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Table of Contents 1.0 Executive Summary ...............................................................................................................1 2.0 Introduction and Program Overview ...................................................................................2 2.1 Background .....................................................................................................................2 2.1.1 The PROCEED PROGRAM...............................................................................2 2.1.2 The AHEAD PROJECT ......................................................................................3 3.0 Program Work: Methods and Assumptions ........................................................................4 3.1 Work Plans and Methods ................................................................................................4 3.1.1 TA 2: Foundations of secure computations.........................................................4 3.1.2 TA 3: Foundations of supporting security technologies .....................................4 3.1.3 TA 4: Implementing fully homomorphic encryption ..........................................4 3.2 Deliverables and Assumptions ........................................................................................4 4.0 Accomplishments and Results ...............................................................................................6 4.1 Summary of Major Contributions ...................................................................................6 4.2 Technology Transitions and Deliverables .......................................................................7 4.3 Publications .....................................................................................................................9 5.0 Conclusions and Recommendations ...................................................................................1 6.0 References .............................................................................................................................1 7.0 Appendix ............................................................................................................................... 8.0 List of Symbols, Abbreviations, and Acronyms ................................................................ 9.0 Public Release Approvals for Papers in Appendix ..…………………………………….2 i
1.0 Executive Summary The ultimate goal of the PROCEED AHEAD project was to advance the study and design of flexible and efficient techniques for processing encrypted data, outsourcing computation and adding robustness to cryptographic computations. Examples of problems that would benefit from such techniques are the verification of policy compliance on encrypted data, spam detection on encrypted data, efficient delegation of computation and more. The contributions of this program are already and will continue to improve the ability to process and manage data, in particular encrypted data, in a robust and fully functional way without sacrificing the confidentiality of information and the privacy of users. We have provided an implementation of Fully Homomorphic Encryption (FHE). The main goal of the implementation was to help us evaluate the different algorithmic approaches, and to innovate on implementation techniques to bypass bottlenecks in the proposed schemes. We stress that building a usable implementation is not just about programming tricks, but rather mathematical innovations that improve the running times in practice. These improvements resulted in large constant factor speed-ups that are ignored in the asymptotic calculations of theoretical research papers, but are essential for a practical implementation. Our approach to making fully homomorphic encryption more efficient using speed-ups improved performance by several orders of magnitude. These improvements come from modifications to the basic schemes of Gentry and others that have little asymptotic impact, but have a large impact in practice. An attractive approach to address issues of privacy is to resort to the area of secure multiparty computations (SMC) as extensively studied in cryptography. However previous solutions in this area are mostly theoretical, and are hardly used in practice due to their complexity. To address this we have proposed a novel framework for the design and analysis of secure computation protocols which allows for much simpler modeling and analysis of cryptographic protocols. Outsourcing computations has also previously suffered from the lack of efficient algorithms. In addition, we created algorithms for delegating computation and spam filters. During just the first two years of the project the PROCEED AHEAD team has developed and implemented new and improved protocols for computing on encrypted data, and a deepened understanding of the foundations of secure computation. The report that follows describes the work of IBM, Stanford University, and University of California, San Diego on the PROCEED AHEAD project from February 2011 through March 2013. Approved for Public Release; Distribution Unlimited. 1
Page 1 and 2: AFRL-RI-RS-TR-2013-191 ADVANCED HOM
Page 3 and 4: REPORT DOCUMENTATION PAGE Form Appr
Page 5: 12. An Equational Approach to Secur
Page 9 and 10: The PROCEED program efforts are bro
Page 11 and 12: Towards the end of the project we i
Page 13 and 14: feature of the framework is that it
Page 15 and 16: 4.3 Publications Figure 1: A block
Page 17 and 18: encrypted data. We introduce a prec
Page 19 and 20: present an attack showing that a pa
Page 21 and 22: alternatives: mis (based on mutual
Page 23 and 24: 5.0 Conclusions and Recommendations
Page 25 and 26: [14] Bogdanov, Andrej, and Lee, Chi
Page 27 and 28: 8.0 List of Symbols, Abbreviations,
Page 29 and 30: Public Affairs Approvals for papers
Page 31 and 32: Contents 1 Introduction 1 1.1 Our M
Page 33 and 34: t j = P (a j ), and finally interpo
Page 35 and 36: apparent problem by replacing the M
Page 37 and 38: As noted above, there is a multilin
Page 39 and 40: f(x 1 , . . . , x n ) ∈ F and eve
Page 41 and 42: Translation. Pick up from the publi
Page 43 and 44: Lemma 4. Let prime p = ω(N 2 ). Th
Page 45 and 46: [vDGHV10] Marten van Dijk, Craig Ge
Page 47 and 48: have u 2 − v 2 = 1 → (u − v)(
Page 49 and 50: inary), and we want to compute g c
Page 51 and 52: Fully Homomorphic Encryption withou
Page 53 and 54: scheme have bit length Ω(D) to all
Page 55 and 56: 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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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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