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187
Fully homomorphic encryption using ideal lattices
 In Proc. STOC
, 2009
"... We propose a fully homomorphic encryption scheme – i.e., a scheme that allows one to evaluate circuits over encrypted data without being able to decrypt. Our solution comes in three steps. First, we provide a general result – that, to construct an encryption scheme that permits evaluation of arbitra ..."
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Cited by 267 (11 self)
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We propose a fully homomorphic encryption scheme – i.e., a scheme that allows one to evaluate circuits over encrypted data without being able to decrypt. Our solution comes in three steps. First, we provide a general result – that, to construct an encryption scheme that permits evaluation of arbitrary circuits, it suffices to construct an encryption scheme that can evaluate (slightly augmented versions of) its own decryption circuit; we call a scheme that can evaluate its (augmented) decryption circuit bootstrappable. Next, we describe a public key encryption scheme using ideal lattices that is almost bootstrappable. Latticebased cryptosystems typically have decryption algorithms with low circuit complexity, often dominated by an inner product computation that is in NC1. Also, ideal lattices provide both additive and multiplicative homomorphisms (modulo a publickey ideal in a polynomial ring that is represented as a lattice), as needed to evaluate general circuits. Unfortunately, our initial scheme is not quite bootstrappable – i.e., the depth that the scheme can correctly evaluate can be logarithmic in the lattice dimension, just like the depth of the decryption circuit, but the latter is greater than the former. In the final step, we show how to modify the scheme to reduce the depth of the decryption circuit, and thereby obtain a bootstrappable encryption scheme, without reducing the depth that the scheme can evaluate. Abstractly, we accomplish this by enabling the encrypter to start the decryption process, leaving less work for the decrypter, much like the server leaves less work for the decrypter in a serveraided cryptosystem.
On Lattices, Learning with Errors, Random Linear Codes, and Cryptography
 In STOC
, 2005
"... Our main result is a reduction from worstcase lattice problems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the ‘learning from parity with error’ problem to higher moduli. It can also be viewed as the problem of decoding from a random linear co ..."
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Cited by 197 (3 self)
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Our main result is a reduction from worstcase lattice problems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the ‘learning from parity with error’ problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for SVP and SIVP. A main open question is whether this reduction can be made classical. We also present a (classical) publickey cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worstcase quantum hardness of SVP and SIVP. Previous latticebased publickey cryptosystems such as the one by Ajtai and Dwork were based only on uniqueSVP, a special case of SVP. The new cryptosystem is much more efficient than previous cryptosystems: the public key is of size Õ(n2) and encrypting a message increases its size by a factor of Õ(n) (in previous cryptosystems these values are Õ(n4) and Õ(n2), respectively). In fact, under the assumption that all parties share a random bit string of length Õ(n2), the size of the public key can be reduced to Õ(n). 1
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 194 (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 Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations
, 1993
"... We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NPhard. 2. If for some ..."
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Cited by 159 (7 self)
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We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NPhard. 2. If for some ffl ? 0 there exists a polynomialtime algorithm that approximates the optimum within a factor of 2 log 0:5\Gammaffl n , then every NP language can be decided in quasipolynomial deterministic time, i.e., NP ` DTIME(n poly(log n) ). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the `1 norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2 log 1\Gammaffl n or n ffl . Improving the factor 2 log 0:5\Gammaffl n to p dimension for either of the lattice problems would imply the hardness of the Shortest Vector Problem in `2 norm; an old open problem. Our proofs use reductions from fewpr...
PublicKey Cryptosystems from Lattice Reduction Problems
, 1996
"... We present a new proposal for a trapdoor oneway function, from whichwe derive publickey encryption and digital signatures. The security of the new construction is based on the conjectured computational difficulty of latticereduction problems, providing a possible alternative to existing publicke ..."
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Cited by 120 (5 self)
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We present a new proposal for a trapdoor oneway function, from whichwe derive publickey encryption and digital signatures. The security of the new construction is based on the conjectured computational difficulty of latticereduction problems, providing a possible alternative to existing publickey encryption algorithms and digital signatures such as RSA and DSS.
Trapdoors for Hard Lattices and New Cryptographic Constructions
, 2007
"... We show how to construct a variety of “trapdoor ” cryptographic tools assuming the worstcase hardness of standard lattice problems (such as approximating the shortest nonzero vector to within small factors). The applications include trapdoor functions with preimage sampling, simple and efficient “ha ..."
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Cited by 104 (20 self)
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We show how to construct a variety of “trapdoor ” cryptographic tools assuming the worstcase hardness of standard lattice problems (such as approximating the shortest nonzero vector to within small factors). The applications include trapdoor functions with preimage sampling, simple and efficient “hashandsign ” digital signature schemes, universally composable oblivious transfer, and identitybased encryption. A core technical component of our constructions is an efficient algorithm that, given a basis of an arbitrary lattice, samples lattice points from a Gaussianlike probability distribution whose standard deviation is essentially the length of the longest vector in the basis. In particular, the crucial security property is that the output distribution of the algorithm is oblivious to the particular geometry of the given basis. ∗ Supported by the Herbert Kunzel Stanford Graduate Fellowship. † This material is based upon work supported by the National Science Foundation under Grants CNS0716786 and CNS0749931. Any opinions, findings, and conclusions or recommedations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. ‡ The majority of this work was performed while at SRI International. 1 1
Las Vegas algorithms for linear and integer programming when the dimension is small
 J. ACM
, 1995
"... Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algor ..."
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Cited by 103 (2 self)
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Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algorlthm N gwen for integer hnear programmmg. Let p bound the number of bits required to specify the ratmnal numbers defmmg an input constraint or the ob~ective function vector. Let n and d be as before. Then, the algorithm requires expected 0(2d dn + S~dm In n) + dc)’d) ~ in H operations on numbers with O(1~p bits d ~ ~ ~z + ~, where the constant factors do not depend on d or p. The expectations are with respect to the random choices made by the algorithms, and the bounds hold for any gwen input. The techmque can be extended to other convex programming problems. For example, m algorlthm for finding the smallest sphere enclosing a set of /z points m Ed has the same t]me bound
Publickey cryptosystems from the worstcase shortest vector problem
, 2008
"... We construct publickey cryptosystems that are secure assuming the worstcase hardness of approximating the length of a shortest nonzero vector in an ndimensional lattice to within a small poly(n) factor. Prior cryptosystems with worstcase connections were based either on the shortest vector probl ..."
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Cited by 84 (18 self)
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We construct publickey cryptosystems that are secure assuming the worstcase hardness of approximating the length of a shortest nonzero vector in an ndimensional lattice to within a small poly(n) factor. Prior cryptosystems with worstcase connections were based either on the shortest vector problem for a special class of lattices (Ajtai and Dwork, STOC 1997; Regev, J. ACM 2004), or on the conjectured hardness of lattice problems for quantum algorithms (Regev, STOC 2005). Our main technical innovation is a reduction from certain variants of the shortest vector problem to corresponding versions of the “learning with errors” (LWE) problem; previously, only a quantum reduction of this kind was known. In addition, we construct new cryptosystems based on the search version of LWE, including a very natural chosen ciphertextsecure system that has a much simpler description and tighter underlying worstcase approximation factor than prior constructions.
Worstcase to averagecase reductions based on Gaussian measures
 SIAM J. on Computing
, 2004
"... We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest indepe ..."
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Cited by 83 (16 self)
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We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem (a variant of the well known closest vector problem). The approximation factor we obtain is nlog O(1) n for all four problems. This greatly improves on all previous work on the subject starting from Ajtai’s seminal paper (STOC, 1996), up to the strongest previously known results by Micciancio (SIAM J. on Computing, 2004). Our results also bring us closer to the limit where the problems are no longer known to be in NP intersect coNP. Our main tools are Gaussian measures on lattices and the highdimensional Fourier transform. We start by defining a new lattice parameter which determines the amount of Gaussian noise that one has to add to a lattice in order to get close to a uniform distribution. In addition to yielding quantitatively much stronger results, the use of this parameter allows us to simplify many of the complications in previous work. Our technical contributions are twofold. First, we show tight connections between this new parameter and existing lattice parameters. One such important connection is between this parameter and the length of the shortest set of linearly independent vectors. Second, we prove that the distribution that one obtains after adding Gaussian noise to the lattice has the following interesting property: the distribution of the noise vector when conditioning on the final value behaves in many respects like the original Gaussian noise vector. In particular, its moments remain essentially unchanged. 1
On the Limits of NonApproximability of Lattice Problems
, 1998
"... We show simple constantround interactive proof systems for problems capturing the approximability, to within a factor of p n, of optimization problems in integer lattices; specifically, the closest vector problem (CVP), and the shortest vector problem (SVP). These interactive proofs are for th ..."
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Cited by 81 (3 self)
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We show simple constantround interactive proof systems for problems capturing the approximability, to within a factor of p n, of optimization problems in integer lattices; specifically, the closest vector problem (CVP), and the shortest vector problem (SVP). These interactive proofs are for the "coNP direction"; that is, we give an interactive protocol showing that a vector is "far" from the lattice (for CVP), and an interactive protocol showing that the shortestlatticevector is "long" (for SVP). Furthermore, these interactive proof systems are HonestVerifier Perfect ZeroKnowledge. We conclude that approximating CVP (resp., SVP) within a factor of p n is in NP " coAM. Thus, it seems unlikely that approximating these problems to within a p n factor is NPhard. Previously, for the CVP (resp., SVP) problem, Lagarias et. al., Hastad and Banaszczyk showed that the gap problem corresponding to approximating CVP (resp., SVP) within n is in NP " coNP . On the other hand, Ar...