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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 366 (6 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
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 153 (22 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.
On ideal lattices and learning with errors over rings
 In Proc. of EUROCRYPT, volume 6110 of LNCS
, 2010
"... The “learning with errors ” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worstcase lattice problems, and in recent years it has served as the foundation for a pleth ..."
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Cited by 126 (18 self)
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The “learning with errors ” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worstcase lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather inefficient due to an inherent quadratic overhead in the use of LWE. A main open question was whether LWE and its applications could be made truly efficient by exploiting extra algebraic structure, as was done for latticebased hash functions (and related primitives). We resolve this question in the affirmative by introducing an algebraic variant of LWE called ringLWE, and proving that it too enjoys very strong hardness guarantees. Specifically, we show that the ringLWE distribution is pseudorandom, assuming that worstcase problems on ideal lattices are hard for polynomialtime quantum algorithms. Applications include the first truly practical latticebased publickey cryptosystem with an efficient security reduction; moreover, many of the other applications of LWE can be made much more efficient through the use of ringLWE. 1
Bonsai Trees, or How to Delegate a Lattice Basis
, 2010
"... We introduce a new latticebased cryptographic structure called a bonsai tree, and use it to resolve some important open problems in the area. Applications of bonsai trees include: • An efficient, stateless ‘hashandsign ’ signature scheme in the standard model (i.e., no random oracles), and • The ..."
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Cited by 124 (6 self)
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We introduce a new latticebased cryptographic structure called a bonsai tree, and use it to resolve some important open problems in the area. Applications of bonsai trees include: • An efficient, stateless ‘hashandsign ’ signature scheme in the standard model (i.e., no random oracles), and • The first hierarchical identitybased encryption (HIBE) scheme (also in the standard model) that does not rely on bilinear pairings. Interestingly, the abstract properties of bonsai trees seem to have no known realization in conventional numbertheoretic cryptography. 1
Predicting lattice reduction
 In Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology, EUROCRYPT’08
, 2008
"... Abstract. Despite their popularity, lattice reduction algorithms remain mysterious cryptanalytical tools. Though it has been widely reported that they behave better than their proved worstcase theoretical bounds, no precise assessment has ever been given. Such an assessment would be very helpful to ..."
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Cited by 98 (1 self)
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Abstract. Despite their popularity, lattice reduction algorithms remain mysterious cryptanalytical tools. Though it has been widely reported that they behave better than their proved worstcase theoretical bounds, no precise assessment has ever been given. Such an assessment would be very helpful to predict the behaviour of latticebased attacks, as well as to select keysizes for latticebased cryptosystems. The goal of this paper is to provide such an assessment, based on extensive experiments performed with the NTL library. The experiments suggest several conjectures on the worst case and the actual behaviour of lattice reduction algorithms. We believe the assessment might also help to design new reduction algorithms overcoming the limitations of current algorithms.
Efficient lattice (H)IBE in the standard model
 In EUROCRYPT 2010, LNCS
, 2010
"... Abstract. We construct an efficient identity based encryption system based on the standard learning with errors (LWE) problem. Our security proof holds in the standard model. The key step in the construction is a family of lattices for which there are two distinct trapdoors for finding short vectors ..."
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Cited by 96 (15 self)
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Abstract. We construct an efficient identity based encryption system based on the standard learning with errors (LWE) problem. Our security proof holds in the standard model. The key step in the construction is a family of lattices for which there are two distinct trapdoors for finding short vectors. One trapdoor enables the real system to generate short vectors in all lattices in the family. The other trapdoor enables the simulator to generate short vectors for all lattices in the family except for one. We extend this basic technique to an adaptivelysecure IBE and a Hierarchical IBE. 1
PublicKey Cryptosystems Resilient to Key Leakage
"... Most of the work in the analysis of cryptographic schemes is concentrated in abstract adversarial models that do not capture sidechannel attacks. Such attacks exploit various forms of unintended information leakage, which is inherent to almost all physical implementations. Inspired by recent sidec ..."
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Cited by 89 (6 self)
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Most of the work in the analysis of cryptographic schemes is concentrated in abstract adversarial models that do not capture sidechannel attacks. Such attacks exploit various forms of unintended information leakage, which is inherent to almost all physical implementations. Inspired by recent sidechannel attacks, especially the “cold boot attacks ” of Halderman et al. (USENIX Security ’08), Akavia, Goldwasser and Vaikuntanathan (TCC ’09) formalized a realistic framework for modeling the security of encryption schemes against a wide class of sidechannel attacks in which adversarially chosen functions of the secret key are leaked. In the setting of publickey encryption, Akavia et al. showed that Regev’s latticebased scheme (STOC ’05) is resilient to any leakage of
New Techniques for Dual System Encryption and Fully Secure HIBE with Short Ciphertexts
"... We construct a fully secure HIBE scheme with short ciphertexts. The previous construction of Boneh, Boyen, and Goh was only proven to be secure in the selective model, under a nonstatic assumption which depended on the depth of the hierarchy. To obtain full security, we apply the dual system encryp ..."
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Cited by 74 (13 self)
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We construct a fully secure HIBE scheme with short ciphertexts. The previous construction of Boneh, Boyen, and Goh was only proven to be secure in the selective model, under a nonstatic assumption which depended on the depth of the hierarchy. To obtain full security, we apply the dual system encryption concept recently introduced by Waters. A straightforward application of this technique is insufficient to achieve short ciphertexts, since the original instantiation of the technique includes tags that do not compress. To overcome this challenge, we design a new method for realizing dual system encryption. We provide a system in composite order groups (of three primes) and prove the security of our scheme under three static assumptions.
Fully Homomorphic Encryption from RingLWE and Security for Key Dependent Messages
 in Advances in Cryptology—CRYPTO 2011, Lect. Notes in Comp. Sci. 6841 (2011
"... Abstract. We present a somewhat homomorphic encryption scheme that is both very simple to describe and analyze, and whose security (quantumly) reduces to the worstcase hardness of problems on ideal lattices. We then transform it into a fully homomorphic encryption scheme using standard “squashing ” ..."
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Cited by 71 (3 self)
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Abstract. We present a somewhat homomorphic encryption scheme that is both very simple to describe and analyze, and whose security (quantumly) reduces to the worstcase hardness of problems on ideal lattices. We then transform it into a fully homomorphic encryption scheme using standard “squashing ” and “bootstrapping ” techniques introduced by Gentry (STOC 2009). One of the obstacles in going from “somewhat ” to full homomorphism is the requirement that the somewhat homomorphic scheme be circular secure, namely, the scheme can be used to securely encrypt its own secret key. For all known somewhat homomorphic encryption schemes, this requirement was not known to be achievable under any cryptographic assumption, and had to be explicitly assumed. We take a step forward towards removing this additional assumption by proving that our scheme is in fact secure when encrypting polynomial functions of the secret key. Our scheme is based on the ring learning with errors (RLWE) assumption that was recently introduced by Lyubashevsky, Peikert and Regev (Eurocrypt 2010). The RLWE assumption is reducible to worstcase problems on ideal lattices, and allows us to completely abstract out the lattice interpretation, resulting in an extremely simple scheme. For example, our secret key is s, and our public key is (a, b = as + 2e), where s, a, e are all degree (n − 1) integer polynomials whose coefficients are independently drawn from easy to sample distributions. 1