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mod q l . A polynomial a ∈ A q is represented as the (l + 1) × φ(m) matrix of its evaluation at the roots of Φ m (X) modulo p i for i = 0, . . . , l: ( ) DoubleCRT l (a) = a(ζ j i ) mod p i . 0≤i≤l, j∈Z ∗ m Addition and multiplication in A q can be computed as component-wise addition and multiplication of the entries in the two tables (modulo the appropriate primes p i ), DoubleCRT l (a + b) = DoubleCRT l (a) + DoubleCRT l (b), DoubleCRT l (a · b) = DoubleCRT l (a) · DoubleCRT l (b). Also, for an element of the Galois group κ ∈ Gal, mapping a(X) ∈ A to a(X k ) mod Φ m (X), we can evaluate κ(a) on the double-CRT representation of a just by permuting the columns in the matrix, sending each column j to column j · k mod m. 1.3 Modules in our Library Very roughly, our HE library consists of four layers: in the bottom layer we have modules for implementing mathematical structures and various other utilities, the second layer implements our Double-CRT representation of polynomials, the third layer implements the cryptosystem itself (with the “native” plaintext space of binary polynomials), and the top layer provides interfaces for using the cryptosystem to operate on arrays of plaintext values (using the plaintext slots as described in Section 1.1). We think of the bottom two layers as the “math layers”, and the top two layers as the “crypto layers”, and describe then in detail in Sections 2 and 3, respectively. A block-diagram description of the library is given in Figure 1. Roughly, the modules NumbTh, timing, bluestein, PAlgebra, PAlgebraModTwo, PAlgebraMod2r, Cmodulus, IndexSet and IndexMap belong to the bottom layer, FHEcontext, SingleCRT and DoubleCRT belong to the second layer, FHE, Ctxt and KeySwitching are in the third layer, and EncryptedArray and EncryptedArrayMod2r are in the top layer. 2 The Math Layers 2.1 The timing module This module contains some utility function for measuring the time that various methods take to execute. To use it, we insert the macro FHE TIMER START at the beginning of the method(s) that we want to time and FHE TIMER STOP at the end, then the main program needs to call the function setTimersOn() to activate the timers and setTimersOff() to pause them. We can have at most one timer per method/function, and the timer is called by the same name as the function itself (using the pre-defiend variable func ). To obtain the value of a given timer (in seconds), the application can use the function double getTime4func(const char *fncName), and the function printAllTimers() prints the values of all timers to the standard output. 2.2 NumbTh: Miscellaneous Utilities This module started out as an implementation of some number-theoretic algorithms (hence the name), but since then it grew to include many different little utility functions. For example, CRT- 3 16. Design and Implementation of a Homomorphic-Encryption Library
EncryptedArray/EncrytedArrayMod2r Routing plaintext slots, §4.1 Crypto KeySwitching Matrices for key-switching, §3.3 FHE KeyGen/Enc/Dec, §3.2 Ctxt Ciphertext operations, §3.1 FHEcontext parameters, §2.7 SingleCRT/DoubleCRT polynomial arithmetic, §2.8 Math CModulus polynomials mod p, §2.3 PAlgebra2/PAlgebra2r plaintext-slot algebra, §2.5 IndexSet/IndexMap Indexing utilities, §2.6 bluestein FFT/IFFT, §2.3 PAlgebra Structure of Zm*, §2.4 NumbTh miscellaneous utilities, §2.2 timing §2.1 Figure 1: A block diagram of the Homomorphic-Encryption library reconstruction of polynomials in coefficient representation, conversion functions between different types, procedures to sample at random from various distributions, etc. 2.3 bluestein and Cmodulus: Polynomials in FFT Representation The bluestein module implements a non-power-of-two FFT over a prime field Z p , using the Bluestein FFT algorithm [1]. We use modulo-p polynomials to encode the FFTs inputs and outputs. Specifically this module builds on Shoup’s NTL library [9], and contains both a bigint version with types ZZ p and ZZ pX, and a smallint version with types zz p and zz pX. We have the following functions: void BluesteinFFT(ZZ_pX& x, const ZZ_pX& a, long n, const ZZ_p& root, ZZ_pX& powers, FFTRep& Rb); void BluesteinFFT(zz_pX& x, const zz_pX& a, long n, const zz_p& root, zz_pX& powers, fftRep& Rb); These functions compute length-n FFT of the coefficient-vector of a and put the result in x. If the degree of a is less than n then it treats the top coefficients as 0, and if the degree is more than n then the extra coefficients are ignored. Similarly, if the top entries in x are zeros then x will have degree smaller than n. The argument root needs to be a 2n-th root of unity in Z p . The inverse-FFT is obtained just by calling BluesteinFFT(...,root −1 ,...), but this procedure is NOT SCALED. Hence calling BluesteinFFT(x,a,n,root,...) and then BluesteinFFT(b,x,n,root −1 ,...) will result in having b = n × a. In addition to the size-n FFT of a which is returned in x, this procedure also returns the powers of root in the powers argument, powers = ( 1, root, root 4 , root 9 , . . . , root (n−1)2 ) . In the Rb argument it returns the size-N FFT representation of the negative powers, for some N ≥ 2n−1, N a power of two: Rb = F F T N ( 0, . . . , 0, root −(n−1) 2 , . . . , root −4 , root −1 , 1, root −1 , root −4 , . . . , root −(n−1)2 0, . . . , 0 ) . 4 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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10 Mihir Bellare, Stefano Tessaro,
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Page 371 and 372: Multi-Instance Security and its App
Page 373 and 374: 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: ‖a‖ canon 2 . The basic operati
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 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
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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