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[5] A. Blum, A. Kalai, and H. Wasserman. Noise-tolerant learning, the parity problem, and the statistical query model. J. ACM, 50(4):506–519, 2003. [6] D. Boneh and D. M. Freeman. Linearly homomorphic signatures over binary fields and new tools for lattice-based signatures. In Public Key Cryptography, pages 1–16, 2011. [7] J.-Y. Cai and A. Nerurkar. An improved worst-case to average-case connection for lattice problems. In FOCS, pages 468–477, 1997. [8] Y. Chen and P. Q. Nguyen. BKZ 2.0: Better lattice security estimates. In ASIACRYPT, pages 1–20, 2011. [9] D. Dadush, C. Peikert, and S. Vempala. Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In FOCS, pages 580–589, 2011. [10] N. Döttling and J. Müller-Quade. Lossy codes and a new variant of the learning-with-errors problem. Manuscript. To appear in Eurocrypt 2013, 2013. [11] N. Gama and P. Q. Nguyen. Predicting lattice reduction. In EUROCRYPT, pages 31–51, 2008. [12] C. Gentry, C. Peikert, and V. Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In STOC, pages 197–206, 2008. [13] A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515–534, December 1982. [14] D. Micciancio. Almost perfect lattices, the covering radius problem, and applications to Ajtai’s connection factor. SIAM J. Comput., 34(1):118–169, 2004. [15] D. Micciancio. Duality in lattice cryptography. In Public Key Cryptography, 2010. Invited talk. [16] 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. [17] D. Micciancio and P. Mol. Pseudorandom knapsacks and the sample complexity of LWE search-todecision reductions. In CRYPTO, pages 465–484, 2011. [18] D. Micciancio and C. Peikert. Trapdoors for lattices: Simpler, tighter, faster, smaller. In EUROCRYPT, pages 700–718, 2012. [19] 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. [20] D. Micciancio and O. Regev. Lattice-based cryptography. In Post Quantum Cryptography, pages 147–191. Springer, February 2009. [21] C. Peikert. Public-key cryptosystems from the worst-case shortest vector problem. In STOC, pages 333–342, 2009. [22] C. Peikert. An efficient and parallel Gaussian sampler for lattices. In CRYPTO, pages 80–97, 2010. 23 10. Hardness of SIS and LWE with Small Parameters
[23] C. Peikert and B. Waters. Lossy trapdoor functions and their applications. In STOC, pages 187–196, 2008. [24] O. Regev. On lattices, learning with errors, random linear codes, and cryptography. J. ACM, 56(6):1–40, 2009. Preliminary version in STOC 2005. [25] D. Wagner. A generalized birthday problem. In CRYPTO, pages 288–303, 2002. 24 10. Hardness of SIS and LWE with Small Parameters
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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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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 +
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
Page 269 and 270: and distribute reprints for Governm
Page 271 and 272: A Simple Dynamic PoR with Square-Ro
Page 273 and 274: from Section 6 that is NRPH secure.
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: k = n/2, and it suffices to set the
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
Page 323 and 324: leading to concrete implementations
Page 325 and 326: y modeling each block by an interac
Page 327 and 328: z i P 2 Q 1 y i P 1 s 1 r 1 x i w i
Page 329 and 330: 2 Distributed Systems, Composition,
Page 331 and 332: [G | H](y) = (w, x): z = y w = y x[
Page 333 and 334: infinite chains, such as (M ∗ ;
Page 335 and 336: 2.3 Multi-party computation, securi
Page 337 and 338: BCast(x) = (y 1 , . . . , y n ): y
Page 339 and 340: Sim(y A , s A ) = (y ′ A , r A):
Page 341 and 342: WCast(x ′ , w) = (y 1 ′ , . . .
Page 343 and 344: s ′ 0 s ′ A r ′ A y ′ H s
Page 345 and 346: 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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