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1.4 Paper Organization The remainder of this paper is organized as follows. In Section 2 we present the basic tools that are used in our constructions. In Section 3 we formalize the notion of targeted malleability. In Sections 4 and 5 we present our constructions. Finally, in Section 6 we discuss possible extensions of our work and several open problems. 2 Preliminaries In this section we present the basic tools that are used in our constructions: public-key encryption and homomorphic encryption, succinct non-interactive arguments, and non-interactive zeroknowledge proofs. 2.1 Public-Key Encryption A public-key encryption scheme is a triplet Π = (KeyGen, Enc, Dec) of probabilistic polynomialtime algorithms, where KeyGen is the key-generation algorithm, Enc is the encryption algorithm, and Dec is the decryption algorithm. The key-generation algorithm KeyGen receives as input the security parameter, and outputs a public key pk and a secret key sk. The encryption algorithm Enc receives as input a public key pk and a message m, and outputs a ciphertext c. The decryption algorithm Dec receives as input a ciphertext c and a secret key sk, and outputs a message m or the symbol ⊥. Functionality. In terms of functionality, in this paper we require the property of almost-all-keys perfect decryption [DNR04], defined as follows: Definition 2.1. A public-key encryption scheme Π = (KeyGen, Enc, Dec) has almost-all-keys perfect decryption if there exists a negligible function ν(k) such that for all sufficiently large k with probability 1 − ν(k) over the choice of (sk, pk) ← KeyGen(1 k ), for any message m it holds that Pr [Dec sk (Enc pk (m)) = m] = 1 , where the probability is taken over the internal randomness of Enc and Dec. We note that Dwork, Naor, and Reingold [DNR04] proposed a general transformation turning any encryption scheme into one that has almost-all-keys perfect decryption. When starting with a scheme that has a very low error probability, their transformation only changes the random bits used by the encryption algorithm, and in our setting this is important as it preserves the homomorphic operations. When starting with a scheme that has a significant error probability, we note that the error probability can be reduced exponentially by encrypting messages under several independently chosen public keys, and decrypting according to the majority. This again preserves the homomorphic operations. Security. In terms of security, we consider the most basic notion of semantic-security against chosen-plaintext attacks, asking that any efficient adversary has only a negligible advantage in distinguishing between encryptions of different messages. This is formalized as follows: Definition 2.2. A public-key encryption scheme Π = (KeyGen, Enc, Dec) is semantically secure against chosen-plaintext attacks if for any probabilistic polynomial-time adversary A = (A 1 , A 2 ) it holds that [ ] [ Adv CPA Π,A(k) def ∣ = ∣Pr Expt CPA Π,A,0(k) = 1 − Pr Expt CPA ∣∣ Π,A,1(k) = 1]∣ 6 3. Targeted Malleability
is negligible in k, where Expt CPA Π,A,b (k) is defined as follows: 1. (sk, pk) ← KeyGen(1 k ). 2. (m 0 , m 1 , state) ← A 1 (1 k , pk) such that |m 0 | = |m 1 |. 3. c ∗ ← Enc pk (m b ). 4. b ′ ← A 2 (c ∗ , state) 5. Output b ′ . Homomorphic encryption. A public-key encryption scheme Π = (KeyGen, Enc, Dec) is homomorphic with respect to a set of efficiently computable functions F if there exists a homomorphic evaluation algorithm HomEval that receives as input a public key pk, an encryption of a message m, and a function f ∈ F, and outputs an encryption of the message f(m). Formally, with overwhelming probability over the choice of (sk, pk) ← KeyGen(1 k ) (as in Definition 2.1), for any ciphertext c such that Dec sk (c) ≠ ⊥ and for any function f ∈ F it holds that Dec sk (HomEval pk (c, f)) = f (Dec sk (c)) with probability 1 over the internal randomness of HomEval and Dec. The main property that is typically required from a homomorphic encryption scheme is compactness, asking that the length of the ciphertext does not trivially grow with the number of repeated homomorphic operations. In our setting, given an upper bound t on the number of repeated homomorphic operations that can be applied to a ciphertext produced by the encryption algorithm, we are interested in minimizing the dependency of the length of the ciphertext on t. An additional property, that we do not consider in this paper, is of function privacy. Informally, this property asks that the homomorphic evaluation algorithm does not reveal which function from the set F it receives as input. We refer the reader to [GHV10] for a formal definition. We note that in our setting, where function privacy is not taken into account, we can assume without loss of generality that the homomorphic evaluation algorithm is deterministic. 2.2 Non-Interactive Extractable Arguments A non-interactive argument system for a language L = ∪ ∪ k∈N L(k) with a witness relation R = k∈N R(k) consists of a triplet of algorithms (CRSGen, P, V), where CRSGen is an algorithm generating a common reference string crs, and P and V are the prover and verifier algorithms, respectively. The prover takes as input a triplet (x, w, crs), where (x, w) ∈ R, and outputs an argument π. The verifier takes as input a triplet (x, π, crs) and either accepts or rejects. In this paper we consider a setting where all three algorithms run in polynomial time, CRSGen and P may be probabilistic, and V is deterministic. We require three properties from such a system. The first property is perfect completeness: for every (x, w, crs) such that (x, w) ∈ R, the prover always generates an argument that is accepted by the verifier. The second property is knowledge extraction: for every efficient malicious prover P ∗ there exists an efficient “knowledge extractor” Ext P ∗, such that whenever P ∗ outputs (x, π) that is accepted by the verifier, Ext P ∗ when given the random coins of P ∗ can in fact produce a witness w such that (x, w) ∈ R with all but a negligible probability. We note that this implies, in particular, soundness against efficient provers. The perfect completeness and knowledge extraction properties are in fact trivial to satisfy: the prover can output the witness w as an argument, and the verifier checks that (x, w) ∈ R (unlike for CS proofs [Mic00, Val08] we do not impose any non-trivial efficiency requirement on the verifier). The third property that we require from the argument system is that of having rather succinct 7 3. Targeted Malleability
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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
Page 33 and 34: t j = P (a j ), and finally interpo
Page 35 and 36: apparent problem by replacing the M
Page 37 and 38: As noted above, there is a multilin
Page 39 and 40: f(x 1 , . . . , x n ) ∈ F and eve
Page 41 and 42: Translation. Pick up from the publi
Page 43 and 44: Lemma 4. Let prime p = ω(N 2 ). Th
Page 45 and 46: [vDGHV10] Marten van Dijk, Craig Ge
Page 47 and 48: have u 2 − v 2 = 1 → (u − v)(
Page 49 and 50: inary), and we want to compute g c
Page 51 and 52: Fully Homomorphic Encryption withou
Page 53 and 54: scheme have bit length Ω(D) to all
Page 55 and 56: fast. Thus, the actual magnitude of
Page 57 and 58: Definition 3 (B-bounded distributio
Page 59 and 60: achieve 2 λ security against known
Page 61 and 62: Proof. 〈c 2 , s 2 〉 = BitDecomp
Page 63 and 64: • FHE.Add(pk, c 1 , c 2 ): Takes
Page 65 and 66: The computation needed to compute t
Page 67 and 68: If a and b are large enough, B domi
Page 69 and 70: 5.1.2 How Batching Works Deploying
Page 71 and 72: We have the following lemma. Lemma
Page 73 and 74: Unpack(c, {τ σi→1 (s 1 )→s 2
Page 75 and 76: Since ‖e q1 ‖ < r q1 ,in, e q1
Page 77 and 78: [22] Damien Stehlé and Ron Steinfe
Page 79 and 80: 1 Introduction Fully homomorphic en
Page 81 and 82: encryptions of the same message usi
Page 83: and Tromer [CT10] in a completely d
Page 87 and 88: 2.3 Non-Interactive Simulation-Soun
Page 89 and 90: is defined as (state 1 , m 1 , . .
Page 91 and 92: scheme has almost-all-keys perfect
Page 93 and 94: The algorithm S 1 . The algorithm S
Page 95 and 96: The distribution D 6 . This distrib
Page 97 and 98: We resolve this issue using the app
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Page 101 and 102: • Homomorphic evaluation: The hom
Page 103 and 104: The number of repeated homomorphic
Page 105 and 106: [Gro04] [Gro10] J. Groth. Rerandomi
Page 107 and 108: Trapdoors for Lattices: Simpler, Ti
Page 109 and 110: mentioned, two central objects are
Page 111 and 112: Dimension m Quality s Length β ≈
Page 113 and 114: defined by G (or the semi-random ma
Page 115 and 116: 1.4 Other Related Work Concrete par
Page 117 and 118: Cryptographic problems. For β > 0,
Page 119 and 120: andom over G m s , for uniformly ra
Page 121 and 122: 3 Search to Decision Reduction Here
Page 123 and 124: • Preimage sampling for f G (x) =
Page 125 and 126: Note that for x ∈ {0, . . . , q
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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
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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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