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2.0 Introduction and Program Overview 2.1 Background Homomorphic Encryption is a form of encryption where a specified algebraic operation is performed on the plaintext and another (possibly different) algebraic operation is performed on the ciphertext. There are several forms of homomorphic encryption that allow an addition or multiplication operation on the plaintext, but to preserve the ring structure of plaintext both addition and multiplication operations must be supported. Using these methods, any circuit could be homomorphically evaluated, effectively allowing the construction of programs which may be run on encryptions of their inputs to produce an encryption of their output. Since the program would never decrypt its input, it could be run by an untrusted party, or transmitted over an untrusted media, without revealing its inputs or internal state. The utility of this scheme is well known, but the algorithms and computation complexity of current implementations are burdensome. To be useful an efficient scheme must be developed and integrated into modern computing. 2.1.1 The PROCEED PROGRAM PROgramming Computation on EncryptEd Data (PROCEED) is a program focused on creating practical methods for computing on encrypted data and is made up of the six Technical Areas (TA) listed below. The PROCEED AHEAD program which we report on here covers Technical Areas 2, 3, and 4 only. • TA1. Mathematical Foundations of Fully Homomorphic Encryption • TA2. Mathematical Foundations of Computation on Encrypted Data via Secure Multiparty Computation • TA3. Mathematical Foundations of Supporting Security Technologies • TA4. Implementation/Measurement/Optimization of Homomorphic Cryptography and Secure Multiparty Protocols • TA5. Algorithms for Computation on Encrypted Data • TA6. Programming Languages The scope of the PROCEED effort is to design, develop, evaluate, integrate, demonstrate and deliver: new mathematical foundations for efficient secure multiparty computation; new mathematical foundations for efficient computation on encrypted data and supporting technologies/techniques; implementations of known and new schemes/protocols, measure and optimize these implementations; develop libraries of efficient algorithms and data structures; develop new programming languages and accompanying compilers and; provide input to the Integration Contractor for the development of a common Application Programmers Interface (API) and integration and evaluation of the research areas. Approved for Public Release; Distribution Unlimited. 2
The PROCEED program efforts are broken up into four program phases as described below. • Program Phase I Initial Capabilities. Phase I will focus on developing the initial capabilities for the mathematical foundations, measurement, algorithms for computation, and programming languages. An API will be developed and coordinated across all performers at the first Principal Investigator (PI) meeting. • Program Phase II Alpha. Phase II will focus on development of alpha quality implementations of algorithms, optimized implementations, programming languages, an initial demonstration of remote regular expression matching and spam filter, and a refined definition of program metrics. • Program Phase III Beta. Phase III will focus on the beta development of core algorithms, interoperability integration, and development of optimized implementations tested against defined program metrics. • Program Phase IV Research Prototype. Phase IV will focus on the development of the research prototype and embedded application prototypes. Final demonstrations will include a functional spam filter prototype. 2.1.2 The AHEAD PROJECT Advancing Homomorphic Encryption its Applications and Derivatives (AHEAD) is a subproject within the broader PROCEED program focused on conducting research in Technical Areas 2, 3, and 4. The AHEAD research team has an extensive background in secure multiparty computation and homomorphic Encryption made up of IBM Research’s Cryptography Group (Prime), and the Departments of Computer Science at Stanford University and the University of California, San Diego (UCSD). The AHEAD research team has worked to advance the study and design of flexible and efficient techniques for processing encrypted data, outsourcing computation and to add robustness to cryptographic computations including protection against accidental or malicious leakage of secret information. Examples of problems that will benefit from our work are the verification of policy compliance on encrypted data, spam detection on encrypted data, efficient delegation of computation, leakage resilient computation and more. Approved for Public Release; Distribution Unlimited. 3
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: 1.0 Executive Summary The ultimate
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
Page 57 and 58: 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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