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Fully homomorphic encryption with polylog overhead
"... We show that homomorphic evaluation of (wide enough) arithmetic circuits can be accomplished with only polylogarithmic overhead. Namely, we present a construction of fully homomorphic encryption (FHE) schemes that for security parameter λ can evaluate any widthΩ(λ) circuit with t gates in time t · ..."
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Cited by 64 (4 self)
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We show that homomorphic evaluation of (wide enough) arithmetic circuits can be accomplished with only polylogarithmic overhead. Namely, we present a construction of fully homomorphic encryption (FHE) schemes that for security parameter λ can evaluate any widthΩ(λ) circuit with t gates in time t · polylog(λ). To get low overhead, we use the recent batch homomorphic evaluation techniques of SmartVercauteren and BrakerskiGentryVaikuntanathan, who showed that homomorphic operations can be applied to “packed” ciphertexts that encrypt vectors of plaintext elements. In this work, we introduce permuting/routing techniques to move plaintext elements across these vectors efficiently. Hence, we are able to implement general arithmetic circuit in a batched fashion without ever needing to “unpack” the plaintext vectors. We also introduce some other optimizations that can speed up homomorphic evaluation in certain cases. For example, we show how to use the Frobenius map to raise plaintext elements to powers of p at the “cost” of a linear operation.
Reusable garbled circuits and succinct functional encryption
, 2013
"... 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 42 (3 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.
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 39 (12 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.
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 35 (10 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
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 27 (7 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
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 19 (2 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
Fully Homomorphic SIMD Operations
"... Abstract. At PKC 2010 Smart and Vercauteren presented a variant of Gentry’s fully homomorphic public key encryption scheme and mentioned that the scheme could support SIMD style operations. The slow key generation process of the Smart–Vercauteren system was then addressed in a paper by Gentry and Ha ..."
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Cited by 14 (0 self)
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Abstract. At PKC 2010 Smart and Vercauteren presented a variant of Gentry’s fully homomorphic public key encryption scheme and mentioned that the scheme could support SIMD style operations. The slow key generation process of the Smart–Vercauteren system was then addressed in a paper by Gentry and Halevi, but their key generation method appears to exclude the SIMD style operation alluded to by Smart and Vercauteren. In this paper, we show how to select parameters to enable such SIMD operations, whilst still maintaining practicality of the key generation technique of Gentry and Halevi. As such, we obtain a somewhat homomorphic scheme supporting both SIMD operations and operations on large finite fields of characteristic two. This somewhat homomorphic scheme can be made fully homomorphic in a naive way by recrypting all data elements seperately. However, we show that the SIMD operations can be used to perform the recrypt procedure in parallel, resulting in a substantial speedup. Finally, we demonstrate how such SIMD operations can be used to perform various tasks by studying two use cases: implementing AES homomorphically and encrypted database lookup. 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 13 (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
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