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[DGHV10] Marten van Dijk, Craig Gentry, Shai Halevi, and Vinod Vaikuntanathan. Fully homomorphic encryption over the integers. In EUROCRYPT, pages 24–43, 2010. Full Version in http://eprint.iacr.org/2009/616.pdf. [Gen09a] Craig Gentry. A fully homomorphic encryption scheme. PhD thesis, Stanford University, 2009. http://crypto.stanford.edu/craig. [Gen09b] Craig Gentry. Fully homomorphic encryption using ideal lattices. In STOC, pages 169–178, 2009. [GGH97] [GH11] [GHS11a] [GHS11b] [GPV08] [LPR10] [MM11] [MP11] [MV10] [Ost11] [Pei09] [RAD78] Oded Goldreich, Shafi Goldwasser, and Shai Halevi. Eliminating decryption errors in the ajtai-dwork cryptosystem. In Burton S. Kaliski Jr., editor, CRYPTO, volume 1294 of Lecture Notes in Computer Science, pages 105–111. Springer, 1997. Craig Gentry and Shai Halevi. Fully homomorphic encryption without squashing using depth-3 arithmetic circuits. In Ostrovsky [Ost11], pages 107–109. Craig Gentry, Shai Halevi, and Nigel P. Smart. Better bootstrapping in fully homomorphic encryption. IACR Cryptology ePrint Archive, 2011:680, 2011. Craig Gentry, Shai Halevi, and Nigel P. Smart. Fully homomorphic encryption with polylog overhead. IACR Cryptology ePrint Archive, 2011:566, 2011. Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In Cynthia Dwork, editor, STOC, pages 197–206. ACM, 2008. Vadim Lyubashevsky, Chris Peikert, and Oded Regev. On ideal lattices and learning with errors over rings. In EUROCRYPT, pages 1–23, 2010. Draft of full version was provided by the authors. Daniele Micciancio and Petros Mol. Pseudorandom knapsacks and the sample complexity of lwe search-to-decision reductions. In Rogaway [Rog11], pages 465–484. Daniele Micciancio and Chris Peikert. Trapdoors for lattices: Simpler, tighter, faster, smaller. IACR Cryptology ePrint Archive, 2011:501, 2011. To appear in Eurocrypt 2012. Daniele Micciancio and Panagiotis Voulgaris. A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations. In Leonard J. Schulman, editor, STOC, pages 351–358. ACM, 2010. Rafail Ostrovsky, editor. IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011. IEEE, 2011. Chris Peikert. Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In STOC, pages 333–342, 2009. R. Rivest, L. Adleman, and M. Dertouzos. On data banks and privacy homomorphisms. In Foundations of Secure Computation, pages 169–177. Academic Press, 1978. 18 6. FHE without Modulus Switching
[Reg03] Oded Regev. New lattice based cryptographic constructions. In Lawrence L. Larmore and Michel X. Goemans, editors, STOC, pages 407–416. ACM, 2003. Full version in [Reg04]. [Reg04] Oded Regev. New lattice-based cryptographic constructions. J. ACM, 51(6):899–942, 2004. [Reg05] [Reg09] [Rog11] [Sch87] [SV10] Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. In Harold N. Gabow and Ronald Fagin, editors, STOC, pages 84–93. ACM, 2005. Full version in [Reg09]. Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. J. ACM, 56(6), 2009. Phillip Rogaway, editor. Advances in Cryptology - CRYPTO 2011 - 31st Annual Cryptology Conference, Santa Barbara, CA, USA, August 14-18, 2011. Proceedings, volume 6841 of Lecture Notes in Computer Science. Springer, 2011. Claus-Peter Schnorr. A hierarchy of polynomial time lattice basis reduction algorithms. Theor. Comput. Sci., 53:201–224, 1987. Nigel P. Smart and Frederik Vercauteren. Fully homomorphic encryption with relatively small key and ciphertext sizes. In Phong Q. Nguyen and David Pointcheval, editors, Public Key Cryptography, volume 6056 of Lecture Notes in Computer Science, pages 420–443. Springer, 2010. 19 6. FHE without Modulus Switching
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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.
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
Page 197 and 198: We can now plug in what we know abo
Page 199: In particular (recall that E < ⌊q
Page 203 and 204: 1 Introduction An encryption scheme
Page 205 and 206: need to have errors. Since the expe
Page 207 and 208: Again we allow some “slackness”
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
Page 221 and 222: the success probability away from 1
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
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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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