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3.0 Program Work: Methods and Assumptions 3.1 Work Plans and Methods We outline below the pl anned work to be accomplished within the three PROCEED program Technical Areas covered by AHEAD as defined at the onset of the program. Program work has been carried out jointly between IBM Research, Stanford University, and UCSD. In addition to the technical work, the AHEAD research team has participated in DARPA PI meetings and contributed to related DARPA events. All program results have been made available on the project website hosted by the PROCEED program integrator, Galois. 3.1.1 TA 2: Foundations of secure computations • Programming models for secure computation. Extend Yao’s garbled circuits to handle arithmetic functions efficiently. Design protocols for repeated executions and for programs with loops. • Develop relations among different execution models and construct general transformations for transferring desirable protocol properties from one model to another. • Design dedicated solutions for problems, e.g., pattern matching. 3.1.2 TA 3: Foundations of supporting security technologies • Design homomorphic encryption for certain function families to enable restricted computation delegation. • Verifying computation. Remove the need for FHE to delegate computation. Introduce proxy re-signatures in order to control malicious servers. • Prevent side channel attacks using leakage resilience. Attempt to remove the reliance on leakage resilient hardware. 3.1.3 TA 4: Implementing fully homomorphic encryption • Optimize Fully Homomorphic Encryption by speeding up key generation and encryption. Shrink the public key and ciphertext size. • Explore fast two-party computation via fast Yao Circuits 3.2 Deliverables and Assumptions Through the course of the planned four year PROCEED AHEAD program IBM, Stanford University and UCSD planned to deliver the following: • An optimized implementation of fully homomorphic encryption. • For all three tasks we delivered white papers and technical papers describing the results of the work. • As we made progress on the theoretical underpinnings of these tasks we experimented with prototype implementations when appropriate. Approved for Public Release; Distribution Unlimited. 4
Towards the end of the project we initiated technology transfer discussions with product groups within IBM and devoted resources to supporting a successful transition to real-world products. Completion criteria for all three tasks was planned to be the development of the required cryptographic systems along with theoretical optimizations and prototype software if appropriate. The following is an outline of the planned schedule for the PROCEED AHEAD program. When the schedule for program work and deliverables was defined, it was assumed that IBM, Stanford, and USCD would all jointly contribute to program efforts for the duration of the planned four year program. This final report covers year one and year two contributions by the AHEAD team. At the close of year two IBM took leave of the program. Stanford and UCSD have continued their work under a new contract put in place with the Office of Naval Research and DARPA. Year 1: Develop initial protocols for pattern matching on encrypted data. Begin investigating improved 2-party computation techniques of the tasks outlines in the proposal. Experiment with optimizations to our fully homomorphic encryption implementation. Year 2: Continue developing our techniques for 2-party computation. Examine multiple executions of the same protocol in a multiparty setting. Tune protocols for pattern matching to the specific tasks of network guards and mail filtering on encrypted data. Continue experimenting with optimizations and modifications to the fully homomorphic system based on the results for year 1. Begin investigating the problem of delegating computation. Write and publish technical papers on intermediate results for all three tasks. After Year 2 the program was cancelled. The Year 3 and 4 planned work is outlined below but was not completed. Year 3: Build on our work from year 2 to and begin prototyping an application for our 2-party protocols, extend the investigation to include loops. Work on additional aspects of computation delegation. Use our optimized fully homomorphic encryption scheme for real world tasks such as network guards and other computations on encrypted data such as curve fitting (e.g., leastsquares fit) on encrypted data. Initiate discussions with products within IBM to promote technology transfer. Submit additional papers for publication in leading conferences. Year 4: Tune the research results to address the needs of product groups within IBM and to support technology transfer. As appropriate, release open source tools to enable other researchers to build on our work. Continue publishing technical papers on results of our research. Approved for Public Release; Distribution Unlimited. 5
Page 1 and 2: AFRL-RI-RS-TR-2013-191 ADVANCED HOM
Page 3 and 4: REPORT DOCUMENTATION PAGE Form Appr
Page 5 and 6: 12. An Equational Approach to Secur
Page 7 and 8: 1.0 Executive Summary The ultimate
Page 9: The PROCEED program efforts are bro
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
Page 57 and 58: Definition 3 (B-bounded distributio
Page 59 and 60: 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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