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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
Efficient Fully Homomorphic Encryption from (Standard) LWE
 LWE, FOCS 2011, IEEE 52ND ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, IEEE
, 2011
"... We present a fully homomorphic encryption scheme that is based solely on the (standard) learning with errors (LWE) assumption. Applying known results on LWE, the security of our scheme is based on the worstcase hardness of “short vector problems ” on arbitrary lattices. Our construction improves on ..."
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Cited by 117 (6 self)
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We present a fully homomorphic encryption scheme that is based solely on the (standard) learning with errors (LWE) assumption. Applying known results on LWE, the security of our scheme is based on the worstcase hardness of “short vector problems ” on arbitrary lattices. Our construction improves on previous works in two aspects: 1. We show that “somewhat homomorphic” encryption can be based on LWE, using a new relinearization technique. In contrast, all previous schemes relied on complexity assumptions related to ideals in various rings. 2. We deviate from the “squashing paradigm” used in all previous works. We introduce a new dimensionmodulus reduction technique, which shortens the ciphertexts and reduces the decryption complexity of our scheme, without introducing additional assumptions. Our scheme has very short ciphertexts and we therefore use it to construct an asymptotically efficient LWEbased singleserver private information retrieval (PIR) protocol. The communication complexity of our protocol (in the publickey model) is k · polylog(k) + log DB  bits per singlebit query (here, k is a security parameter).
Can Homomorphic Encryption be Practical?
"... Abstract. The prospect of outsourcing an increasing amount of data storage and management to cloud services raises many new privacy concerns for individuals and businesses alike. The privacy concerns can be satisfactorily addressed if users encrypt the data they send to the cloud. If the encryption ..."
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Cited by 80 (6 self)
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Abstract. The prospect of outsourcing an increasing amount of data storage and management to cloud services raises many new privacy concerns for individuals and businesses alike. The privacy concerns can be satisfactorily addressed if users encrypt the data they send to the cloud. If the encryption scheme is homomorphic, the cloud can still perform meaningful computations on the data, even though it is encrypted. In fact, we now know a number of constructions of fully homomorphic encryption schemes that allow arbitrary computation on encrypted data. In the last two years, solutions for fully homomorphic encryption have been proposed and improved upon, but it is hard to ignore the elephant in the room, namely efficiency – can homomorphic encryption ever be efficient enough to be practical? Certainly, it seems that all known fully homomorphic encryption schemes have a long way to go before they can be used in practice. Given this state of affairs, our contribution is twofold. First, we exhibit a number of realworld applications, in the medical, financial, and the advertising domains, which require only that the encryption scheme is “somewhat ” homomorphic. Somewhat homomorphic encryption schemes, which support a limited number of homomorphic operations, can be much faster, and more compact than fully homomorphic encryption schemes. Secondly, we show a proofofconcept implementation of the recent somewhat homomorphic encryption scheme of Brakerski and Vaikuntanathan, whose security relies on the “ring learning with errors ” (Ring LWE) problem. The system is very efficient, and has reasonably short ciphertexts. Our unoptimized implementation in magma enjoys comparable efficiency to even optimized pairingbased schemes with the same level of security and homomorphic capacity. We also show a number of applicationspecific optimizations to the encryption scheme, most notably the ability to convert between different message encodings in a ciphertext.
Lattice Signatures Without Trapdoors
"... We provide an alternative method for constructing latticebased digital signatures which does not use the “hashandsign” methodology of Gentry, Peikert, and Vaikuntanathan (STOC 2008). Our resulting signature scheme is secure, in the random oracle model, based on the worstcase hardness of the Õ(n ..."
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Cited by 43 (9 self)
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We provide an alternative method for constructing latticebased digital signatures which does not use the “hashandsign” methodology of Gentry, Peikert, and Vaikuntanathan (STOC 2008). Our resulting signature scheme is secure, in the random oracle model, based on the worstcase hardness of the Õ(n1.5)SIVP problem in general lattices. The secret key, public key, and the signature size of our scheme are smaller than in all previous instantiations of the hashandsign signature, and our signing algorithm is also quite simple, requiring just a few matrixvector multiplications and rejection samplings. We then also show that by slightly changing the parameters, one can get even more efficient signatures that are based on the hardness of the Learning With Errors problem. Our construction naturally transfers to the ring setting, where the size of the public and secret keys can be significantly shrunk, which results in the most practical todate provably secure signature scheme based on lattices.
Improved Security for a RingBased Fully Homomorphic Encryption Scheme
"... Abstract. In 1996, Hoffstein, Pipher and Silverman introduced an efficient lattice based encryption scheme dubbed NTRUEncrypt. Unfortunately, this scheme lacks a proof of security. However, in 2011, Stehlé and Steinfeld showed how to modify NTRUEncrypt to reduce security to standard problems in idea ..."
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Cited by 26 (6 self)
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Abstract. In 1996, Hoffstein, Pipher and Silverman introduced an efficient lattice based encryption scheme dubbed NTRUEncrypt. Unfortunately, this scheme lacks a proof of security. However, in 2011, Stehlé and Steinfeld showed how to modify NTRUEncrypt to reduce security to standard problems in ideal lattices. At STOC 2012, LópezAlt, Tromer and Vaikuntanathan proposed a fully homomorphic scheme based on this modified system. However, to allow homomorphic operations and prove security, a nonstandard assumption is required in their scheme. In this paper, we show how to remove this nonstandard assumption via techniques introduced by Brakerski at CRYPTO 2012 and construct a new fully homomorphic encryption scheme from the Stehlé and Steinfeld version based on standard lattice assumptions and a circular security assumption. The scheme is scaleinvariant and therefore avoids modulus switching, it eliminates ciphertext expansion in homomorphic multiplication, and the size of ciphertexts is one ring element. Moreover, we present a practical variant of our scheme, which is secure under stronger assumptions, along with parameter recommendations and promising implementation results. Finally, we present a novel approach for encrypting larger input sizes by applying a CRT approach on the input space.
A toolkit for ringLWE cryptography
 In EUROCRYPT
, 2013
"... Recent advances in lattice cryptography, mainly stemming from the development of ringbased primitives such as ringLWE, have made it possible to design cryptographic schemes whose efficiency is competitive with that of more traditional numbertheoretic ones, along with entirely new applications lik ..."
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Cited by 21 (7 self)
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Recent advances in lattice cryptography, mainly stemming from the development of ringbased primitives such as ringLWE, have made it possible to design cryptographic schemes whose efficiency is competitive with that of more traditional numbertheoretic ones, along with entirely new applications like fully homomorphic encryption. Unfortunately, realizing the full potential of ringbased cryptography has so far been hindered by a lack of practical algorithms and analytical tools for working in this context. As a result, most previous works have focused on very special classes of rings such as poweroftwo cyclotomics, which significantly restricts the possible applications. We bridge this gap by introducing a toolkit of fast, modular algorithms and analytical techniques that can be used in a wide variety of ringbased cryptographic applications, particularly those built around ringLWE. Our techniques yield applications that work in arbitrary cyclotomic rings, with no loss in their underlying worstcase hardness guarantees, and very little loss in computational efficiency, relative to poweroftwo cyclotomics. To demonstrate the toolkit’s applicability, we develop a few illustrative applications: two variant publickey cryptosystems, and a “somewhat homomorphic ” symmetric encryption scheme. Both apply to arbitrary cyclotomics, have tight parameters, and very efficient implementations. 1
Faster Gaussian lattice sampling using lazy floatingpoint arithmetic
 FULL VERSION OF THE ASIACRYPT ’12 ARTICLE
, 2013
"... Many lattice cryptographic primitives require an efficient algorithm to sample lattice points according to some Gaussian distribution. All algorithms known for this task require longinteger arithmetic at some point, which may be problematic in practice. We study how much lattice sampling can be sp ..."
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Cited by 17 (1 self)
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Many lattice cryptographic primitives require an efficient algorithm to sample lattice points according to some Gaussian distribution. All algorithms known for this task require longinteger arithmetic at some point, which may be problematic in practice. We study how much lattice sampling can be sped up using floatingpoint arithmetic. First, we show that a direct floatingpoint implementation of these algorithms does not give any asymptotic speedup: the floatingpoint precision needs to be greater than the security parameter, leading to an overall complexity Õ(n 3) where n is the lattice dimension. However, we introduce a laziness technique that can significantly speed up these algorithms. Namely, in certain cases such as NTRUSign lattices, laziness can decrease the complexity to Õ(n2) or even Õ(n). Furthermore, our analysis is practical: for typical parameters, most of the floatingpoint operations only require the doubleprecision IEEE standard.
Sampling from discrete Gaussians for latticebased cryptography on a constrained device
 Appl. Algebra Eng. Commun. Comput
"... ABSTRACT. Modern latticebased publickey cryptosystems require sampling from discrete Gaussian (normal) distributions. The paper surveys algorithms to implement such sampling efficiently, with particular focus on the case of constrained devices with small onboard storage and without access to larg ..."
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Cited by 15 (0 self)
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ABSTRACT. Modern latticebased publickey cryptosystems require sampling from discrete Gaussian (normal) distributions. The paper surveys algorithms to implement such sampling efficiently, with particular focus on the case of constrained devices with small onboard storage and without access to large numbers of external random bits. We review latticebased encryption schemes and signature schemes and their requirements for sampling from discrete Gaussians. Finally, we make some remarks on challenges and potential solutions for practical latticebased cryptography.
Circular and KDM security for identitybased encryption
 In Public Key Cryptography
, 2012
"... We initiate the study of security for keydependent messages (KDM), sometimes also known as “circular ” or “clique ” security, in the setting of identitybased encryption (IBE). Circular/KDM security requires that ciphertexts preserve secrecy even when they encrypt messages that may depend on the se ..."
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Cited by 12 (3 self)
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We initiate the study of security for keydependent messages (KDM), sometimes also known as “circular ” or “clique ” security, in the setting of identitybased encryption (IBE). Circular/KDM security requires that ciphertexts preserve secrecy even when they encrypt messages that may depend on the secret keys, and arises in natural usage scenarios for IBE. We construct an IBE system that is circular secure for affine functions of users ’ secret keys, based on the learning with errors (LWE) problem (and hence on worstcase lattice problems). The scheme is secure in the standard model, under a natural extension of a selectiveidentity attack. Our three main technical contributions are (1) showing the circular/KDMsecurity of a “dual”style LWE publickey cryptosystem, (2) proving the hardness of a version of the “extended LWE ” problem due to O’Neill, Peikert and Waters (CRYPTO’11), and (3) building an IBE scheme around the dualstyle system using a novel latticebased “allbutd ” trapdoor function. 1
Trapdoors for lattices: Simpler, tighter, faster, smaller
 In EUROCRYPT
, 2012
"... We give new methods for generating and using “strong trapdoors ” in cryptographic lattices, which are simultaneously simple, efficient, easy to implement (even in parallel), and asymptotically optimal with very small hidden constants. Our methods involve a new kind of trapdoor, and include specializ ..."
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Cited by 12 (3 self)
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We give new methods for generating and using “strong trapdoors ” in cryptographic lattices, which are simultaneously simple, efficient, easy to implement (even in parallel), and asymptotically optimal with very small hidden constants. Our methods involve a new kind of trapdoor, and include specialized algorithms for inverting LWE, randomly sampling SIS preimages, and securely delegating trapdoors. These tasks were previously the main bottleneck for a wide range of cryptographic schemes, and our techniques substantially improve upon the prior ones, both in terms of practical performance and quality of the produced outputs. Moreover, the simple structure of the new trapdoor and associated algorithms can be exposed in applications, leading to further simplifications and efficiency improvements. We exemplify the applicability of our methods with new digital signature schemes and CCAsecure encryption schemes, which have better efficiency and security than the previously known latticebased constructions. 1