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Design and Implementation of a Homomorphic-Encryption Library Shai Halevi Victor Shoup November 30, 2012 Abstract We describe the design and implementation of a software library that implements the Brakerski-Gentry-Vaikuntanathan (BGV) homomorphic encryption scheme, along with many optimizations to make homomorphic evaluation runs faster, focusing mostly on effective use of the Smart-Vercauteren ciphertext packing techniques. Our library is written in C++ and uses the NTL mathematical library. Partially supported 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. Also paritally supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center (DoI/NBC) contract number D11PC20202. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: 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 IARPA, DoI/NBC, or the U.S. Government. 1 16. Design and Implementation of a Homomorphic-Encryption Library
Contents 1 The BGV Homomorphic Encryption Scheme 1 1.1 Plaintext Slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Our Modulus Chain and Double-CRT Representation . . . . . . . . . . . . . . . . . 2 1.3 Modules in our Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 The Math Layers 3 2.1 The timing module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 NumbTh: Miscellaneous Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 bluestein and Cmodulus: Polynomials in FFT Representation . . . . . . . . . . . . . . 4 2.4 PAlgebra: The Structure of Z ∗ m and Z ∗ m/ 〈2〉 . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 PAlgebraModTwo/PAlgebraMod2r: Plaintext Slots . . . . . . . . . . . . . . . . . . . . 6 2.6 IndexSet and IndexMap: Sets and Indexes . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6.1 The IndexSet class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6.2 The IndexMap class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 FHEcontext: Keeping the parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.8 DoubleCRT: Efficient Polynomial Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 10 3 The Crypto Layer 13 3.1 The Ctxt module: Ciphertexts and homomorphic operations . . . . . . . . . . . . . . 13 3.1.1 The SKHandle class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.2 The CtxtPart class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.3 The Ctxt class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.4 Noise estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.5 Modulus-switching operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.6 Key-switching/re-linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.7 Native arithmetic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.8 More Ctxt methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 The FHE module: Keys and key-switching matrices . . . . . . . . . . . . . . . . . . . 24 3.2.1 The KeySwitch class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.2 The FHEPubKey class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.3 The FHESecKey class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 The KeySwitching module: What matrices to generate . . . . . . . . . . . . . . . . . 28 4 The Data-Movement Layer 29 4.1 The classes EncryptedArray and EncryptedArrayMod2r . . . . . . . . . . . . . . . . . . 29 5 Using the Library 33 5.1 Homomorphic Operations over GF (2 8 ) . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Homomorphic Operations over Z 2 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 A Proof of noise-estimate 37 16. Design and Implementation of a Homomorphic-Encryption Library
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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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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 373 and 374: Multi-instance Security and Passwor
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Page 393 and 394: guisher queries (our lower bounds r
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Page 399 and 400: main POW H[P],n,l1 : P 1 P 2 X 2
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Page 407 and 408: This material is based on research
Page 409: 24. Ari Juels and John G. Brainard.
Page 413 and 414: ‖a‖ canon 2 . The basic operati
Page 415 and 416: EncryptedArray/EncrytedArrayMod2r R
Page 417 and 418: g i ’s and h i ’s in one list,
Page 419 and 420: 2.6 IndexSet and IndexMap: Sets and
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Page 423 and 424: DoubleCRT& operator-=(const ZZ &num
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Page 427 and 428: For reasons that will be discussed
Page 429 and 430: where D i is the size of the i’th
Page 431 and 432: When key-switching a ciphertext ⃗
Page 433 and 434: constant (i.e. |poly(τ m )| 2 ), o
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Page 437 and 438: For the noise estimate in the new c
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Page 441 and 442: EncryptedArray(const FHEcontext& co
Page 443 and 444: The MUX operation is implemented by
Page 445 and 446: 5.1 Homomorphic Operations over GF
Page 447 and 448: EncryptedArrayMod2r ea(context); lo
Page 449 and 450: H/2m each and 0 with probability 1
Page 451 and 452: So now consider the inner sum in (7
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