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[Vai11] Vinod Vaikuntanathan. Computing blindfolded: New developments in fully homomorphic encryption. In Rafail Ostrovsky, editor, FOCS, pages 5–16. IEEE, 2011. [Val84] Leslie G. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134–1142, 1984. Preliminary version in STOC 84. [vDGHV10] Marten van Dijk, Craig Gentry, Shai Halevi, and Vinod Vaikuntanathan. Fully homomorphic encryption over the integers. In EUROCRYPT, pages 24–43, 2010. Full Version in http://eprint.iacr.org/2009/616.pdf. 16 7. When Homomorphism Becomes a Liability
Quantum-Secure Message Authentication Codes Dan Boneh Mark Zhandry Stanford University {dabo,zhandry}@cs.stanford.edu Abstract We construct the first Message Authentication Codes (MACs) that are existentially unforgeable against a quantum chosen message attack. These chosen message attacks model a quantum adversary’s ability to obtain the MAC on a superposition of messages of its choice. We begin by showing that a quantum secure PRF is sufficient for constructing a quantum secure MAC, a fact that is considerably harder to prove than its classical analogue. Next, we show that a variant of Carter-Wegman MACs can be proven to be quantum secure. Unlike the classical settings, we present an attack showing that a pair-wise independent hash family is insufficient to construct a quantum secure one-time MAC, but we prove that a four-wise independent family is sufficient for one-time security. Keywords: Quantum computing, MAC,chosen message attacks, post-quantum security 1 Introduction Message Authentication Codes (MACs) are an important building block in cryptography used to ensure data integrity. A MAC system is said to be secure if an efficient attacker capable of mounting a chosen message attack cannot produce an existential MAC forgery (see Section 2.2). With the advent of quantum computing there is a strong interest in post-quantum cryptography, that is systems that remain secure even when the adversary has access to a quantum computer. There are two natural approaches to defining security of a MAC system against a quantum adversary. One approach is to restrict the adversary to issue classical chosen message queries, but then allow the adversary to perform quantum computations between queries. Security in this model can be achieved by basing the MAC construction on a quantum intractable problem. The other more conservative approach to defining quantum MAC security is to model the entire security game as a quantum experiment and allow the adversary to issue quantum chosen message queries. That is, the adversary can submit a superposition of messages from the message space and in response receive a superposition of MAC tags on those messages. Informally, a quantum chosen message query performs the following transformation on a given superposition of messages: ∑ ∣ ψ m m 〉 −→ ∑ ∣ ψ m m, S(k, m) 〉 m m where S(k, m) is a tag on the message m with secret key k. To define security, let q be the number of queries that the adversary issues by the end of the game. Clearly it can now produce q classical message-tag pairs by sampling the q superpositions 1 8. Quantum-Secure Message Authentication Codes
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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
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
Page 177 and 178: The Smallest Modulus. After evaluat
Page 179 and 180: down by a factor of p t . Let’s r
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
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: [BV11b] [Gen09] [Gen10] [GH11] [GHS
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
Page 231 and 232: 4.1 A Tight Upper Bound Theorem 4.1
Page 233 and 234: The algorithm is similar to the alg
Page 235 and 236: As we have already seen, for expone
Page 237 and 238: where S i (m) = (R i (m), PRF(k, (R
Page 239 and 240: To prove this theorem, we treat Y a
Page 241 and 242: Now we use this attack to obtain an
Page 243 and 244: Using this lemma we show that (3q +
Page 245 and 246: [KK11] Daniel M. Kane and Samuel A.
Page 247 and 248: 1 Introduction Cloud storage system
Page 249 and 250: subset of the server’s storage, t
Page 251 and 252: security does not suffice. The main
Page 253 and 254: Static Data. PoR and PDP schemes fo
Page 255 and 256: For any efficient adversarial serve
Page 257 and 258: Overview of Construction. Let (Enc,
Page 259 and 260: means that one of the audit protoco
Page 261 and 262: Now we claim that an execution of E
Page 263 and 264: having higher capacity. The tables
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Page 267 and 268: 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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