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We say that an ORAM protocol is pattern hiding if for all efficient adversaries A we have: ∣ ∣Pr[ORAMGame 0 A(λ) = 1] − Pr[ORAMGame 1 A(λ) = 1] ∣ ∣ ≤ negl(λ). Sometimes we also want to achieve a stronger notion of security where we also wish to hide whether each operation is a read or a write. This can be done generically by always first executing a read for the desired location and then executing a write to either just write-back the read value (when we only wanted to do a read) or writing in a new value. C Standard ORAM Security Does not Suffice for PORAM In this section we construct an ORAM that is secure in the usual sense but is not next-read pattern hiding. In fact, we will show something stronger: If the ORAM below were used to instantiate our PORAM scheme then the resulting dynamic PoR scheme is not secure. This shows that some notion of security beyond regular ORAM is necessary for the security PORAM. Counterexample construction. We can take any ORAM scheme (e.g., the one in Section 6 for concreteness) and modify it by “packing” multiple consecutive logical addresses into a single slot of the ORAM. In particular, if the client initializes the modified ORAM (called MORAM within this section) with alphabet Σ = {0, 1} w , it will translate this into initializing the original ORAM with the alphabet Σ n = {0, 1} nw , where each symbol in the modified alphabet “packs” together n symbols of the original alphabet. Assume this is the same n as the codeword length in our PORAM protocol. Whenever the client wants to read some address i using MORAM, the modified scheme looks up where it was packed by computing j = ⌊i/n⌋, uses the original ORAM scheme to execute ORead(j), and then parses the resulting output as (v 0 , . . . , v n−1 ) ∈ Σ n , and returns v i mod n . To write v to address i, MORAM runs ORAM scheme’s ORead(⌊i/n⌋) to get (v 0 , . . . , v n−1 ) as before, then sets v i mod n ← v and writes the data back via ORAM scheme’s OWrite(⌊i/n⌋, (v 0 , . . . , v n−1 )). It is not hard to show that this modified scheme retains standard ORAM security, since it hides which locations are being read/written. We next discuss why this modification causes the MORAM to not be NRPH secure. Consider what happens if the client issues a read for an address, say i = 0, and then is rewound and reads another address that was packed into the same ORAM slot, say i+1. Both operations will cause the client to issue ORead(0). And since our MORAM was deterministic, the client will access exactly same table indices at every level on the server on both runs. But, if these addresses were permuted to not be packed together (e.g., blocks were packed using equivalence classes of their indices ( mod l/n)), then the client will issue ORead commands on different addresses, reading different table positions (with high probability), thus allowing the server to distinguish which case it was in and break NRPH security. This establishes that the modified scheme is not NRPH secure. To see why PORAM is not secure with MORAM, consider an adversary that, after a sequence of many read/write operations, randomly deletes one block of its storage (say, from the lowest level cuckoo table). If this block happens to contain a non-dummy ciphertext that contains actual data (which occurs with reasonable probability), then this attack corresponds to deleting some codeword block in full (because all codeword blocks corresponding to a message block was packed in the same ORAM storage location), even though the server does not necessarily know which one. Therefore, the underlying message block can never be recovered from the attacker. But this adversary can still pass an audit with good probability, because the audit would only catch the adversary if it happened to access the deleted block during its reads either by (1) selecting exactly this location to check during the audit, (2) reading this location in the cuckoo table slot as a dummy read. This happens with relatively low probability, around 1/l, where l is the number of addresses in the client memory. To provide some more intuition, we can also examine why this same attack (deleting a random location in the lowest level cuckoo table) does not break PORAM when instantiated with the ORAM implementation 26 9. Dynamic Proofs of Retrievability via Oblivious RAM
from Section 6 that is NRPH secure. After this attack, the adversary still maintains a good probability of passing a subsequent audit. However, by deleting only a single ciphertext in one of the cuckoo tables, the attacker now deleted only a single codeword symbol, not a full block of n of them. And now we can show that our extractor can still recover enough of the other symbols of the codeword block so that the erasure code will enable recovery of the original data. Of course, the server could start deleting more of the locations in the lowest level cuckoo table, but he cannot selectively target codeword symbols belonging to a single codeword block, since it has no idea where those reside. If he starts to delete too many of them just to make sure a message block is not recoverable, then he will lose his ability to pass an audit. 27 9. Dynamic Proofs of Retrievability via Oblivious RAM
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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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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
Page 249 and 250: subset of the server’s storage, t
Page 251 and 252: security does not suffice. The main
Page 253 and 254: Static Data. PoR and PDP schemes fo
Page 255 and 256: For any efficient adversarial serve
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
Page 265 and 266: OWrite(i 1 , . . . , i t ; v 1 , .
Page 267 and 268: j-th level table again. With this o
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Page 271: A Simple Dynamic PoR with Square-Ro
Page 275 and 276: The LWE problem was introduced in t
Page 277 and 278: 1.2 Techniques and Comparison to Re
Page 279 and 280: (large) uniform errors as in [10],
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
Page 303 and 304: the clients jointly run a secure co
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
Page 311 and 312: 2n ciphertexts in order to be sure
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
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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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