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Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP Zvika Brakerski ∗ Abstract We present a new tensoring technique for LWE-based fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically (B → B 2 · poly(n)) with every multiplication (before “refreshing”), our noise only grows linearly (B → B · poly(n)). We use this technique to construct a scale-invariant fully homomorphic encryption scheme, whose properties only depend on the ratio between the modulus q and the initial noise level B, and not on their absolute values. Our scheme has a number of advantages over previous candidates: It uses the same modulus throughout the evaluation process (no need for “modulus switching”), and this modulus can take arbitrary form. In addition, security can be classically reduced from the worst-case hardness of the GapSVP problem (with quasi-polynomial approximation factor), whereas previous constructions could only exhibit a quantum reduction from GapSVP. ∗ Stanford University, zvika@stanford.edu. Supported by a Simons Postdoctoral Fellowship and by DARPA. 6. FHE without Modulus Switching
1 Introduction Fully homomorphic encryption has been the focus of extensive study since the first candidate scheme was introduced by Gentry [Gen09b]. In a nutshell, fully homomorphic encryption allows to perform arbitrary computation on encrypted data. It can thus be used, for example, to outsource a computation to a remote server without compromising data privacy. The first generation of fully homomorphic schemes [Gen09b, DGHV10, SV10, BV11a, CMNT11, GH11] that started with Gentry’s seminal work, all followed a similar and fairly complicated methodology, often relying on relatively strong computational assumptions. A second generation of schemes started with the work of Brakerski and Vaikuntanathan [BV11b], who established full homomorphism in a simpler way, based on the learning with errors (LWE) assumption. Using known reductions [Reg05, Pei09], the security of their construction is based on the (often quantum) hardness of approximating some short vector problems in worst-case lattices. Their scheme was then improved by Brakerski, Gentry and Vaikuntanathan [BGV12], as we describe below. In LWE-based schemes such as [BV11b, BGV12], ciphertexts are represented as vectors in Z q , for some modulus q. The decryption process is essentially computing an inner product of the ciphertext and the secret key vector, which produces a noisy version of the message (the noise is added at encryption for security purposes). The noise increases with every homomorphic operation, and correct decryption is guaranteed if the final noise magnitude is below q/4. Homomorphic addition roughly doubles the noise, while homomorphic multiplication roughly squares it. In the [BV11b] scheme, after L levels of multiplication (e.g. evaluating a depth L multiplication tree), the noise grows from an initial magnitude of B, to B 2L . Hence, to enable decryption, a very large modulus q ≈ B 2L was required. This affected both efficiency and security (the security of the scheme depends inversely on the ratio q/B, so bigger q for the same B means less security). The above was improved by [BGV12], who suggested to scale down the ciphertext vector after every multiplication (they call this “modulus switching”, see below). 1 That is, to go from a vector c over Z q , into the vector c/w over Z q/w (for some scaling factor w). Scaling “switches” the modulus q to a smaller q/w, but also reduces the noise by the same factor (from B to B/w). To see why this change of scale is effective, consider scaling by a factor B after every multiplication (as indeed suggested by [BGV12]): After the first multiplication, the noise goes up to B 2 , but scaling brings it back down to B, at the cost of reducing the modulus to q/B. With more multiplications, the noise magnitude always goes back to B, but the modulus keeps reducing. After L levels of multiplication-and-scaling, the noise magnitude is still B, but the modulus is down to q/B L . Therefore it is sufficient to use q ≈ B L+1 , which is significantly lower than before. However, this process results in a complicated homomorphic evaluation process that “climbs down the ladder of moduli”. The success of the scaling methodology teaches us that perspective matters: scaling does not change the ratio between the modulus and noise, but it still manages the noise better by changing the perspective in which we view the ciphertext. In this work, we suggest to work in an invariant perspective where only the ratio q/B matters (and not the absolute values of q, B as in previous works). We derive a scheme that is superior to the previous best known in simplicity, noise management and security. Details follow. 1 A different scaling technique was already suggested in [BV11b] as a way to simplify decryption and improve efficiency, but not to manage noise. 1 6. FHE without Modulus Switching
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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
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8.0 List of Symbols, Abbreviations,
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Public Affairs Approvals for papers
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Contents 1 Introduction 1 1.1 Our M
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t j = P (a j ), and finally interpo
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apparent problem by replacing the M
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As noted above, there is a multilin
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f(x 1 , . . . , x n ) ∈ F and eve
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Translation. Pick up from the publi
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Lemma 4. Let prime p = ω(N 2 ). Th
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[vDGHV10] Marten van Dijk, Craig Ge
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have u 2 − v 2 = 1 → (u − v)(
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inary), and we want to compute g c
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Fully Homomorphic Encryption withou
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scheme have bit length Ω(D) to all
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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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Page 137 and 138: scheme is su-scma-secure if Adv su-
Page 139 and 140: a discrete Gaussian with parameter
Page 141 and 142: • G ∈ Z n×nk q is a gadget mat
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Page 145 and 146: [Gen09b] C. Gentry. Fully homomorph
Page 147 and 148: [vGHV10] M. van Dijk, C. Gentry, S.
Page 149 and 150: Contents 1 Introduction 1 2 Backgro
Page 151 and 152: 1 Introduction In his breakthrough
Page 153 and 154: Then in Section 3 we describe our
Page 155 and 156: In some more detail, the BGV key sw
Page 157 and 158: itself has low norm. In BGV [5] thi
Page 159 and 160: 3.4 Randomized Multiplication by Co
Page 161 and 162: One subtle implementation detail to
Page 163 and 164: ciphertexts. Producing the encrypte
Page 165 and 166: [24] Benny Pinkas, Thomas Schneider
Page 167 and 168: A.4 Sampling From A q At various po
Page 169 and 170: To maintain the noise estimate, the
Page 171 and 172: variance σ 2 φ(m), so 2d 2 (ζ)ɛ
Page 173 and 174: We stress that the only place where
Page 175 and 176: C.1.1 LWE with Sparse Key The analy
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Page 181: to apply a map κ i we express it a
Page 185 and 186: 4. No restrictions on the secret ke
Page 187 and 188: We point out that an alternative so
Page 189 and 190: Recall that GapSVP γ is the (promi
Page 191 and 192: Proof. By definition ⟨c, (1, s)
Page 193 and 194: • SI-HE.Enc pk (m): Identical to
Page 195 and 196: Proof of Theorem 4.2. Consider the
Page 197 and 198: We can now plug in what we know abo
Page 199 and 200: In particular (recall that E < ⌊q
Page 201 and 202: [Reg03] Oded Regev. New lattice bas
Page 203 and 204: 1 Introduction An encryption scheme
Page 205 and 206: need to have errors. Since the expe
Page 207 and 208: Again we allow some “slackness”
Page 209 and 210: Learning Linear Functions. the clas
Page 211 and 212: Let us first outline the proof of T
Page 213 and 214: P ′ . 4 Namely H n ′ :n = P ′
Page 215 and 216: In conclusion, we get a sub-scheme
Page 217 and 218: [BV11b] [Gen09] [Gen10] [GH11] [GHS
Page 219 and 220: Quantum-Secure Message Authenticati
Page 221 and 222: the success probability away from 1
Page 223 and 224: Functions will be denoted by capito
Page 225 and 226: Quantum chosen message queries. In
Page 227 and 228: Since the w are the possible result
Page 229 and 230: number of distinct component values
Page 231 and 232: 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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