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When Homomorphism Becomes a Liability Zvika Brakerski ∗ Abstract We show that an encryption scheme cannot have a simple decryption function and be homomorphic at the same time, even with added noise. Specifically, if a scheme can homomorphically evaluate the majority function, then its decryption cannot be weakly-learnable (in particular, linear), even if large decryption error is allowed. (In contrast, without homomorphism, such schemes do exist and are presumed secure, e.g. based on LPN.) An immediate corollary is that known schemes that are based on the hardness of decoding in the presence of low hamming-weight noise cannot be fully homomorphic. This applies to known schemes such as LPN-based symmetric or public key encryption. Using these techniques, we show that the recent candidate fully homomorphic encryption, suggested by Bogdanov and Lee (ePrint ’11, henceforth BL), is insecure. In fact, we show two attacks on the BL scheme: One that uses homomorphism, and another that directly attacks a component of the scheme. ∗ Stanford University, zvika@stanford.edu. Supported by a Simons Postdoctoral Fellowship and by DARPA. 7. When Homomorphism Becomes a Liability
1 Introduction An encryption scheme is called homomorphic if there is an efficient transformation that given Enc(m) for some message m, and a function f, produces Enc(f(m)) using only public information. A scheme that is homomorphic w.r.t all efficient f is called fully homomorphic (FHE). Homomorphic encryption is a useful tool in both theory and practice and is extensively researched in recent years (see [Vai11] for survey), and a few candidates for full homomorphism are known. Most of these candidates [Gen09, Gen10, SV10, BV11a, BV11b, GH11, BGV12, GHS12, Bra12] are based (either explicitly or implicitly) on lattice assumptions (the hardness of approximating short vectors in certain lattices). In particular, the learning with errors (LWE) assumption proved to be very useful in the design of such schemes. The one notable exception is [vDGHV10], but even that could be thought of as working over an appropriately defined lattice over the integers. An important open problem is, therefore, to diversify and base fully homomorphic encryption on different assumptions (so as to not put all the eggs in one basket). One appealing direction is to try to use the learning parity with noise (LPN) problem, which is very similar in syntax to LWE: Making a vast generalization, LWE can be interpreted as a decoding problem for a linear code, where the noise comes from a family of low norm vectors. Namely, each coordinate in the code suffers from noise, but this noise is relatively small (this requires that the code is defined over a large alphabet). The LPN assumption works over the binary alphabet and requires that the noise has low hamming weight, namely that only a small number of coordinates are noisy, but in these coordinates the noise amplitude can be large. While similar in syntax, a direct connection between these two types of assumptions is not known. While an LPN-based construction is not known, recently Bogdanov and Lee [BL11] presented a candidate, denoted by BL throughout this manuscript, that is based on a different low hammingweight decoding problem: They consider a carefully crafted code over a large alphabet and assume that decoding in the presence of low-hamming-weight noise is hard. In this work, we show that not only that BL’s construction is insecure, but rather the entire approach of constructing code based homomorphic encryption analogously to the LWE construction cannot work. We stress that we don’t show that FHE cannot be based on LPN (or other code based assumptions), but rather that the decryption algorithm of such scheme cannot take the naïve form. (In particular this applies to the attempt to add homomorphism to schemes such as [Ale03, GRS08, ACPS09].) 1.1 Our Results Our main result shows that encryption schemes with learnable decryption functions cannot be homomorphic, even if large decryption error is allowed. In particular, such schemes cannot evaluate the majority function. This extends the result of Kearns and Valiant [KV94] (slightly extended by Klivans and Sherstov [KS09]) that learnability breaks security for schemes with negligible decryption error. In other words, homomorphic capabilities can sometimes make noisy learning become no harder than noiseless learning. We use a simplified notion of learning, which essentially requires that given polynomially many labeled samples (from an arbitrary distribution), the learner’s hypothesis correctly computes the label for the next sample with probability, say, 0.9. We show that this notion, that we call sclearning, is equivalent to weak learning defined in [KV94]. This allows us to prove the following theorem (in Section 3). 1 7. When Homomorphism Becomes a Liability
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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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Page 155 and 156: In some more detail, the BGV key sw
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Page 163 and 164: ciphertexts. Producing the encrypte
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Page 167 and 168: A.4 Sampling From A q At various po
Page 169 and 170: To maintain the noise estimate, the
Page 171 and 172: variance σ 2 φ(m), so 2d 2 (ζ)ɛ
Page 173 and 174: We stress that the only place where
Page 175 and 176: C.1.1 LWE with Sparse Key The analy
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Page 183 and 184: 1 Introduction Fully homomorphic en
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Page 187 and 188: We point out that an alternative so
Page 189 and 190: Recall that GapSVP γ is the (promi
Page 191 and 192: Proof. By definition ⟨c, (1, s)
Page 193 and 194: • SI-HE.Enc pk (m): Identical to
Page 195 and 196: Proof of Theorem 4.2. Consider the
Page 197 and 198: We can now plug in what we know abo
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Page 201: [Reg03] Oded Regev. New lattice bas
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Page 209 and 210: Learning Linear Functions. the clas
Page 211 and 212: Let us first outline the proof of T
Page 213 and 214: P ′ . 4 Namely H n ′ :n = P ′
Page 215 and 216: In conclusion, we get a sub-scheme
Page 217 and 218: [BV11b] [Gen09] [Gen10] [GH11] [GHS
Page 219 and 220: Quantum-Secure Message Authenticati
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Page 223 and 224: Functions will be denoted by capito
Page 225 and 226: Quantum chosen message queries. In
Page 227 and 228: Since the w are the possible result
Page 229 and 230: number of distinct component values
Page 231 and 232: 4.1 A Tight Upper Bound Theorem 4.1
Page 233 and 234: The algorithm is similar to the alg
Page 235 and 236: As we have already seen, for expone
Page 237 and 238: where S i (m) = (R i (m), PRF(k, (R
Page 239 and 240: To prove this theorem, we treat Y a
Page 241 and 242: Now we use this attack to obtain an
Page 243 and 244: Using this lemma we show that (3q +
Page 245 and 246: [KK11] Daniel M. Kane and Samuel A.
Page 247 and 248: 1 Introduction Cloud storage system
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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
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Now we claim that an execution of E
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having higher capacity. The tables
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OWrite(i 1 , . . . , i t ; v 1 , .
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j-th level table again. With this o
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and distribute reprints for Governm
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A Simple Dynamic PoR with Square-Ro
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from Section 6 that is NRPH secure.
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The LWE problem was introduced in t
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1.2 Techniques and Comparison to Re
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(large) uniform errors as in [10],
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Proof. Let A be an algorithm runnin
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that 1 + ɛ = (1 + ɛ 1 )(1 + ɛ 2
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Because we are concerned with small
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For each i, since gcd(z i , q) = 1
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Proof. Let ω n = ω( √ log n) sa
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Theorem 3.8. Let m, n be integers,
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σ · O( √ m), except with probab
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k = n/2, and it suffices to set the
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[23] C. Peikert and B. Waters. Loss
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1 Introduction Consider the followi
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In the online phase clients individ
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the clients jointly run a secure co
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to compute F from scratch. The work
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I. Pre-processing phase. In this ph
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Online Phase: 1. Each client D i ge
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2n ciphertexts in order to be sure
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[KMR11] [KMR12] [Lip12] [LTV12] Sen
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For any set of adversarial parties
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C.5 Secure Computation We make use
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Experiment H 4 . This experiment is
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of A only in the case when A guesse
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leading to concrete implementations
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y modeling each block by an interac
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z i P 2 Q 1 y i P 1 s 1 r 1 x i w i
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2 Distributed Systems, Composition,
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[G | H](y) = (w, x): z = y w = y x[
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infinite chains, such as (M ∗ ;
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2.3 Multi-party computation, securi
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BCast(x) = (y 1 , . . . , y n ): y
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Sim(y A , s A ) = (y ′ A , r A):
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WCast(x ′ , w) = (y 1 ′ , . . .
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s ′ 0 s ′ A r ′ A y ′ H s
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p r ′ A Net’ s ′ r ′ H r A
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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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