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References [Aar02] [Amb00] [Amb06] Scott Aaronson. Quantum lower bound for the collision problem. In STOC, pages 635–642, 2002. Andris Ambainis. Quantum lower bounds by quantum arguments. In STOC, pages 636–643, 2000. Andris Ambainis. Polynomial degree vs. quantum query complexity. J. Comput. Syst. Sci., 72(2):220–238, 2006. [ASdW09] Andris Ambainis, Robert Spalek, and Ronald de Wolf. A new quantum lower bound method, with applications to direct product theorems and time-space tradeoffs. Algorithmica, 55(3):422–461, 2009. [BBC + 01] Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum Lower Bounds by Polynomials. Journal of the ACM (JACM), 48(4):778–797, July 2001. [BCK96] Mihir Bellare, Ran Canetti, and Hugo Krawczyk. Pseudorandom functions revisited: The cascade construction and its concrete security. In FOCS, pages 514–523, 1996. [BDF + 11] Dan Boneh, Özgür Dagdelen, Marc Fischlin, Anja Lehmann, Christian Schaffner, and Mark Zhandry. Random Oracles in a Quantum World. In Advances in Cryptology — ASIACRYPT 2011, 2011. [BHK + 11] Gilles Brassard, Peter Høyer, Kassem Kalach, Marc Kaplan, Sophie Laplante, and Louis Salvail. Merkle Puzzles in a Quantum World. Advances in Cryptology - CRYPTO 2011, pages 391–410, 2011. [BKR00] [BS08] Mihir Bellare, Joe Kilian, and Phillip Rogaway. The security of the cipher block chaining message authentication code. J. Comput. Syst. Sci., 61(3), 2000. Gilles Brassard and Louis Salvail. Quantum Merkle Puzzles. Second International Conference on Quantum, Nano and Micro Technologies (ICQNM 2008), pages 76–79, February 2008. [Can01] Ran Canetti. Universally composable security: A new paradigm for cryptographic protocols. In Proc. of FOCS. IEEE, 2001. [DFNS11] Ivan Damgård, Jakob Funder, Jesper Buus Nielsen, and Louis Salvail. Superposition attacks on cryptographic protocols. CoRR, abs/1108.6313, 2011. [GGM86] Oded Goldreich, Shafi Goldwasser, and Silvio Micali. How to Construct Random Functions. Journal of the ACM (JACM), 33(4):792–807, 1986. [HSS11] [KdW03] Sean Hallgren, Adam Smith, and Fang Song. Classical cryptographic protocols in a quantum world. In Proc. of Crypto, LNCS. Springer, 2011. Iordanis Kerenidis and Ronald de Wolf. Exponential lower bound for 2-query locally decodable codes via a quantum argument. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pages 106–115, 2003. 26 8. Quantum-Secure Message Authentication Codes
[KK11] Daniel M. Kane and Samuel A. Kutin. Quantum interpolation of polynomials. Quantum Information & Computation, 11(1&2):95–103, 2011. First published in 2009. [Sha79] Adi Shamir. How to share a secret. Commun. ACM, 22(11):612–613, 1979. [Unr10] [vD98] [WC81] [Zha12a] [Zha12b] Dominique Unruh. Universally Composable Quantum Multi-Party Computation. Advances in Cryptology — EUROCRYPT 2010, pages 486–505, 2010. Wim van Dam. Quantum oracle interrogation: Getting all information for almost half the price. In FOCS, pages 362–367, 1998. Mark N. Wegman and Larry Carter. New hash functions and their use in authentication and set equality. J. Comput. Syst. Sci., 22(3):265–279, 1981. Mark Zhandry. Secure Identity-Based Encryption in the Quantum Random Oracle Model. In Advances in Cryptology — CRYPTO, 2012. Full version available at the Cryptology ePrint Archives: http://eprint.iacr.org/2012/076/. Mark Zhandry. How to Construct Quantum Random Functions. In Proceedings of FOCS, 2012. Full version available at the Cryptology ePrint Archives: http://eprint.iacr. org/2012/182/. 27 8. Quantum-Secure Message Authentication Codes
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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)
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 and 200: In particular (recall that E < ⌊q
Page 201 and 202: [Reg03] Oded Regev. New lattice bas
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: Using this lemma we show that (3q +
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
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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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