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space modulus p), we need to convert ⃗c into another ciphertext vector ⃗c ′ satisfying (a) (q ′ ) −1 ⃗c ′ ≡ q −1 ⃗c (mod p), and (b) the “rounding error term” ɛ def = ⃗c ′ − (q ′ /q)⃗c is small. As described in [5], we apply the following optimized procedure: 1. Let ⃗ δ = ⃗c mod ∆, 2. Add or subtract multiples of ∆ from the coefficients in ⃗ δ until it is divisible by p, 3. Set ⃗c ∗ = ⃗c − ⃗ δ, // ⃗c ∗ divisible by ∆, and ⃗c ∗ ≡ ⃗c (mod p) 4. Output ⃗c ′ = ⃗c/∆. An argument similar to the proof of [2, Lemma 4] shows that if before the transformation we had m = [〈⃗c, ⃗s〉] q ≡ q · m (mod p), then after the transformation we have m ′ = [〈⃗c ′ , ⃗s〉] q ′ ≡ q ′ · m (mod p), as needed. (The difference from [2, Lemma 4] is that we do not assume that q, q ′ ≡ 1 (mod p).) Considering the noise magnitude, we can write ⃗c ′ = ⃗c/∆ + ⃗ɛ where ⃗ɛ is the rounding error (i.e., the terms that are added in Step 2 above, divided by ∆). The noise polynomial is thus scaled down by a ∆ factor, then increased by the additive term a def = 〈⃗ɛ, ⃗s〉 = ∑ j ɛ j(X) · s r j j (X t j ) (with a ∈ A). We make the heuristic assumption that the coefficients in all the ɛ j ’s behave as if they are chosen uniformly in the interval −[p/2, p/2). Under this assumption, we have [ E |ɛ j (τ m )| 2] = φ(m) · p 2 /12, since the variance of a uniform random variable in −[p/2, p/2) is p 2 /12, and ɛ j (τ m ) is a sum of φ(m) such variables, scaled by different magnitude-1 complex constants. Assuming heuristically that the ɛ j ’s are independent of the public key, we have [ E |a(τ m )| 2] = ∑ j [ E |ɛ j (ρ m )| 2] [ ∣∣∣s r · E j j (X t j ) ∣ 2] ≈ ∑ j (φ(m) · p 2 /12) · (r j )! · H r j j , where p is the plaintext-space modulus, H j is the Hamming weight of the secret key for the j’th part, and r j is the power of that secret key. 3.1.6 Key-switching/re-linearization The re-linearization operation ensures that all the ciphertext parts have handles that point to either the constant 1 or a base secret-key: Any ciphertext part j with a handle pointing to s r j j (X t j ) with either r j > 1 or r j = 1 and t j > 1, is replace by two adding two parts, one that points to 1 and the other than points to s j (X), using some key-switching matrices from the public key. Also, a side-effect of re-linearization is that we add all the “special primes” to the prime-set of the resulting ciphertext. To explain the re-linearization procedure, we begin by recalling that the “ciphertext primes” that define our moduli-chain are partitioned into some number n ≥ 1 of “digits”, of roughly equal size. (For example, say that we have 15 small primes in the chain and we partition them to three digits, then we may take the first five primes to be the first digit, the next five primes to be the second, and the last five primes to be the third.) The size of a digit is the product of all the primes that are associated with it, and below we denote by D i the size of the i’th digit. 19 16. Design and Implementation of a Homomorphic-Encryption Library
When key-switching a ciphertext ⃗c using n > 1 digits, we begin by breaking ⃗c into a collection of (at most) n lower-norm ciphertexts ⃗c i . First we remove all the special primes from the prime-set of the ciphertext by modulus-DOWN, if needed. Then we determine the smallest n ′ such that the product of of the first n ′ digits exceeds the current modulus q, and then we set 1. d1 ⃗ := ⃗c 2. For i = 1 to n ′ do: 3. ⃗c i := d ⃗ i mod D i // |⃗c i | < D i /2 4. di+1 ⃗ := ( d ⃗ i − ⃗c i )/D i // ( d ⃗ i − ⃗c i ) divisible by D i Note that ⃗c = ∑ n ′ i=1 ⃗c i · ∏j 1 or r j = 1 and t j > 1. Let us denote by c i,j the ciphertext-parts corresponding to c j in the low-norm ciphertexts ⃗c i . That is, we have c j = ∑ n ′ i=1 c i,j · ∏j
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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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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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Page 379 and 380: Multi-instance Security and Passwor
Page 391 and 392: a hash-of-hash construction: H 2 (M
Page 393 and 394: guisher queries (our lower bounds r
Page 395 and 396: in so doing they accidentally intro
Page 397 and 398: of D is defined as [ [ ] AdvH[P],R,
Page 399 and 400: main POW H[P],n,l1 : P 1 P 2 X 2
Page 401 and 402: now,thinkforconvenienceofw = l).The
Page 403 and 404: aroundO(q 2 )queries.Ideally, wewou
Page 405 and 406: HMAC l (K,Y 0) Y 0 F K Y 0 ′ G K
Page 407 and 408: This material is based on research
Page 409 and 410: 24. Ari Juels and John G. Brainard.
Page 411 and 412: Contents 1 The BGV Homomorphic Encr
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
Page 421 and 422: turn, are sometimes partitioned fur
Page 423 and 424: DoubleCRT& operator-=(const ZZ &num
Page 425 and 426: homomorphic automorphism operation
Page 427 and 428: For reasons that will be discussed
Page 429: where D i is the size of the i’th
Page 433 and 434: constant (i.e. |poly(τ m )| 2 ), o
Page 435 and 436: ool isCorrect() const; and double l
Page 437 and 438: For the noise estimate in the new c
Page 439 and 440: The first time that ImportSecKey is
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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