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Three-way Multiplications. Sometime we need to multiply several ciphertexts together, and if their number is not a power of two then we do not have a complete binary tree of multiplications, which means that at some point in the process we will have three ciphertexts that we need to multiply together. The standard way of implementing this 3-way multiplication is via two 2-argument multiplications, e.g., x · (y · z). But it turns out that here it is better to use “raw multiplication” to multiply these three ciphertexts (as done in [7]), thus getting an “extended” ciphertext with four elements, then apply key-switching (and later modulus switching) to this ciphertext. This takes only six ring-multiplication operations (as opposed to eight according to the standard approach), three modulus switching (as opposed to four), and only one key switching (applied to this 4-element ciphertext) rather than two (which are applied to 3-element extended ciphertexts). All in all, this three-way multiplication takes roughly 1.5 times a standard two-element multiplication. We stress that this technique is not useful for larger products, since for more than three multiplicands the noise begins to grow too large. But with only three multiplicands we get noise of roughly B 3 after the multiplication, which can be reduced to noise ≈ B by dropping two levels, and this is also what we get by using two standard two-element multiplications. Commuting Automorphisms and Multiplications. Recalling that the automorphisms X ↦→ X i commute with the arithmetic operations, we note that some ordering of these operations can sometimes be better than others. For example, it may be better perform the multiplication-by-constant before the automorphism operation whenever possible. The reason is that if we perform the multiply-by-constant after the key-switching that follows the automorphism, then added noise term due to that key-switching is multiplied by the same constant, thereby making the noise slightly larger. We note that to move the multiplication-byconstant before the automorphism, we need to multiply by a different constant. Switching to higher-level moduli. We note that it may be better to perform automorphisms at a higher level, in order to make the added noise term due to key-switching small with respect to the modulus. On the other hand operations at high levels are more expensive than the same operations at a lower level. A good rule of thumb is to perform the automorphism operations one level above the lowest one. Namely, if the naive strategy that never switches to higher-level moduli would perform some Frobenius operation at level q i , then we perform the key-switching following this Frobenius operation at level Q i+1 , and then switch back to level q i+1 (rather then using Q i and q i ). Commuting Addition and Modulus-switching. When we need to add many terms that were obtained from earlier operations (and their subsequent key-switching), it may be better to first add all of these terms relative to the large modulus Q i before switching the sum down to the smaller q i (as opposed to switching all the terms individually to q i and then adding). Reducing the number of key-switching matrices. When using many different automorphisms κ i : X ↦→ X i we need to keep many different key-switching matrices in the public key, one for every value of i that we use. We can reduces this memory requirement, at the expense of taking longer to perform the automorphisms. We use the fact that the Galois group Gal that contains all the maps κ i (which is isomorphic to (Z/mZ) ∗ ) is generated by a relatively small number of generators. (Specifically, for our choice of parameters the group (Z/mZ) ∗ has two or three generators.) It is therefore enough to store in the public key only the key-switching matrices corresponding to κ gj ’s for these generators g j of the group Gal. Then in order 30 5. Homomorphic Evaluation of the AES Circuit
to apply a map κ i we express it as a product of the generators and apply these generators to get the effect of κ i . (For example, if i = g 2 1 · g 2 then we need to apply κ g1 twice followed by a single application of κ g2 .) 31 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
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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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Page 135 and 136: is a coset of the lattice Λ = L( [
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 179: down by a factor of p t . Let’s r
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
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
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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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