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The use of O(·) just hides small constants. The proof is given in the full version. Combining Theorem 3 with the indifferentiability composition theorem allows us to conclude security for HMAC d [H] for underyling hash function H that is, itself, indifferentiable from a RO. For example, should H be one of the proven-indifferentiable SHA-3 candidates. This does not, however, give us a security guarantee should H not be indifferentiable from a RO, as is the case with MD based hash functions. We therefore also prove, in the full version, the following theorem that establishes indifferentiability of HMAC using an underlying hash function built via the strengthened Merkle-Damgård (SMD) domain extension transform. Theorem 4. Fix d,k,n > 0 with k < d − 1 and d ≥ n. Let f : {0,1} n × {0,1} d → {0,1} n be a RO and consider HMAC d [SMD[f]] restricted to k-bit keys. Let R: {0,1} ∗ ×{0,1} ∗ → {0,1} n be a keyed RO. Then there exists a simulator S such that for any distinguisher A whose total query cost is σ ≤ 2 n−2 it holds that ( ) σ Adv indiff 2 HMAC d [SMD[f]],R,S(A) ≤ O 2 n S makes at most q 2 queries and runs in time O(q 2 logq 2 ) where q 2 is the number of Prim queries by A. □ We note that the restriction to σ ≤ 2 n−2 in the theorem statement is just a technicality to make the bound simpler and likewise the use of O(·) in the advantage statement hides just a small constant. Unlike our positive results about H 2 , the bounds provided by Theorems 3 and 4 match, up to small constants, results for other now-standard indifferentiable constructions (c.f., [13]). First, the advantage bounds both hold up to the birthday bound, namely σ ≈ 2 n/2 . Second, the simulators are efficient and, specifically, make at most one query per invocation. All this enables use of the indifferentiability composition theorem in a way that yields strong, standard concrete security bounds. Acknowledgments The authors thank Hugo Krawczyk for providing significant feedback and suggestions, in particular encouraging the authors to include positive results for the indifferentiability of HMAC; Niels Ferguson for in-depth discussions regarding the security of H 2 ; and the anonymous reviewers for their helpful suggestions. Dodis was supported in part by NSF grants CNS-1065134, CNS-1065288, CNS-1017471, CNS-0831299. Ristenpart was supported in part by NSF grant CNS-1065134. Steinberger is supported by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174, and by NSF grant0994380.Tessarowassupported in partby NSF grantsCCF-0915675, CCF-1018064. 17 15. To Hash or Not to Hash Again?
This material is based on research sponsored by DARPA under agreement numbers FA8750-11-C-0096and FA8750-11-2-0225.The U.S. Government is authorizedtoreproduceanddistributereprintsforGovernmentalpurposesnotwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA or the U.S. Government. References 1. Elena Andreeva, Bart Mennink, and Bart Preneel. On the indifferentiability of the Grøstl hash function. In Juan A. Garay and Roberto De Prisco, editors, SCN 10: 7th International Conference on Security in Communication Networks, volume 6280 of Lecture Notes in Computer Science, pages 88–105. Springer, September 2010. 2. Elena Andreeva, Gregory Neven, Bart Preneel, and Thomas Shrimpton. Sevenproperty-preserving iterated hashing: ROX. In Kaoru Kurosawa, editor, Advances in Cryptology – ASIACRYPT 2007, volume 4833 of Lecture Notes in Computer Science, pages 130–146. Springer, December 2007. 3. Mihir Bellare. New proofs for NMAC and HMAC: Security without collisionresistance. In Cynthia Dwork, editor, Advances in Cryptology – CRYPTO 2006, volume 4117 of Lecture Notes in Computer Science, pages 602–619. Springer, August 2006. 4. Mihir Bellare, Zvika Brakerski, Moni Naor, Thomas Ristenpart, Gil Segev, Hovav Shacham, and Scott Yilek. Hedged public-key encryption: How to protect against bad randomness. In Mitsuru Matsui, editor, Advances in Cryptology – ASIACRYPT 2009, volume 5912 of Lecture Notes in Computer Science, pages 232–249. Springer, December 2009. 5. Mihir Bellare, Ran Canetti, and Hugo Krawczyk. Keying hash functions for message authentication. In Neal Koblitz, editor, Advances in Cryptology – CRYPTO’96, volume 1109 of Lecture Notes in Computer Science, pages 1–15. Springer, August 1996. 6. Mihir Bellare and Tadayoshi Kohno. A theoretical treatment of related-key attacks: RKA-PRPs, RKA-PRFs, and applications. In Eli Biham, editor, Advances in Cryptology – EUROCRYPT 2003, volume 2656 of Lecture Notes in Computer Science, pages 491–506. Springer, May 2003. 7. Mihir Bellare and Thomas Ristenpart. Multi-property-preserving hash domain extensionandtheEMD transform. InXuejiaLaiandKefeiChen,editors, Advances in Cryptology – ASIACRYPT 2006, volume 4284 of Lecture Notes in Computer Science, pages 299–314. Springer, December 2006. 8. Mihir Bellare and Thomas Ristenpart. Hash functions in the dedicated-key setting: Design choices and MPP transforms. In Lars Arge, Christian Cachin, Tomasz Jurdzinski, and Andrzej Tarlecki, editors, ICALP 2007: 34th International Colloquium on Automata, Languages and Programming, volume 4596 of Lecture Notes in Computer Science, pages 399–410. Springer, July 2007. 9. Mihir Bellare, Thomas Ristenpart, and Stefano Tessaro. Multi-instance security and its application to password-based cryptography. In Advances in Cryptology – CRYPTO ‘12, Lecture Notes in Computer Science. Springer, 2012. 18 15. To Hash or Not to Hash Again?
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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
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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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Page 371 and 372: Multi-Instance Security and its App
Page 373 and 374: 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: HMAC l (K,Y 0) Y 0 F K Y 0 ′ G K
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
Page 425 and 426: homomorphic automorphism operation
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
Page 435 and 436: ool isCorrect() const; and double l
Page 437 and 438: For the noise estimate in the new c
Page 439 and 440: The first time that ImportSecKey is
Page 441 and 442: EncryptedArray(const FHEcontext& co
Page 443 and 444: The MUX operation is implemented by
Page 445 and 446: 5.1 Homomorphic Operations over GF
Page 447 and 448: EncryptedArrayMod2r ea(context); lo
Page 449 and 450: H/2m each and 0 with probability 1
Page 451 and 452: So now consider the inner sum in (7
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