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Targeted Malleability: Homomorphic Encryption for Restricted Computations Dan Boneh ∗ Gil Segev † Brent Waters ‡ Abstract We put forward the notion of targeted malleability: given a homomorphic encryption scheme, in various scenarios we would like to restrict the homomorphic computations one can perform on encrypted data. We introduce a precise framework, generalizing the foundational notion of non-malleability introduced by Dolev, Dwork, and Naor (SICOMP ’00), ensuring that the malleability of a scheme is targeted only at a specific set of “allowable” functions. In this setting we are mainly interested in the efficiency of such schemes as a function of the number of repeated homomorphic operations. Whereas constructing a scheme whose ciphertext grows linearly with the number of such operations is straightforward, obtaining more realistic (or merely non-trivial) length guarantees is significantly more challenging. We present two constructions that transform any homomorphic encryption scheme into one that offers targeted malleability. Our constructions rely on standard cryptographic tools and on succinct non-interactive arguments, which are currently known to exist in the standard model based on variants of the knowledge-of-exponent assumption. The two constructions offer somewhat different efficiency guarantees, each of which may be preferable depending on the underlying building blocks. Keywords: Homomorphic encryption, Non-malleable encryption. ∗ Stanford University. Supported by NSF, DARPA, and AFOSR. † Microsoft Research, Mountain View, CA 94043, USA. ‡ University of Texas at Austin. Supported by NSF CNS-0716199, CNS-0915361, and CNS-0952692, DARPA PROCEED, Air Force Office of Scientific Research (AFO SR) MURI, DHS Grant 2006-CS-001-000001-02, and the Sloan Foundation. 3. Targeted Malleability
1 Introduction Fully homomorphic encryption [RAD78, Gen09, SV10, vDGH + 10] is a remarkable development in cryptography enabling anyone to compute arbitrary functions on encrypted data. In many settings, however, the data owner may wish to restrict the class of homomorphic computations to a certain set F of allowable functions. In this paper we put forward the notion of “targeted malleability”: given an encryption scheme that supports homomorphic operations with respect to some set of functions F, we would like to ensure that the malleability of the scheme is targeted only at the set F. That is, it should not be possible to apply any homomorphic operation other than the ones in F. Enforcing targeted malleability can be simply done by requiring the entity performing the homomorphic operation to embed a proof in the ciphertext showing that the ciphertext was computed using an allowable function. The decryptor then verifies the proof before decrypting the ciphertext, and outputs ⊥ if the proof is invalid. Unfortunately, as the homomorphic operation is repeated the number of proofs grows making the ciphertext grow at least linearly with the number of repeated homomorphic operations. It is not difficult to see that targeted malleability with a linear-size ciphertext is trivial to construct: Use any non-malleable encryption scheme, and embed in the ciphertext a description of all the functions being computed. The decryptor decrypts the original ciphertext and applies the embedded functions to it (verifying, of course, that these functions are in the allowable set). Minimizing ciphertext expansion. Targeted malleability is much harder to construct once we require that ciphertext growth is at most sub-linear in the number of repeated homomorphic operations. Our goal is to construct systems where even after t applications of the homomorphic operation the ciphertext length does not increase much. In our main construction we are able to completely shift the dependence on t from the ciphertext to the public key: the ciphertext size is essentially independent of t. This is a natural goal since public keys are typically much more static than ciphertexts which are frequently generated and transmitted. Motivation. While targeted malleability is an interesting concept in its own right, it has many applications in cryptography and beyond. We give a few illustrative examples: • A spam filter implemented in a mail server adds a spam tag to encrypted emails whose content satisfies a certain spam predicate. The filter should be allowed to run the spam predicate, but should not modify the email contents. In this case, the set of allowable functions F would be the set of allowable spam predicates and nothing else. As email passes from one server to the next each server homomorphically computes its spam predicate on the encrypted output of the previous server. Each spam filter in the chain can run its chosen spam predicate and nothing else. • More generally, in a distributed system users initiate encrypted requests to various servers. To service a request a server may need to contact another server and that server may need to contact another, resulting in a chain of messages from server to server until the transaction is fulfiled. Each server along the way has an allowed set of operations it can apply to a recieved message and it should be unable to apply any operation outside this approved set. • In a voting system based on homomorphic encryption (e.g. [CGS97]) voters take turns incrementing an encrypted vote tally using a homomorphic operation. They are only allowed to increase the encrypted tally by 1 (indicating a vote for the candidate) or by 0 (indicating a 1 3. Targeted Malleability
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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
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
Page 61 and 62: Proof. 〈c 2 , s 2 〉 = BitDecomp
Page 63 and 64: • FHE.Add(pk, c 1 , c 2 ): Takes
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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
Page 77: [22] Damien Stehlé and Ron Steinfe
Page 81 and 82: encryptions of the same message usi
Page 83 and 84: and Tromer [CT10] in a completely d
Page 85 and 86: is negligible in k, where Expt CPA
Page 87 and 88: 2.3 Non-Interactive Simulation-Soun
Page 89 and 90: is defined as (state 1 , m 1 , . .
Page 91 and 92: scheme has almost-all-keys perfect
Page 93 and 94: The algorithm S 1 . The algorithm S
Page 95 and 96: The distribution D 6 . This distrib
Page 97 and 98: We resolve this issue using the app
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Page 101 and 102: • Homomorphic evaluation: The hom
Page 103 and 104: The number of repeated homomorphic
Page 105 and 106: [Gro04] [Gro10] J. Groth. Rerandomi
Page 107 and 108: Trapdoors for Lattices: Simpler, Ti
Page 109 and 110: mentioned, two central objects are
Page 111 and 112: Dimension m Quality s Length β ≈
Page 113 and 114: defined by G (or the semi-random ma
Page 115 and 116: 1.4 Other Related Work Concrete par
Page 117 and 118: Cryptographic problems. For β > 0,
Page 119 and 120: andom over G m s , for uniformly ra
Page 121 and 122: 3 Search to Decision Reduction Here
Page 123 and 124: • Preimage sampling for f G (x) =
Page 125 and 126: Note that for x ∈ {0, . . . , q
Page 127 and 128: 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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