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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 46 (8 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
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 37 (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
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 32 (2 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.
Making NTRU as secure as worstcase problems over ideal lattices
 In Proc. of EUROCRYPT, volume 6632 of LNCS
, 2011
"... Abstract. NTRUEncrypt, proposed in 1996 by Ho stein, Pipher and Silverman, is the fastest known latticebased encryption scheme. Its moderate keysizes, excellent asymptotic performance and conjectured resistance to quantum computers could make it a desirable alternative to factorisation and discret ..."
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Cited by 15 (5 self)
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Abstract. NTRUEncrypt, proposed in 1996 by Ho stein, Pipher and Silverman, is the fastest known latticebased encryption scheme. Its moderate keysizes, excellent asymptotic performance and conjectured resistance to quantum computers could make it a desirable alternative to factorisation and discretelog based encryption schemes. However, since its introduction, doubts have regularly arisen on its security. In the present work, we show how to modify NTRUEncrypt to make it provably secure in the standard model, under the assumed quantum hardness of standard worstcase lattice problems, restricted to a family of lattices related to some cyclotomic elds. Our main contribution is to show that if the secret key polynomials are selected by rejection from discrete Gaussians, then the public key, which is their ratio, is statistically indistinguishable from uniform over its domain. The security then follows from the already proven hardness of the RLWE problem.
Pseudorandom Functions and Lattices
, 2011
"... We give direct constructions of pseudorandom function (PRF) families based on conjectured hard lattice problems and learning problems. Our constructions are asymptotically efficient and highly parallelizable in a practical sense, i.e., they can be computed by simple, relatively small lowdepth arith ..."
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Cited by 13 (3 self)
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We give direct constructions of pseudorandom function (PRF) families based on conjectured hard lattice problems and learning problems. Our constructions are asymptotically efficient and highly parallelizable in a practical sense, i.e., they can be computed by simple, relatively small lowdepth arithmetic or boolean circuits (e.g., in NC 1 or even TC 0). In addition, they are the first lowdepth PRFs that have no known attack by efficient quantum algorithms. Central to our results is a new “derandomization ” technique for the learning with errors (LWE) problem which, in effect, generates the error terms deterministically. 1 Introduction and Main Results The past few years have seen significant progress in constructing publickey, identitybased, and homomorphic cryptographic schemes using lattices, e.g., [Reg05, PW08, GPV08, Gen09, CHKP10, ABB10a] and many more. Part of their appeal stems from provable worstcase hardness guarantees (starting with the seminal work of Ajtai [Ajt96]), good asymptotic efficiency and parallelism, and apparent resistance to quantum
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 13 (4 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.
New generic algorithms for hard knapsacks
 of Lecture Notes in Computer Science
, 2010
"... Abstract. In this paper, we study the complexity of solving hard knapsack problems, i.e., knapsacks with a density close to 1 where latticebased low density attacks are not an option. For such knapsacks, the current stateoftheart is a 31year old algorithm by Schroeppel and Shamir which is based ..."
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Cited by 11 (2 self)
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Abstract. In this paper, we study the complexity of solving hard knapsack problems, i.e., knapsacks with a density close to 1 where latticebased low density attacks are not an option. For such knapsacks, the current stateoftheart is a 31year old algorithm by Schroeppel and Shamir which is based on birthday paradox techniques and yields a running time of Õ(2n/2) for knapsacks of n elements and uses Õ(2n/4) storage. We propose here two new algorithms which improve on this bound, finally
FiatShamir with aborts: Applications to lattice and factoringbased signatures
, 2009
"... Abstract. We demonstrate how the framework that is used for creating efficient numbertheoretic ID and signature schemes can be transferred into the setting of lattices. This results in constructions of the most efficient todate identification and signature schemes with security based on the worst ..."
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Abstract. We demonstrate how the framework that is used for creating efficient numbertheoretic ID and signature schemes can be transferred into the setting of lattices. This results in constructions of the most efficient todate identification and signature schemes with security based on the worstcase hardness of problems in ideal lattices. In particular, our ID scheme has communication complexity of around 65, 000 bits and the length of the signatures produced by our signature scheme is about 50, 000 bits. All prior latticebased identification schemes required on the order of millions of bits to be transferred, while all previous latticebased signature schemes were either stateful, too inefficient, or produced signatures whose lengths were also on the order of millions of bits. The security of our identification scheme is based on the hardness of finding the approximate shortest vector to within a factor of Õ(n2) in the standard model, while the security of the signature scheme is based on the same assumption in the random oracle model. Our protocols are very efficient, with all operations requiring Õ(n) time. We also show that the technique for constructing our latticebased schemes can be used to improve certain numbertheoretic schemes. In particular, we are able to shorten the length of the signatures that are produced by Girault’s factoringbased digital signature scheme ([10, 11, 31]). 1
How Risky is the RandomOracle Model?
"... Abstract. RSAFDH and many other schemes secure in the RandomOracle Model (ROM) require a hash function with output size larger than standard sizes. We show that the randomoracle instantiations proposed in the literature for such cases are weaker than a random oracle, including the proposals by Be ..."
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Abstract. RSAFDH and many other schemes secure in the RandomOracle Model (ROM) require a hash function with output size larger than standard sizes. We show that the randomoracle instantiations proposed in the literature for such cases are weaker than a random oracle, including the proposals by Bellare and Rogaway from 1993 and 1996, and the ones implicit in IEEE P1363 and PKCS standards: for instance, we obtain a practical preimage attack on BR93 for 1024bit digests (with complexity less than 2 30). Next, we study the security impact of hash function defects for ROM signatures. As an extreme case, we note that any hash collision would suffice to disclose the master key in the IDbased cryptosystem by Boneh et al. from FOCS ’07, and the secret key in the RabinWilliams signature for which Bernstein proved tight security at EUROCRYPT ’08. We also remark that collisions can be found as a precomputation for any instantiation of the ROM, and this violates the security definition of the scheme in the standard model. Hence, this gives an example of a natural scheme that is proven secure in the ROM but that in insecure for any instantiation by a single function. Interestingly, for both of these schemes, a slight modification can prevent these attacks, while preserving the ROM security result. We give evidence that in the case of RSA and Rabin/RabinWilliams, an appropriate PSS padding is more robust than all other paddings known. 1