over the lattice. For the “sphere decoding” algorithm of Fincke and Pohst, we find a closedform expression for the expected complexity, both for the infinite and finite lattice.
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 67 (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.
Hardness of approximating the shortest vector problem in high Lp norms
 In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science. IEEE Computer
"... Abstract. Let p> 1beany fixed real. We show that assuming NP ⊆ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓp norm within a constant factor. Under the stronger assumption NP ⊆ RTIME(2poly(log n)), we show that there is no polynomialtime (log n) ..."
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Cited by 63 (2 self)
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Abstract. Let p> 1beany fixed real. We show that assuming NP ⊆ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓp norm within a constant factor. Under the stronger assumption NP ⊆ RTIME(2poly(log n)), we show that there is no polynomialtime (log n)1/2−ɛ algorithm with approximation ratio 2 where n is the dimension of the lattice and ɛ>0isan arbitrarily small constant. We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product,anew (log n)1/2−ɛ variant of tensor product that we introduce. This enables us to boost the hardness factor to 2.
The shortest vector in a lattice is hard to approximate to within some constant
 in Proc. 39th Symposium on Foundations of Computer Science
, 1998
"... Abstract. We show that approximating the shortest vector problem (in any ℓp norm) to within any constant factor less than p √ 2 is hardfor NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (r ..."
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Cited by 51 (4 self)
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Abstract. We show that approximating the shortest vector problem (in any ℓp norm) to within any constant factor less than p √ 2 is hardfor NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NPhardness result (i.e., hardness under deterministic manyone reductions) under a reasonable number theoretic conjecture on the distribution of squarefree smooth numbers. As part of our proof, we give an alternative construction of Ajtai’s constructive variant of Sauer’s lemma that greatly simplifies Ajtai’s original proof. Key words. NPhardness, shortest vector problem, point lattices, geometry of numbers, sphere packing
A knapsacktype public key cryptosystem based on arithmetic in finite fields
 IEEE Trans. Inform. Theory
, 1988
"... AbstractA new knapsacktype public key cryptosystem is introduced. The system is based on a novel application of arithmetic in finite fields, following a construction by Bose and Chowla. By appropriately choosing the parameters, one can control the density of the resulting knapsack, which is the ra ..."
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Cited by 40 (0 self)
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AbstractA new knapsacktype public key cryptosystem is introduced. The system is based on a novel application of arithmetic in finite fields, 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 sue in bits. In particular, the density can be made high enough to foil “lowdensity ” attacks against our system. At the moment, no attacks capable of “breaking ” this system in a reasonable amount of time are known. I.
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.
Latticebased Cryptography
, 2008
"... In this chapter we describe some of the recent progress in latticebased cryptography. Latticebased cryptographic constructions hold a great promise for postquantum cryptography, as they enjoy very strong security proofs based on worstcase hardness, relatively efficient implementations, as well a ..."
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Cited by 35 (5 self)
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In this chapter we describe some of the recent progress in latticebased cryptography. Latticebased cryptographic constructions hold a great promise for postquantum cryptography, as they enjoy very strong security proofs based on worstcase hardness, relatively efficient implementations, as well as great simplicity. In addition, latticebased cryptography is believed to be secure against quantum computers. Our focus here