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[MG02] [Mic02] [MM11] [MR04] [MR09] [Pei09a] [Pei09b] D. Micciancio and S. Goldwasser. Complexity of Lattice Problems: a cryptographic perspective, volume 671 of The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, Massachusetts, 2002. D. Micciancio. Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Computational Complexity, 16(4):365–411, 2007. Preliminary version in FOCS 2002. D. Micciancio and P. Mol. Pseudorandom knapsacks and the sample complexity of LWE search-to-decision reductions. In CRYPTO, pages 465–484. 2011. D. Micciancio and O. Regev. Worst-case to average-case reductions based on Gaussian measures. SIAM J. Comput., 37(1):267–302, 2007. Preliminary version in FOCS 2004. D. Micciancio and O. Regev. Lattice-based cryptography. In Post Quantum Cryptography, pages 147–191. Springer, February 2009. C. Peikert. Bonsai trees (or, arboriculture in lattice-based cryptography). Cryptology ePrint Archive, Report 2009/359, July 2009. http://eprint.iacr.org/. C. Peikert. Public-key cryptosystems from the worst-case shortest vector problem. In STOC, pages 333–342. 2009. [Pei10] C. Peikert. An efficient and parallel Gaussian sampler for lattices. In CRYPTO, pages 80–97. 2010. [PR06] [PV08] C. Peikert and A. Rosen. Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices. In TCC, pages 145–166. 2006. C. Peikert and V. Vaikuntanathan. Noninteractive statistical zero-knowledge proofs for lattice problems. In CRYPTO, pages 536–553. 2008. [PVW08] C. Peikert, V. Vaikuntanathan, and B. Waters. A framework for efficient and composable oblivious transfer. In CRYPTO, pages 554–571. 2008. [PW08] [Reg05] [RS10] [Rüc10] [ST01] C. Peikert and B. Waters. Lossy trapdoor functions and their applications. In STOC, pages 187–196. 2008. O. Regev. On lattices, learning with errors, random linear codes, and cryptography. J. ACM, 56(6):1–40, 2009. Preliminary version in STOC 2005. M. Rückert and M. Schneider. Selecting secure parameters for lattice-based cryptography. Cryptology ePrint Archive, Report 2010/137, 2010. http://eprint.iacr.org/. M. Rückert. Strongly unforgeable signatures and hierarchical identity-based signatures from lattices without random oracles. In PQCrypto, pages 182–200. 2010. A. Shamir and Y. Tauman. Improved online/offline signature schemes. In CRYPTO, pages 355–367. 2001. [Ver11] R. Vershynin. Introduction to the non-asymptotic analysis of random matrices, January 2011. Available at http://www-personal.umich.edu/˜romanv/papers/ non-asymptotic-rmt-plain.pdf, last accessed 4 Feb 2011. 40 4. Trapdoors for Lattices
[vGHV10] M. van Dijk, C. Gentry, S. Halevi, and V. Vaikuntanathan. Fully homomorphic encryption over the integers. In EUROCRYPT, pages 24–43. 2010. [Wat05] B. Waters. Efficient identity-based encryption without random oracles. In EUROCRYPT, pages 114–127. 2005. 41 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
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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 and 144: must return this e (but possibly di
Page 145: [Gen09b] C. Gentry. Fully homomorph
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
Page 195 and 196: 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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