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62
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
From extractable collision resistance to succinct noninteractive arguments of knowledge, and back again
, 2011
"... The existence of succinct noninteractive arguments for NP (i.e., noninteractive computationallysound proofs where the verifier’s work is essentially independent of the complexity of the NP nondeterministic verifier) has been an intriguing question for the past two decades. Other than CS proofs in ..."
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Cited by 37 (14 self)
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The existence of succinct noninteractive arguments for NP (i.e., noninteractive computationallysound proofs where the verifier’s work is essentially independent of the complexity of the NP nondeterministic verifier) has been an intriguing question for the past two decades. Other than CS proofs in the random oracle model [Micali, FOCS ’94], the only existing candidate construction is based on an elaborate assumption that is tailored to a specific protocol [Di Crescenzo and Lipmaa, CiE ’08]. We formulate a general and relatively natural notion of an extractable collisionresistant hash function (ECRH) and show that, if ECRHs exist, then a modified version of Di Crescenzo and Lipmaa’s protocol is a succinct noninteractive argument for NP. Furthermore, the modified protocol is actually a succinct noninteractive adaptive argument of knowledge (SNARK). We then propose several candidate constructions for ECRHs and relaxations thereof. We demonstrate the applicability of SNARKs to various forms of delegation of computation, to succinct noninteractive zero knowledge arguments, and to succinct twoparty secure computation. Finally, we show that SNARKs essentially imply the existence of ECRHs, thus demonstrating the necessity of
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 36 (2 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
Fully homomorphic encryption without modulus switching from classical GapSVP
 In Advances in Cryptology  Crypto 2012, volume 7417 of Lecture
"... We present a new tensoring technique for LWEbased fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically (B → B 2 · poly(n)) with every multiplication (before “refreshing”), our noise only grows linearly (B → B · poly(n)). We use this technique to constr ..."
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Cited by 33 (5 self)
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We present a new tensoring technique for LWEbased fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically (B → B 2 · poly(n)) with every multiplication (before “refreshing”), our noise only grows linearly (B → B · poly(n)). We use this technique to construct a scaleinvariant fully homomorphic encryption scheme, whose properties only depend on the ratio between the modulus q and the initial noise level B, and not on their absolute values. Our scheme has a number of advantages over previous candidates: It uses the same modulus throughout the evaluation process (no need for “modulus switching”), and this modulus can take arbitrary form. In addition, security can be classically reduced from the worstcase hardness of the GapSVP problem (with quasipolynomial approximation factor), whereas previous constructions could only exhibit a quantum reduction from GapSVP. Fully homomorphic encryption has been the focus of extensive study since the first candidate scheme was introduced by Gentry [Gen09b]. In a nutshell, fully homomorphic encryption allows to
Delegation of computation without rejection problem from designated verifier CSproofs. Cryptology ePrint Archive, Report 2011/456
, 2011
"... We present a designated verifier CS proof system for polynomial time computations. The proof system can only be verified by a designated verifier: one who has published a publickey for which it knows a matching secret key unknown to the prover. Whereas Micali’s CS proofs require the existence of ra ..."
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Cited by 24 (0 self)
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We present a designated verifier CS proof system for polynomial time computations. The proof system can only be verified by a designated verifier: one who has published a publickey for which it knows a matching secret key unknown to the prover. Whereas Micali’s CS proofs require the existence of random oracles, we can base soundness on computational assumptions: the existence of leveled fully homomorphic encryption (FHE) schemes, the DDH assumption and a new knowledge of exponent assumption. Using our designated verifier CS proof system, we construct two schemes for delegating (polynomialtime) computation. In such schemes, a delegator outsources the computation of a function F on input x to a polynomial time worker, who computes the output y = F (x) and proves to the delegator the correctness of the output. Let T be the complexity of computing F on inputs of length n = x  and let k be a security parameter. Our first scheme calls for an onetime offline stage where the delegator sends a message to the worker, and a noninteractive online stage where the worker sends the output together with a certificate of correctness to the prover per input x. The total computational
Functional Encryption for Inner Product Predicates from Learning with Errors
, 2011
"... We propose a latticebased functional encryption scheme for inner product predicates whose security follows from the difficulty of the learning with errors (LWE) problem. This construction allows us to achieve applications such as range and subset queries, polynomial evaluation, and CNF/DNF formulas ..."
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Cited by 24 (9 self)
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We propose a latticebased functional encryption scheme for inner product predicates whose security follows from the difficulty of the learning with errors (LWE) problem. This construction allows us to achieve applications such as range and subset queries, polynomial evaluation, and CNF/DNF formulas on encrypted data. Our scheme supports inner products over small fields, in contrast to earlier works based on bilinear maps. Our construction is the first functional encryption scheme based on lattice techniques that goes beyond basic identitybased encryption. The main technique in our scheme is a novel twist to the identitybased encryption scheme of Agrawal, Boneh and Boyen (Eurocrypt 2010). Our scheme is weakly attribute hiding in the standard model.
Reusable garbled circuits and succinct functional encryption. Cryptology ePrint Archive, Report 2012/733
, 2012
"... Garbled circuits, introduced by Yao in the mid 80s, allow computing a function f on an input x without leaking anything about f or x besides f(x). Garbled circuits found numerous applications, but every known construction suffers from one limitation: it offers no security if used on multiple inputs ..."
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Cited by 15 (2 self)
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Garbled circuits, introduced by Yao in the mid 80s, allow computing a function f on an input x without leaking anything about f or x besides f(x). Garbled circuits found numerous applications, but every known construction suffers from one limitation: it offers no security if used on multiple inputs x. In this paper, we construct for the first time reusable garbled circuits. The key building block is a new succinct singlekey functional encryption scheme. Functional encryption is an ambitious primitive: given an encryption Enc(x) of a value x, and a secret key skf for a function f, anyone can compute f(x) without learning any other information about x. We construct, for the first time, a succinct functional encryption scheme for any polynomialtime function f where succinctness means that the ciphertext size does not grow with the size of the circuit for f, but only with its depth. The security of our construction is based on the intractability of the Learning with Errors (LWE) problem and holds as long as an adversary has access to a single key skf (or even an a priori bounded number of keys for different functions). Building on our succinct singlekey functional encryption scheme, we show several new applications in addition to reusable garbled circuits, such as a paradigm for general function obfuscation which we call tokenbased obfuscation, homomorphic encryption for a class of Turing machines where the evaluation runs in inputspecific time rather than worstcase time, and a scheme for delegating computation which is publicly verifiable and maintains the privacy of the computation.
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
Faster Algorithms for Approximate Common Divisors: Breaking FullyHomomorphicEncryption Challenges over the Integers
 In Eurocrypto 2012
"... At EUROCRYPT ’10, van Dijk, Gentry, Halevi and Vaikuntanathan presented simple fullyhomomorphic encryption (FHE) schemes based on the hardness of approximate integer common divisors problems, which were introduced in 2001 by HowgraveGraham. There are two versions for these problems: the partial ve ..."
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Cited by 10 (0 self)
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At EUROCRYPT ’10, van Dijk, Gentry, Halevi and Vaikuntanathan presented simple fullyhomomorphic encryption (FHE) schemes based on the hardness of approximate integer common divisors problems, which were introduced in 2001 by HowgraveGraham. There are two versions for these problems: the partial version (PACD) and the general version (GACD). The seemingly easier problem PACD was recently used by Coron, Mandal, Naccache and Tibouchi at CRYPTO ’11 to build a more efficient variant of the FHE scheme by van Dijk et al.. We present a new PACD algorithm whose running time is essentially the “square root ” of that of exhaustive search, which was the best attack in practice. This allows us to experimentally break the FHE challenges proposed by Coron et al. Our PACD algorithm directly gives rise to a new GACD algorithm, which is exponentially faster than exhaustive search: namely, the running time is essentially the 3/4th root of that of exhaustive search. Interestingly, our main technique can also be applied to other settings, such as noisy factoring, fault attacks on CRTRSA signatures, and attacking lowexponent RSA encryption. 1
Multiparty Computation with Low Communication, Computation and Interaction via Threshold FHE ⋆
, 2012
"... Abstract. Fully homomorphic encryption (FHE) enables secure computation over the encrypted data of a single party. We explore how to extend this to multiple parties, using threshold fully homomorphic encryption (TFHE). In such scheme, the parties jointly generate a common FHE public key along with a ..."
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Cited by 8 (1 self)
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Abstract. Fully homomorphic encryption (FHE) enables secure computation over the encrypted data of a single party. We explore how to extend this to multiple parties, using threshold fully homomorphic encryption (TFHE). In such scheme, the parties jointly generate a common FHE public key along with a secret key that is shared among them; they can later cooperatively decrypt ciphertexts without learning anything but the plaintext. We show how to instantiate this approach efficiently, by extending the recent FHE schemes of Brakerski, Gentry and Vaikuntanathan (CRYPTO ’11, FOCS ’11, ITCS ’12) based on the (ring) learning with errors assumption. Our main tool is to exploit the property that such schemes are additively homomorphic over their keys. Using TFHE, we construct simple multiparty computation protocols secure against fully malicious attackers, tolerating any number of corruptions, and providing security in the universal composability framework. Our protocols have the following properties: Low interaction: 3 rounds of interaction given a common random string, or 2 rounds with a publickey infrastructure. Low communication: independent of the function being computed (proportional to just input and output sizes). Cloudassisted computation: the bulk of the computation can be efficiently outsourced to an external entity (e.g. a cloud service) so that the computation of all other parties is independent of the complexity of the evaluated function. 1