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[BGV12] Zvika Brakerski, Craig Gentry, and Vinod Vaikuntanathan. Fully homomorphic encryption without bootstrapping. In ITCS, 2012. [BOGW88] Michael Ben-Or, Shafi Goldwasser, and Avi Wigderson. Completeness theorems for noncryptographic fault-tolerant distributed computation (extended abstract). In STOC, pages 1–10, 1988. [BV11] [CCD88] [CKKC12] [CKKC13] [CKV10] [DFH12] Zvika Brakerski and Vinod Vaikuntanathan. Efficient fully homomorphic encryption from (standard) lwe. In FOCS, pages 97–106, 2011. David Chaum, Claude Crépeau, and Ivan Damgård. Multiparty unconditionally secure protocols (extended abstract). In STOC, pages 11–19, 1988. Seung Geol Choi, Jonathan Katz, Ranjit Kumaresan, and Carlos Cid. Multi-user non-interactive verifiable computation. ACITA Conference, 2012. Seung Geol Choi, Jonathan Katz, Ranjit Kumaresan, and Carlos Cid. Multi-client non-interactive verifiable computation. In TCC, pages 499–518, 2013. Kai-Min Chung, Yael Tauman Kalai, and Salil P. Vadhan. Improved delegation of computation using fully homomorphic encryption. In CRYPTO, pages 483–501, 2010. Ivan Damgård, Sebastian Faust, and Carmit Hazay. Secure two-party computation with low communication. In TCC, pages 54–74, 2012. [Gen09] Craig Gentry. Fully homomorphic encryption using ideal lattices. In STOC, pages 169–178, 2009. [GGP10] [GGPR12] [GKR08] [GLR11] [GMW87] [Gro10] [GW11] [HPS98] [Kil92] Rosario Gennaro, Craig Gentry, and Bryan Parno. Non-interactive verifiable computing: Outsourcing computation to untrusted workers. In CRYPTO, pages 465–482, 2010. Rosario Gennaro, Craig Gentry, Bryan Parno, and Mariana Raykova. Quadratic span programs and succinct nizks without pcps. IACR Cryptology ePrint Archive, 2012:215, 2012. Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: interactive proofs for muggles. In STOC, pages 113–122, 2008. Shafi Goldwasser, Huijia Lin, and Aviad Rubinstein. Delegation of computation without rejection problem from designated verifier CS-proofs. IACR Cryptology ePrint Archive, 2011:456, 2011. Oded Goldreich, Silvio Micali, and Avi Wigderson. How to play any mental game or a completeness theorem for protocols with honest majority. In STOC, pages 218–229, 1987. Jens Groth. Short pairing-based non-interactive zero-knowledge arguments. In ASIACRYPT, pages 321–340, 2010. Craig Gentry and Daniel Wichs. Separating succinct non-interactive arguments from all falsifiable assumptions. In STOC, pages 99–108, 2011. Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman. Ntru: A ring-based public key cryptosystem. In ANTS, pages 267–288, 1998. Joe Kilian. A note on efficient zero-knowledge proofs and arguments (extended abstract). In STOC, pages 723–732, 1992. 14 11. How to Delegate Secure Multiparty Computation to the Cloud
[KMR11] [KMR12] [Lip12] [LTV12] Seny Kamara, Payman Mohassel, and Mariana Raykova. Outsourcing multi-party computation. IACR Cryptology ePrint Archive, 2011:272, 2011. Seny Kamara, Payman Mohassel, and Ben Riva. Salus: a system for server-aided secure function evaluation. In ACM Conference on Computer and Communications Security, pages 797–808, 2012. Helger Lipmaa. Progression-free sets and sublinear pairing-based non-interactive zero-knowledge arguments. In TCC, volume 7194 of Lecture Notes in Computer Science, pages 169–189. Springer, 2012. Adriana López-Alt, Eran Tromer, and Vinod Vaikuntanathan. On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In STOC, pages 1219–1234, 2012. [Mic94] Silvio Micali. Cs proofs (extended abstracts). In FOCS, pages 436–453, 1994. [Nao89] Moni Naor. Bit commitment using pseudo-randomness. In CRYPTO, pages 128–136, 1989. [Nao03] Moni Naor. On cryptographic assumptions and challenges. In CRYPTO, pages 96–109, 2003. [PRV12] [Reg05] [Yao82] Bryan Parno, Mariana Raykova, and Vinod Vaikuntanathan. How to delegate and verify in public: Verifiable computation from attribute-based encryption. In TCC, pages 422–439, 2012. Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. In STOC, pages 84–93, 2005. Andrew Chi-Chih Yao. Protocols for secure computations (extended abstract). In FOCS, pages 160–164, 1982. A Related works Interactive Solutions. Goldwasser et al. [GKR08] show how to build an interactive proof between a client and a server to verify arbitrary polynomial time computations in almost linear time. Because of the interactive nature, this protocol is not suited to the multi-client case (as we discussed above this would require the clients to be all present and interacting with the server during the computation – our model enforces a single message exchanged between server and client during the online phase) 2 . SNARGs. A class of solutions is based on succint non-interactive arguments (or SNARGs): (computationally sound [BCC88]) proofs that are very short and very efficient to verify, regardless of the complexity of the function being evaluated. Solutions of this type are usually constructed using Probabilistically Checkable Proofs (PCPs), long proofs that the verifier can check in only very few places (in particular only a constant number of bits of the proofs are needed for NP languages). Kilian [Kil92] showed how to use PCPs to construct interactive succint arguments by committing to the entire PCP string using a Merkle tree. Micali [Mic94] removed the interaction by use of a random oracle. Recent work [BCCT12, GLR11, DFH12] has replaced the random oracle with an “extractable collision-resistant hash functions” (ECRHs), a non-falsifiable [Nao03], assumption that any algorithm that computes an image of the ECRH must ”know” the corresponding pre-image. There are alternative constructions of SNARGs based on different forms of arithmetization of Boolean computations used together with cryptographic constructions based on bilinear maps (e.g. [Gro10, Lip12, GGPR12]). Those protocols also rely on non-falsifiable ”knowledge” assumptions over the cryptographic groups used in the 2 A non-interactive argument for a restricted class of functions is also presented in [GKR08]. We did not investigate if this could be turned into a multi-client non-interactive protocol (though we suspect it could, when coupled with an FHE), because the focus of this paper is a general solution for arbitrary polynomial computations. 15 11. How to Delegate Secure Multiparty Computation to the Cloud
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AFRL-RI-RS-TR-2013-191 ADVANCED HOM
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REPORT DOCUMENTATION PAGE Form Appr
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12. An Equational Approach to Secur
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1.0 Executive Summary The ultimate
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The PROCEED program efforts are bro
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Towards the end of the project we i
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feature of the framework is that it
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4.3 Publications Figure 1: A block
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encrypted data. We introduce a prec
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present an attack showing that a pa
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alternatives: mis (based on mutual
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5.0 Conclusions and Recommendations
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[14] Bogdanov, Andrej, and Lee, Chi
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8.0 List of Symbols, Abbreviations,
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Public Affairs Approvals for papers
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Contents 1 Introduction 1 1.1 Our M
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t j = P (a j ), and finally interpo
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apparent problem by replacing the M
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As noted above, there is a multilin
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f(x 1 , . . . , x n ) ∈ F and eve
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Translation. Pick up from the publi
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Lemma 4. Let prime p = ω(N 2 ). Th
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[vDGHV10] Marten van Dijk, Craig Ge
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have u 2 − v 2 = 1 → (u − v)(
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inary), and we want to compute g c
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Fully Homomorphic Encryption withou
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scheme have bit length Ω(D) to all
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fast. Thus, the actual magnitude of
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Definition 3 (B-bounded distributio
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achieve 2 λ security against known
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Proof. 〈c 2 , s 2 〉 = BitDecomp
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• FHE.Add(pk, c 1 , c 2 ): Takes
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The computation needed to compute t
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If a and b are large enough, B domi
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5.1.2 How Batching Works Deploying
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We have the following lemma. Lemma
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Unpack(c, {τ σi→1 (s 1 )→s 2
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Since ‖e q1 ‖ < r q1 ,in, e q1
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[22] Damien Stehlé and Ron Steinfe
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1 Introduction Fully homomorphic en
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encryptions of the same message usi
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and Tromer [CT10] in a completely d
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is negligible in k, where Expt CPA
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2.3 Non-Interactive Simulation-Soun
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is defined as (state 1 , m 1 , . .
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scheme has almost-all-keys perfect
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The algorithm S 1 . The algorithm S
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The distribution D 6 . This distrib
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We resolve this issue using the app
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the number of common reference stri
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• Homomorphic evaluation: The hom
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The number of repeated homomorphic
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[Gro04] [Gro10] J. Groth. Rerandomi
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Trapdoors for Lattices: Simpler, Ti
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mentioned, two central objects are
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Dimension m Quality s Length β ≈
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defined by G (or the semi-random ma
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1.4 Other Related Work Concrete par
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Cryptographic problems. For β > 0,
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andom over G m s , for uniformly ra
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3 Search to Decision Reduction Here
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• Preimage sampling for f G (x) =
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Note that for x ∈ {0, . . . , q
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The offline ‘bucketing’ approac
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G is primitive, the tag H in the ab
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conditional distribution of R given
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and parallelism of Algorithm 3 come
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is a coset of the lattice Λ = L( [
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scheme is su-scma-secure if Adv su-
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a discrete Gaussian with parameter
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• G ∈ Z n×nk q is a gadget mat
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must return this e (but possibly di
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[Gen09b] C. Gentry. Fully homomorph
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[vGHV10] M. van Dijk, C. Gentry, S.
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Contents 1 Introduction 1 2 Backgro
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1 Introduction In his breakthrough
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Then in Section 3 we describe our
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In some more detail, the BGV key sw
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itself has low norm. In BGV [5] thi
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3.4 Randomized Multiplication by Co
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One subtle implementation detail to
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ciphertexts. Producing the encrypte
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[24] Benny Pinkas, Thomas Schneider
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A.4 Sampling From A q At various po
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To maintain the noise estimate, the
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variance σ 2 φ(m), so 2d 2 (ζ)ɛ
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We stress that the only place where
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C.1.1 LWE with Sparse Key The analy
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The Smallest Modulus. After evaluat
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down by a factor of p t . Let’s r
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to apply a map κ i we express it a
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1 Introduction Fully homomorphic en
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4. No restrictions on the secret ke
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We point out that an alternative so
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Recall that GapSVP γ is the (promi
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Proof. By definition ⟨c, (1, s)
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• SI-HE.Enc pk (m): Identical to
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Proof of Theorem 4.2. Consider the
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We can now plug in what we know abo
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In particular (recall that E < ⌊q
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[Reg03] Oded Regev. New lattice bas
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1 Introduction An encryption scheme
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need to have errors. Since the expe
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Again we allow some “slackness”
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Learning Linear Functions. the clas
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Let us first outline the proof of T
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P ′ . 4 Namely H n ′ :n = P ′
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In conclusion, we get a sub-scheme
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[BV11b] [Gen09] [Gen10] [GH11] [GHS
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Quantum-Secure Message Authenticati
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the success probability away from 1
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Functions will be denoted by capito
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Quantum chosen message queries. In
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Since the w are the possible result
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number of distinct component values
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4.1 A Tight Upper Bound Theorem 4.1
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The algorithm is similar to the alg
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As we have already seen, for expone
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where S i (m) = (R i (m), PRF(k, (R
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To prove this theorem, we treat Y a
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Now we use this attack to obtain an
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Using this lemma we show that (3q +
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[KK11] Daniel M. Kane and Samuel A.
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1 Introduction Cloud storage system
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subset of the server’s storage, t
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security does not suffice. The main
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Static Data. PoR and PDP schemes fo
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For any efficient adversarial serve
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Overview of Construction. Let (Enc,
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means that one of the audit protoco
Page 261 and 262: Now we claim that an execution of E
Page 263 and 264: having higher capacity. The tables
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Page 267 and 268: j-th level table again. With this o
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Page 271 and 272: A Simple Dynamic PoR with Square-Ro
Page 273 and 274: from Section 6 that is NRPH secure.
Page 275 and 276: The LWE problem was introduced in t
Page 277 and 278: 1.2 Techniques and Comparison to Re
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Page 281 and 282: Proof. Let A be an algorithm runnin
Page 283 and 284: that 1 + ɛ = (1 + ɛ 1 )(1 + ɛ 2
Page 285 and 286: Because we are concerned with small
Page 287 and 288: For each i, since gcd(z i , q) = 1
Page 289 and 290: Proof. Let ω n = ω( √ log n) sa
Page 291 and 292: Theorem 3.8. Let m, n be integers,
Page 293 and 294: σ · O( √ m), except with probab
Page 295 and 296: k = n/2, and it suffices to set the
Page 297 and 298: [23] C. Peikert and B. Waters. Loss
Page 299 and 300: 1 Introduction Consider the followi
Page 301 and 302: In the online phase clients individ
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Page 307 and 308: I. Pre-processing phase. In this ph
Page 309 and 310: Online Phase: 1. Each client D i ge
Page 311: 2n ciphertexts in order to be sure
Page 315 and 316: For any set of adversarial parties
Page 317 and 318: C.5 Secure Computation We make use
Page 319 and 320: Experiment H 4 . This experiment is
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Page 329 and 330: 2 Distributed Systems, Composition,
Page 331 and 332: [G | H](y) = (w, x): z = y w = y x[
Page 333 and 334: infinite chains, such as (M ∗ ;
Page 335 and 336: 2.3 Multi-party computation, securi
Page 337 and 338: BCast(x) = (y 1 , . . . , y n ): y
Page 339 and 340: Sim(y A , s A ) = (y ′ A , r A):
Page 341 and 342: WCast(x ′ , w) = (y 1 ′ , . . .
Page 343 and 344: s ′ 0 s ′ A r ′ A y ′ H s
Page 345 and 346: p r ′ A Net’ s ′ r ′ H r A
Page 347 and 348: Notice that we have already underta
Page 349 and 350: simpler protocol description. For t
Page 351 and 352: [29] Andrew Chi-Chih Yao. Protocols
Page 353 and 354: 2 Mihir Bellare, Stefano Tessaro, a
Page 361 and 362: 10 Mihir Bellare, Stefano Tessaro,
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12 Mihir Bellare, Stefano Tessaro,
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Multi-Instance Security and its App
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Multi-instance Security and Passwor
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a hash-of-hash construction: H 2 (M
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guisher queries (our lower bounds r
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in so doing they accidentally intro
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of D is defined as [ [ ] AdvH[P],R,
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main POW H[P],n,l1 : P 1 P 2 X 2
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now,thinkforconvenienceofw = l).The
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aroundO(q 2 )queries.Ideally, wewou
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HMAC l (K,Y 0) Y 0 F K Y 0 ′ G K
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This material is based on research
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24. Ari Juels and John G. Brainard.
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Contents 1 The BGV Homomorphic Encr
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‖a‖ canon 2 . The basic operati
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EncryptedArray/EncrytedArrayMod2r R
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g i ’s and h i ’s in one list,
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2.6 IndexSet and IndexMap: Sets and
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turn, are sometimes partitioned fur
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DoubleCRT& operator-=(const ZZ &num
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homomorphic automorphism operation
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For reasons that will be discussed
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where D i is the size of the i’th
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When key-switching a ciphertext ⃗
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constant (i.e. |poly(τ m )| 2 ), o
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ool isCorrect() const; and double l
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For the noise estimate in the new c
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The first time that ImportSecKey is
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EncryptedArray(const FHEcontext& co
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The MUX operation is implemented by
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5.1 Homomorphic Operations over GF
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EncryptedArrayMod2r ea(context); lo
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H/2m each and 0 with probability 1
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So now consider the inner sum in (7
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