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Homomorphic Evaluation of the AES Circuit Craig Gentry IBM Research Shai Halevi IBM Research June 15, 2012 Nigel P. Smart University of Bristol Abstract We describe a working implementation of leveled homomorphic encryption (without bootstrapping) that can evaluate the AES-128 circuit in three different ways. One variant takes under over 36 hours to evaluate an entire AES encryption operation, using NTL (over GMP) as our underlying software platform, and running on a large-memory machine. Using SIMD techniques, we can process over 54 blocks in each evaluation, yielding an amortized rate of just under 40 minutes per block. Another implementation takes just over two and a half days to evaluate the AES operation, but can process 720 blocks in each evaluation, yielding an amortized rate of just over five minutes per block. We also detail a third implementation, which theoretically could yield even better amortized complexity, but in practice turns out to be less competitive. For our implementations we develop both AES-specific optimizations as well as several “generic” tools for FHE evaluation. These last tools include (among others) a different variant of the Brakerski- Vaikuntanathan key-switching technique that does not require reducing the norm of the ciphertext vector, and a method of implementing the Brakerski-Gentry-Vaikuntanathan modulus-switching transformation on ciphertexts in CRT representation. Keywords. AES, Fully Homomorphic Encryption, Implementation The first and second authors are sponsored by DARPA under agreement number FA8750-11-C-0096. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA or the U.S. Government. Distribution Statement “A” (Approved for Public Release, Distribution Unlimited). The third author is sponsored by DARPA and AFRL under agreement number FA8750-11-2-0079. The same disclaimers as above apply. He is also supported by the European Commission through the ICT Programme under Contract ICT-2007-216676 ECRYPT II and via an ERC Advanced Grant ERC-2010- AdG-267188-CRIPTO, by EPSRC via grant COED–EP/I03126X, and by a Royal Society Wolfson Merit Award. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the European Commission or EPSRC. 5. Homomorphic Evaluation of the AES Circuit
Contents 1 Introduction 1 2 Background 3 2.1 Notations and Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 BGV-type Cryptosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Computing on Packed Ciphertexts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 General-Purpose Optimizations 6 3.1 A New Variant of Key Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Modulus Switching in Evaluation Representation . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Dynamic Noise Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Randomized Multiplication by Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Homomorphic Evaluation of AES 9 4.1 Homomorphic Evaluation of the Basic Operations . . . . . . . . . . . . . . . . . . . . . . . 10 4.1.1 AddKey and SubBytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.1.2 ShiftRows and MixColumns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.1.3 The Cost of One Round Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Byte- and Bit-Slice Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3 Performance Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References 13 A More Details 15 A.1 Plaintext Slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.2 Canonical Embedding Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.3 Double CRT Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 A.4 Sampling From A q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.5 Canonical embedding norm of random polynomials . . . . . . . . . . . . . . . . . . . . . . 17 B The Basic Scheme 18 B.1 Our Moduli Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 B.2 Modulus Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 B.3 Key Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 B.4 Key-Generation, Encryption, and Decryption . . . . . . . . . . . . . . . . . . . . . . . . . 21 B.5 Homomorphic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 C Security Analysis and Parameter Settings 23 C.1 Lower-Bounding the Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 C.1.1 LWE with Sparse Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 C.2 The Modulus Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 C.3 Putting It Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 D Scale(c, q t , q t−1 ) in dble-CRT Representation 28 5. Homomorphic Evaluation of the AES Circuit
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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
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
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Page 111 and 112: Dimension m Quality s Length β ≈
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Page 115 and 116: 1.4 Other Related Work Concrete par
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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
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Page 137 and 138: scheme is su-scma-secure if Adv su-
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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
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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
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Page 159 and 160: 3.4 Randomized Multiplication by Co
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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
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Page 173 and 174: We stress that the only place where
Page 175 and 176: C.1.1 LWE with Sparse Key The analy
Page 177 and 178: The Smallest Modulus. After evaluat
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Page 181 and 182: to apply a map κ i we express it a
Page 183 and 184: 1 Introduction Fully homomorphic en
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
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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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