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s ′ Net’ r ′ A r ′ H s ArA G A Net Graph rH s H G H G H G A p H r A s A G A G A s ′ Sim p r ′ A p A p H Player[H] VSS Figure 21: Security of the VSS protocol when the dealer is dishonest. Finally, it is easy to see that p(h) = p h for all h ∈ H. Namely, if there exists C ⊆ [n] with clique C (G) = ⊤, then r h [h ′ ] = g h ′(h) = ˜f(h, h ′ ) for all h ′ ∈ C ∩ H, and therefore o h = interpolate C (r h ) = interpolate C∩H (r h ) = f(h, ·), and thus p h = o h (0) = ˜f(h, 0) = p(h). We therefore conclude that the real system is equivalent to (Sim| VSS) where Sim is the simulator defined by the following equations: Sim(s ′ , G A , s A , p A ) = (r A ′ , r A, G A , p): r a ′ = s ′ [a] (a ∈ A) r a [h] = g h (a) (h ∈ H, a ∈ A) r a [a ′ ] = s a ′[a] (a, a ′ ∈ A) G[h, h ′ ] = eq(g h ′(h), h h (h ′ )) (h, h ′ ∈ H) G[h, a] = eq(s a [h], h h (a)) (a ∈ A, h ∈ H) G[a, ∨ j] = G a [j] (a ∈ A, j ∈ [n]) ˜f = clique C (G) ∧ interpolate2 C (s ′ ) C⊆[n] |C|≥n−t p = ˜f(·, 0) 5 Conclusions Recognizing the inherent hardness of delivering security proofs for complex cryptographic protocols that are both precise and intuitive within existing security frameworks, we have presented a new framework to study the security of multi-party computations based on equational descriptions of interactive processes. Our framework allows a simple and intuitive, yet completely formal, description of interactive processes via sets of equations, and its foundations rely on tools from programming-language theory and domain theory. Beyond its simplicity, our framework completely avoids explicit addressing of non-determinism within cryptographic security proofs, making security proofs a matter of simple equational manipulations over precise mathematical structures. As a case study, we have presented simple security analyses of (variants of) two classical asynchronous protocols within our framework, Bracha’s broadcast protocol [8] and the Ben-Or, Canetti, Goldreich VSS protocol [5]. We are convinced that our work will open up the avenue to several directions for future work. First off, while the results in this paper are presented for the special case of perfect security, a natural next step is to extend the framework to statistical and even computational security. Moreover, while the expressiveness of our framework (i.e., the monotonicity restrictions on protocols) remains to be thoroughly investigated, most distributed protocols we examined so far, seemed to admit a representation within our framework, possibly after minor modifications which often resulted in a 27 12. An Equational Approach to Secure Multi-party Computation
simpler protocol description. For this reason, our thesis is that this holds true for all protocols of interest, and that non-monotonicity, as a source of unnecessary complexity and proof mistakes, should be avoided whenever possible. References [1] S. Abramsky and A. Jung. Handbook of Logic in Computer Science, volume III, chapter Domain theory, pages 1–168. Oxford University Press, 1994. [2] Michael Backes, Birgit Pfitzmann, and Michael Waidner. A general composition theorem for secure reactive systems. In TCC 2004, volume 2951 of Lecture Notes in Computer Science, pages 336–354, 2004. [3] Donald Beaver and Shafi Goldwasser. Multiparty computation with faulty majority. In CRYPTO ’89, volume 435 of Lecture Notes in Computer Science, pages 589–590, 1989. [4] Assaf Ben-David, Noam Nisan, and Benny Pinkas. FairplayMP: a system for secure multiparty computation. In ACM Conference on Computer and Communications Security, pages 257–266, 2008. [5] Michael Ben-Or, Ran Canetti, and Oded Goldreich. Asynchronous secure computation. In STOC, pages 52–61, 1993. [6] Michael Ben-Or, Shafi Goldwasser, and Avi Wigderson. Completeness theorems for noncryptographic fault-tolerant distributed computation (extended abstract). In STOC ’88, pages 1–10, 1988. [7] Peter Bogetoft, Dan Lund Christensen, Ivan Damgård, Martin Geisler, Thomas P. Jakobsen, Mikkel Krøigaard, Janus Dam Nielsen, Jesper Buus Nielsen, Kurt Nielsen, Jakob Pagter, Michael I. Schwartzbach, and Tomas Toft. Secure multiparty computation goes live. In Financial Cryptography (FC 2009), volume 5628 of Lecture Notes in Computer Science, pages 325–343, 2009. [8] Gabriel Bracha. An asynchronous [(n-1)/3]-resilient consensus protocol. In PODC ’84, pages 154–162, 1984. [9] Ran Canetti. Security and composition of multiparty cryptographic protocols. J. Cryptology, 13(1):143–202, 2000. [10] Ran Canetti. Universally composable security: A new paradigm for cryptographic protocols. In FOCS ’01, pages 136–145, 2001. [11] Ivan Damgård, Marcel Keller, E. Larraia, C. Miles, and Nigel P. Smart. Implementing aes via an actively/covertly secure dishonest-majority mpc protocol. IACR Cryptology ePrint Archive, 2012. [12] Oded Goldreich, Silvio Micali, and Avi Wigderson. How to play any mental game or a completeness theorem for protocols with honest majority. In STOC ’87, pages 218–229, 1987. 28 12. An Equational Approach to Secure Multi-party Computation
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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
Page 297 and 298: [23] C. Peikert and B. Waters. Loss
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
Page 347: Notice that we have already underta
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 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,
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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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