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[ACPS09] B. Applebaum, D. Cash, C. Peikert, and A. Sahai. Fast cryptographic primitives and circularsecure encryption based on hard learning problems. In CRYPTO, pages 595–618. 2009. [Ajt96] M. Ajtai. Generating hard instances of lattice problems. Quaderni di Matematica, 13:1–32, 2004. Preliminary version in STOC 1996. [Ajt99] M. Ajtai. Generating hard instances of the short basis problem. In ICALP, pages 1–9. 1999. [AP09] [Bab85] [Ban93] J. Alwen and C. Peikert. Generating shorter bases for hard random lattices. Theory of Computing Systems, 48(3):535–553, April 2011. Preliminary version in STACS 2009. L. Babai. On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1–13, 1986. Preliminary version in STACS 1985. W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(4):625–635, 1993. [BCHK07] D. Boneh, R. Canetti, S. Halevi, and J. Katz. Chosen-ciphertext security from identity-based encryption. SIAM J. Comput., 36(5):1301–1328, 2007. [BFKL93] A. Blum, M. L. Furst, M. J. Kearns, and R. J. Lipton. Cryptographic primitives based on hard learning problems. In CRYPTO, pages 278–291. 1993. [BGV11] [Boy10] [BV11a] [BV11b] Z. Brakerski, C. Gentry, and V. Vaikuntanathan. Fully homomorphic encryption without bootstrapping. Cryptology ePrint Archive, Report 2011/277, 2011. http://eprint.iacr.org/. X. Boyen. Lattice mixing and vanishing trapdoors: A framework for fully secure short signatures and more. In Public Key Cryptography, pages 499–517. 2010. Z. Brakerski and V. Vaikuntanathan. Efficient fully homomorphic encryption from (standard) LWE. In FOCS. 2011. To appear. Z. Brakerski and V. Vaikuntanathan. Fully homomorphic encryption from ring-LWE and security for key dependent messages. In CRYPTO, pages 505–524. 2011. [CHKP10] D. Cash, D. Hofheinz, E. Kiltz, and C. Peikert. Bonsai trees, or how to delegate a lattice basis. In EUROCRYPT, pages 523–552. 2010. [CN11] Y. Chen and P. Q. Nguyen. BKZ 2.0: Simulation and better lattice security estimates. In ASIACRYPT. 2011. To appear. [DDN00] D. Dolev, C. Dwork, and M. Naor. Nonmalleable cryptography. SIAM J. Comput., 30(2):391–437, 2000. [DF94] [Feh98] Y. Desmedt and Y. Frankel. Perfect homomorphic zero-knowledge threshold schemes over any finite abelian group. SIAM J. Discrete Math., 7(4):667–679, 1994. S. Fehr. Span Programs over Rings and How to Share a Secret from a Module. Master’s thesis, ETH Zurich, Institute for Theoretical Computer Science, 1998. [Gen09a] C. Gentry. A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University, 2009. crypto.stanford.edu/craig. 38 4. Trapdoors for Lattices
[Gen09b] C. Gentry. Fully homomorphic encryption using ideal lattices. In STOC, pages 169–178. 2009. [GGH97] [GH11] [GHV10] [GKV10] O. Goldreich, S. Goldwasser, and S. Halevi. Public-key cryptosystems from lattice reduction problems. In CRYPTO, pages 112–131. 1997. C. Gentry and S. Halevi. Fully homomorphic encryption without squashing using depth-3 arithmetic circuits. In FOCS. 2011. To appear. C. Gentry, S. Halevi, and V. Vaikuntanathan. A simple BGN-type cryptosystem from LWE. In EUROCRYPT, pages 506–522. 2010. S. D. Gordon, J. Katz, and V. Vaikuntanathan. A group signature scheme from lattice assumptions. In ASIACRYPT, pages 395–412. 2010. [GN08] N. Gama and P. Q. Nguyen. Predicting lattice reduction. In EUROCRYPT, pages 31–51. 2008. [GPV08] [HW09] C. Gentry, C. Peikert, and V. Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In STOC, pages 197–206. 2008. S. Hohenberger and B. Waters. Short and stateless signatures from the RSA assumption. In CRYPTO, pages 654–670. 2009. [Kle00] P. N. Klein. Finding the closest lattice vector when it’s unusually close. In SODA, pages 937–941. 2000. [KMO10] E. Kiltz, P. Mohassel, and A. O’Neill. Adaptive trapdoor functions and chosen-ciphertext security. In EUROCRYPT, pages 673–692. 2010. [KR00] H. Krawczyk and T. Rabin. Chameleon signatures. In NDSS. 2000. [LM06] [LM08] [LM09] V. Lyubashevsky and D. Micciancio. Generalized compact knapsacks are collision resistant. In ICALP (2), pages 144–155. 2006. V. Lyubashevsky and D. Micciancio. Asymptotically efficient lattice-based digital signatures. In TCC, pages 37–54. 2008. V. Lyubashevsky and D. Micciancio. On bounded distance decoding, unique shortest vectors, and the minimum distance problem. In CRYPTO, pages 577–594. 2009. [LMPR08] V. Lyubashevsky, D. Micciancio, C. Peikert, and A. Rosen. SWIFFT: A modest proposal for FFT hashing. In FSE, pages 54–72. 2008. [LP11] [LPR10] [Lyu08] R. Lindner and C. Peikert. Better key sizes (and attacks) for LWE-based encryption. In CT-RSA, pages 319–339. 2011. V. Lyubashevsky, C. Peikert, and O. Regev. On ideal lattices and learning with errors over rings. In EUROCRYPT, pages 1–23. 2010. V. Lyubashevsky. Lattice-based identification schemes secure under active attacks. In Public Key Cryptography, pages 162–179. 2008. 39 4. Trapdoors for Lattices
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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
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
Page 99 and 100: the number of common reference stri
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
Page 127 and 128: The offline ‘bucketing’ approac
Page 129 and 130: G is primitive, the tag H in the ab
Page 131 and 132: conditional distribution of R given
Page 133 and 134: and parallelism of Algorithm 3 come
Page 135 and 136: is a coset of the lattice Λ = L( [
Page 137 and 138: scheme is su-scma-secure if Adv su-
Page 139 and 140: a discrete Gaussian with parameter
Page 141 and 142: • G ∈ Z n×nk q is a gadget mat
Page 143: must return this e (but possibly di
Page 147 and 148: [vGHV10] M. van Dijk, C. Gentry, S.
Page 149 and 150: Contents 1 Introduction 1 2 Backgro
Page 151 and 152: 1 Introduction In his breakthrough
Page 153 and 154: Then in Section 3 we describe our
Page 155 and 156: In some more detail, the BGV key sw
Page 157 and 158: itself has low norm. In BGV [5] thi
Page 159 and 160: 3.4 Randomized Multiplication by Co
Page 161 and 162: One subtle implementation detail to
Page 163 and 164: ciphertexts. Producing the encrypte
Page 165 and 166: [24] Benny Pinkas, Thomas Schneider
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
Page 177 and 178: The Smallest Modulus. After evaluat
Page 179 and 180: down by a factor of p t . Let’s r
Page 181 and 182: to apply a map κ i we express it a
Page 183 and 184: 1 Introduction Fully homomorphic en
Page 185 and 186: 4. No restrictions on the secret ke
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
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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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