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18 Mihir Bellare, Thomas Ristenpart, and Stefano Tessaro 20. R. Impagliazzo and V. Kabanets. Constructive proofs of concentration bounds. In M. J. Serna, R. Shaltiel, K. Jansen, and J. D. P. Rolim, editors, APPROX- RANDOM, volume 6302 of LNCS, pages 617–631. Springer, 2010. 21. J. Kelsey, B. Schneier, C. Hall, and D. Wagner. Secure applications of low-entropy keys. In E. Okamoto, G. I. Davida, and M. Mambo, editors, 1st Information Security Workshop (ISW ’97), volume 1396 of LNCS, pages 121–134. Springer, Sept. 1998. 22. T. Kohno. Attacking and repairing the winZip encryption scheme. In V. Atluri, B. Pfitzmann, and P. McDaniel, editors, ACM CCS 04, pages 72–81. ACM Press, Oct. 2004. 23. H. Krawczyk. Cryptographic extraction and key derivation: The HKDF scheme. In T. Rabin, editor, CRYPTO 2010, volume 6223 of LNCS, pages 631–648. Springer, Aug. 2010. 24. M. Luby and C. Rackoff. A study of password security. In C. Pomerance, editor, CRYPTO’87, volume 293 of LNCS, pages 392–397. Springer, Aug. 1988. 25. U. M. Maurer, K. Pietrzak, and R. Renner. Indistinguishability amplification. In A. Menezes, editor, CRYPTO 2007, volume 4622 of LNCS, pages 130–149. Springer, Aug. 2007. 26. U. M. Maurer, R. Renner, and C. Holenstein. Indifferentiability, impossibility results on reductions, and applications to the random oracle methodology. In M. Naor, editor, TCC 2004, volume 2951 of LNCS, pages 21–39. Springer, Feb. 2004. 27. U. M. Maurer and S. Tessaro. Computational indistinguishability amplification: Tight product theorems for system composition. In S. Halevi, editor, CRYPTO 2009, volume 5677 of LNCS, pages 355–373. Springer, Aug. 2009. 28. U. M. Maurer and S. Tessaro. A hardcore lemma for computational indistinguishability: Security amplification for arbitrarily weak PRGs with optimal stretch. In D. Micciancio, editor, TCC 2010, volume 5978 of LNCS, pages 237–254. Springer, Feb. 2010. 29. R. Morris and K. Thompson. Password security: a case history. Commun. ACM, 22:594–597, November 1979. 30. S. Myers. Efficient amplification of the security of weak pseudo-random function generators. Journal of Cryptology, 16(1):1–24, Jan. 2003. 31. A. Panconesi and A. Srinivasan. Randomized distributed edge coloring via an extensionofthechernoff-hoeffdingbounds. SIAM J. Comput.,26(2):350–368, 1997. 32. PKCS #5: Password-based cryptography standard (rfc 2898). RSA Data Security, Inc., Sept. 2000. Version 2.0. 33. T. Ristenpart, H. Shacham, and T. Shrimpton. Careful with composition: Limitations of the indifferentiability framework. In K. G. Paterson, editor, EURO- CRYPT 2011, volume 6632 of LNCS, pages 487–506. Springer, May 2011. 34. S. Tessaro. Security amplification for the cascade of arbitrarily weak PRPs: Tight bounds via the interactive hardcore lemma. In Y. Ishai, editor, TCC 2011, volume 6597 of LNCS, pages 37–54. Springer, Mar. 2011. 35. F. Unger. A probabilistic inequality with applications to threshold direct-product theorems. In 50th FOCS, pages 221–229. IEEE Computer Society Press, Oct. 2009. 36. D. Wagner and I. Goldberg. Proofs of security for the Unix password hashing algorithm. In T. Okamoto, editor, ASIACRYPT 2000, volume 1976 of LNCS, pages 560–572. Springer, Dec. 2000. 37. A. C. Yao. Theory and applications of trapdoor functions. In 23rd FOCS, pages 80–91. IEEE Computer Society Press, Nov. 1982. 14. Multi-Instance Security
Multi-instance Security and Password-based Cryptography 19 38. F. F. Yao and Y. L. Yin. Design and analysis of password-based key derivation functions. In A. Menezes, editor, CT-RSA 2005, volume 3376 of LNCS, pages 245–261. Springer, Feb. 2005. 14. Multi-Instance Security
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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 +
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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
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
Page 347 and 348: Notice that we have already underta
Page 349 and 350: simpler protocol description. For t
Page 351 and 352: [29] Andrew Chi-Chih Yao. Protocols
Page 353 and 354: 2 Mihir Bellare, Stefano Tessaro, a
Page 361 and 362: 10 Mihir Bellare, Stefano Tessaro,
Page 371 and 372: Multi-Instance Security and its App
Page 373 and 374: Multi-instance Security and Passwor
Page 387: Multi-instance Security and Passwor
Page 391 and 392: a hash-of-hash construction: H 2 (M
Page 393 and 394: guisher queries (our lower bounds r
Page 395 and 396: in so doing they accidentally intro
Page 397 and 398: of D is defined as [ [ ] AdvH[P],R,
Page 399 and 400: main POW H[P],n,l1 : P 1 P 2 X 2
Page 401 and 402: now,thinkforconvenienceofw = l).The
Page 403 and 404: aroundO(q 2 )queries.Ideally, wewou
Page 405 and 406: HMAC l (K,Y 0) Y 0 F K Y 0 ′ G K
Page 407 and 408: This material is based on research
Page 409 and 410: 24. Ari Juels and John G. Brainard.
Page 411 and 412: Contents 1 The BGV Homomorphic Encr
Page 413 and 414: ‖a‖ canon 2 . The basic operati
Page 415 and 416: EncryptedArray/EncrytedArrayMod2r R
Page 417 and 418: g i ’s and h i ’s in one list,
Page 419 and 420: 2.6 IndexSet and IndexMap: Sets and
Page 421 and 422: turn, are sometimes partitioned fur
Page 423 and 424: DoubleCRT& operator-=(const ZZ &num
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Page 427 and 428: For reasons that will be discussed
Page 429 and 430: where D i is the size of the i’th
Page 431 and 432: When key-switching a ciphertext ⃗
Page 433 and 434: constant (i.e. |poly(τ m )| 2 ), o
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Page 437 and 438: 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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