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How to Delegate Secure Multiparty Computation to the Cloud Abstract We study the problem of verifiable computation in the presence of many clients who rely on a server to perform computations over inputs privately held by each client. This generalizes the single-client model for verifiable outsourced computation previously studied in the literature. We put forward a computational model and strong simulation-based security for this task. We then present a new protocol that allow the clients to securely outsource an arbitrary polynomial-time computation over privately held inputs to a powerful server. At the end, the clients will be assured that the result of the computation is correct, while at the same time protecting their data from the server and each other. Our new protocol satisfies the crucial efficiency requirement of outsourced computation where the work of the client is substantially smaller than what is required to compute the function. We use the Gennaro et al. amortized model, where the clients are allowed to invest in a one-time computationally expensive preprocessing phase. Our protocol is secure in the real/ideal paradigm even when dishonest clients can collude with the server in order to learn honest party’s inputs or in order to maliciously change the output of the computation. Keywords: verifiable computation, secure multi-party computation 11. How to Delegate Secure Multiparty Computation to the Cloud
1 Introduction Consider the following scenario: a set of computationally weak devices holding private inputs, wish to jointly compute a function F over those inputs. Each device does not have the power to compute F on its own, let alone engaging in a secure multiparty computation protocol such as [Yao82, GMW87, BOGW88, CCD88] for computing F . They can however access the services of a “computation provider” who can “help” them compute F . To maintain the privacy of their input, the clients need to engage in a cryptographic protocol where the provider does the bulk of the computation (i.e., computes F ), while the computation and communication of each client is “low” (in particular, less than the time it takes to compute F ). Nevertheless the clients must be able to verify the correctness of the output of this protocol, even under the assumption that some corrupted clients might cooperate with a malicious provider to fool them into accepting an incorrect ouput or to learn their input. One can think of this problem, called multi-client verifiable computation or multi-client delegation of computation, as a secure multi-party computation protocol between the clients and the provider (the cloud service), where however only the provider’s work is allowed to be proportional to the complexity of the function being computed (the function that computes the joint statistics). In this paper, we solve the above problem, by relying on only standard polynomial time cryptographic assumptions. We present a protocol that allows many computationally weak clients to securely outsource a computation over privately held inputs to a powerful server, in the presence of the most powerful adversarial model that can be considered, and by minimizing the “on-line” computation by the clients. The round complexity being of paramount importance in this line of work, we follow the convention of obtaining a solution in which the clients delegates the computation non-interactively (i.e., the clients and the server exchange a single message). While a lot of work has been devoted to secure outsourced computation in the case of a single client interacting with a single server (see for example [Mic94, GKR08, GGP10, CKV10, AIK10]), the research effort for the multi-client case is still in the preliminary stages with very few works that consider much weaker models of security (we shall discuss these works in detail later on). 1.1 Our Model Before we state our main results, we shall first take a closer look at the model in which we work in. In a nutshell, we obtain our results a) based on standard cryptographic hardness assumptions; b) in the strongest adversarial model - i.e., simulation based security in the ideal/real paradigm when malicious clients may collude with the server; and c) with minimal communication between the clients and that too only when verifying the results of the computation. In explaining our model, we consider three important design principles that influence our choice - first, the cryptographic hardness assumption that we make; second, the corruption model (which parties can the adversary corrupt), and finally, the communication model (how much do clients need to interact with each other and with the server). Hardness assumptions: standard cryptographic assumptions. Following the standard convention in cryptography, we are interested in constructing multi-client verifiable computation protocols based on standard cryptographic assumptions (i.e., without resorting to random oracles or non-falsifiable hardness assumptions). Furthermore, we are interested in obtaining solutions where the interaction between the clients and the server is minimal, i.e., only one message is sent in each direction between the client and the server. We note that obtaining such solutions is a difficult problem even in the single-client setting, exemplified by the small number of known solutions [GGP10, CKV10, AIK10]. In particular, all known single-client non-interactive solutions based on standard assumptions work in an “amortized” computational model (also known as the pre-processing model) [GGP10]. In view of the above, in this work, we will also work in (a natural extension of) the pre-processing model, which we discuss later on in this section. Corruption model: simulation based security. As we discussed earlier, it is quite natural to have a situation where one of the clients might collude with a corrupt server in order to either learn something about the honest client’s inputs or to force the output of the computation to some value. Naturally, it would be highly desirable to 1 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.
Page 247 and 248: 1 Introduction Cloud storage system
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Page 251 and 252: security does not suffice. The main
Page 253 and 254: Static Data. PoR and PDP schemes fo
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Page 257 and 258: Overview of Construction. Let (Enc,
Page 259 and 260: 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
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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: [23] C. Peikert and B. Waters. Loss
Page 301 and 302: In the online phase clients individ
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Page 305 and 306: to compute F from scratch. The work
Page 307 and 308: I. Pre-processing phase. In this ph
Page 309 and 310: Online Phase: 1. Each client D i ge
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Page 313 and 314: [KMR11] [KMR12] [Lip12] [LTV12] Sen
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
Page 321 and 322: of A only in the case when A guesse
Page 323 and 324: leading to concrete implementations
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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 ′ , . . .
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Page 345 and 346: p r ′ A Net’ s ′ r ′ H r A
Page 347 and 348: Notice that we have already underta
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simpler protocol description. For t
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[29] Andrew Chi-Chih Yao. Protocols
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2 Mihir Bellare, Stefano Tessaro, a
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10 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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