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7.0 Appendix Please see Attachment 1 – Publications (PROCEED AHEAD Final Report Full Papers Attachment 1 Mar 2013.pdf) for the following publications: 1. Fully Homomorphic Encryption without Squashing Using Depth-3 Arithmetic Circuits 2. Fully Homomorphic Encryption without Bootstrapping 3. Targeted Malleability: Homomorphic Encryption for Restricted Computations 4. Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller 5. Homomorphic Evaluation of the AES Circuit 6. Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP 7. When Homomorphism Becomes a Liability 8. Quantum-Secure Message Authentication Codes 9. Dynamic Proofs of Retrievability via Oblivious RAM 10. Hardness of SIS and LWE with Small Parameters 11. How to Delegate Secure Multiparty Computation to the Cloud 12. An Equational Approach to Secure Multi-party Computation 13. Semantic Security for the Wiretap Channel 14. Multi-Instance Security and its Application to Password-Based Cryptography 15. To Hash or Not to Hash Again (In)differentiability Results for H 2 and HMAC 16. FHE Library - Design and Implementation of a Homomorphic-Encryption Library 17. Using Homomorphic Encryption for Large Scale Statistical Analysis Approved for Public Release; Distribution Unlimited. 20
8.0 List of Symbols, Abbreviations, and Acronyms Abbreviation AES AHEAD API BGV BL CCA CRT DARPA FHE GapSVP GMP HE IBM LPN LWE MAC MHE NTL PI PRF PROCEED RAM RLWE SIMD SIS SSSP SWHE TA UCSD Definition Advanced Encryption Standard Advancing Homomorphic Encryption its Applications and Derivatives Application Programmers Interface Brakerski-Gentry-Vaikuntanathan Bogdanov and Lee Chosen Cyphertext Attack Chinese Remainder Theorem Defense Advanced Research Projects Agency Fully Homomorphic Encryption GapShort Vector Problem The GNU MP Bignum Library Homomorphic Encryption International Business Machines Corporation Learning Parity with Noise Learning With Errors Message Authentication Code Multiplicatively Homomorphic Encryption Number Theory Library Principal Investigator Pseudo Random Function PROgramming Computation on EncryptEd Data Random Access Memory Ring Learning With Errors Single Instruction Multiple Data Short Integer Solution Sparse Subset Sum Problem SomeWhat Homomorphic Encryption Technical Area University of California, San Diego Approved for Public Release; Distribution Unlimited. 21
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 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: [14] Bogdanov, Andrej, and Lee, Chi
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
Page 61 and 62: Proof. 〈c 2 , s 2 〉 = BitDecomp
Page 63 and 64: • FHE.Add(pk, c 1 , c 2 ): Takes
Page 65 and 66: The computation needed to compute t
Page 67 and 68: If a and b are large enough, B domi
Page 69 and 70: 5.1.2 How Batching Works Deploying
Page 71 and 72: We have the following lemma. Lemma
Page 73 and 74: Unpack(c, {τ σi→1 (s 1 )→s 2
Page 75 and 76: 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 361 and 362:
10 Mihir Bellare, Stefano Tessaro,
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Page 369 and 370:
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Multi-Instance Security and its App
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Multi-instance Security and Passwor
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Page 387 and 388:
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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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