a590003

More documents

Recommendations

Info

[6] Marten van Dijk, Craig Gentry, Shai Halevi, and Vinod Vaikuntanathan. Fully homomorphic encryption over the integers. In Advances in Cryptology - EUROCRYPT’10, volume 6110 of Lecture Notes in Computer Science, pages 24–43. Springer, 2010. Full version available on-line from http://eprint.iacr.org/2009/616. [7] Craig Gentry. A fully homomorphic encryption scheme. PhD thesis, Stanford University, 2009. crypto.stanford.edu/craig. [8] Craig Gentry. Fully homomorphic encryption using ideal lattices. In Michael Mitzenmacher, editor, STOC, pages 169–178. ACM, 2009. [9] Craig Gentry and Shai Halevi. Fully homomorphic encryption without squashing using depth-3 arithmetic circuits. Manuscript, to appear in FOCS 2011, available at http://eprint.iacr.org/2011/279. [10] Craig Gentry and Shai Halevi. Implementing gentry’s fully-homomorphic encryption scheme. In EUROCRYPT, volume 6632 of Lecture Notes in Computer Science, pages 129–148. Springer, 2011. [11] Shai Halevi, 2011. Personal communication. [12] Yuval Ishai and Anat Paskin. Evaluating branching programs on encrypted data. In Salil P. Vadhan, editor, TCC, volume 4392 of Lecture Notes in Computer Science, pages 575–594. Springer, 2007. [13] Kristin Lauter, Michael Naehrig, and Vinod Vaikuntanathan. Can homomorphic encryption be practical? Manuscript at http://eprint.iacr.org/2011/405, 2011. [14] Vadim Lyubashevsky, Chris Peikert, and Oded Regev. On ideal lattices and learning with errors over rings. In EUROCRYPT, volume 6110 of Lecture Notes in Computer Science, pages 1–23, 2010. [15] Carlos Aguilar Melchor, Philippe Gaborit, and Javier Herranz. Additively homomorphic encryption with -operand multiplications. In Tal Rabin, editor, CRYPTO, volume 6223 of Lecture Notes in Computer Science, pages 138–154. Springer, 2010. [16] Chris Peikert. Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In STOC, pages 333–342. ACM, 2009. [17] Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. In Harold N. Gabow and Ronald Fagin, editors, STOC, pages 84–93. ACM, 2005. [18] Oded Regev. The learning with errors problem (invited survey). In IEEE Conference on Computational Complexity, pages 191–204. IEEE Computer Society, 2010. [19] Ron Rivest, Leonard Adleman, and Michael L. Dertouzos. On data banks and privacy homomorphisms. In Foundations of Secure Computation, pages 169–180, 1978. [20] Nigel P. Smart and Frederik Vercauteren. Fully homomorphic encryption with relatively small key and ciphertext sizes. In Public Key Cryptography - PKC’10, volume 6056 of Lecture Notes in Computer Science, pages 420–443. Springer, 2010. [21] Nigel P. Smart and Frederik Vercauteren. Fully homomorphic SIMD operations. Manuscript at http://eprint.iacr.org/2011/133, 2011. 25 2. Fully Homomorphic Encryption without Bootstrapping
[22] Damien Stehlé and Ron Steinfeld. Faster fully homomorphic encryption. In ASIACRYPT, volume 6477 of Lecture Notes in Computer Science, pages 377–394. Springer, 2010. 26 2. Fully Homomorphic Encryption without Bootstrapping
Page 1 and 2:
AFRL-RI-RS-TR-2013-191 ADVANCED HOM
Page 3 and 4:
REPORT DOCUMENTATION PAGE Form Appr
Page 5 and 6:
12. An Equational Approach to Secur
Page 7 and 8:
1.0 Executive Summary The ultimate
Page 9 and 10:
The PROCEED program efforts are bro
Page 11 and 12:
Towards the end of the project we i
Page 13 and 14:
feature of the framework is that it
Page 15 and 16:
4.3 Publications Figure 1: A block
Page 17 and 18:
encrypted data. We introduce a prec
Page 19 and 20:
present an attack showing that a pa
Page 21 and 22:
alternatives: mis (based on mutual
Page 23 and 24:
5.0 Conclusions and Recommendations
Page 25 and 26: [14] Bogdanov, Andrej, and Lee, Chi
Page 27 and 28: 8.0 List of Symbols, Abbreviations,
Page 29 and 30: Public Affairs Approvals for papers
Page 31 and 32: Contents 1 Introduction 1 1.1 Our M
Page 33 and 34: t j = P (a j ), and finally interpo
Page 35 and 36: apparent problem by replacing the M
Page 37 and 38: As noted above, there is a multilin
Page 39 and 40: f(x 1 , . . . , x n ) ∈ F and eve
Page 41 and 42: Translation. Pick up from the publi
Page 43 and 44: Lemma 4. Let prime p = ω(N 2 ). Th
Page 45 and 46: [vDGHV10] Marten van Dijk, Craig Ge
Page 47 and 48: have u 2 − v 2 = 1 → (u − v)(
Page 49 and 50: inary), and we want to compute g c
Page 51 and 52: Fully Homomorphic Encryption withou
Page 53 and 54: scheme have bit length Ω(D) to all
Page 55 and 56: fast. Thus, the actual magnitude of
Page 57 and 58: Definition 3 (B-bounded distributio
Page 59 and 60: achieve 2 λ security against known
Page 61 and 62: Proof. 〈c 2 , s 2 〉 = BitDecomp
Page 63 and 64: • FHE.Add(pk, c 1 , c 2 ): Takes
Page 65 and 66: The computation needed to compute t
Page 67 and 68: If a and b are large enough, B domi
Page 69 and 70: 5.1.2 How Batching Works Deploying
Page 71 and 72: We have the following lemma. Lemma
Page 73 and 74: Unpack(c, {τ σi→1 (s 1 )→s 2
Page 75: Since ‖e q1 ‖ < r q1 ,in, e q1
Page 79 and 80: 1 Introduction Fully homomorphic en
Page 81 and 82: encryptions of the same message usi
Page 83 and 84: and Tromer [CT10] in a completely d
Page 85 and 86: is negligible in k, where Expt CPA
Page 87 and 88: 2.3 Non-Interactive Simulation-Soun
Page 89 and 90: is defined as (state 1 , m 1 , . .
Page 91 and 92: scheme has almost-all-keys perfect
Page 93 and 94: The algorithm S 1 . The algorithm S
Page 95 and 96: The distribution D 6 . This distrib
Page 97 and 98: We resolve this issue using the app
Page 99 and 100: the number of common reference stri
Page 101 and 102: • Homomorphic evaluation: The hom
Page 103 and 104: The number of repeated homomorphic
Page 105 and 106: [Gro04] [Gro10] J. Groth. Rerandomi
Page 107 and 108: Trapdoors for Lattices: Simpler, Ti
Page 109 and 110: mentioned, two central objects are
Page 111 and 112: Dimension m Quality s Length β ≈
Page 113 and 114: defined by G (or the semi-random ma
Page 115 and 116: 1.4 Other Related Work Concrete par
Page 117 and 118: Cryptographic problems. For β > 0,
Page 119 and 120: andom over G m s , for uniformly ra
Page 121 and 122: 3 Search to Decision Reduction Here
Page 123 and 124: • Preimage sampling for f G (x) =
Page 125 and 126: Note that for x ∈ {0, . . . , q
Page 127 and 128:
The offline ‘bucketing’ approac
Page 129 and 130:
G is primitive, the tag H in the ab
Page 131 and 132:
conditional distribution of R given
Page 133 and 134:
and parallelism of Algorithm 3 come
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
Page 143 and 144:
must return this e (but possibly di
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
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 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
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
Page 265 and 266:
OWrite(i 1 , . . . , i t ; v 1 , .
Page 267 and 268:
j-th level table again. With this o
Page 269 and 270:
and distribute reprints for Governm
Page 271 and 272:
A Simple Dynamic PoR with Square-Ro
Page 273 and 274:
from Section 6 that is NRPH secure.
Page 275 and 276:
The LWE problem was introduced in t
Page 277 and 278:
1.2 Techniques and Comparison to Re
Page 279 and 280:
(large) uniform errors as in [10],
Page 281 and 282:
Proof. Let A be an algorithm runnin
Page 283 and 284:
that 1 + ɛ = (1 + ɛ 1 )(1 + ɛ 2
Page 285 and 286:
Because we are concerned with small
Page 287 and 288:
For each i, since gcd(z i , q) = 1
Page 289 and 290:
Proof. Let ω n = ω( √ log n) sa
Page 291 and 292:
Theorem 3.8. Let m, n be integers,
Page 293 and 294:
σ · O( √ m), except with probab
Page 295 and 296:
k = n/2, and it suffices to set the
Page 297 and 298:
[23] C. Peikert and B. Waters. Loss
Page 299 and 300:
1 Introduction Consider the followi
Page 301 and 302:
In the online phase clients individ
Page 303 and 304:
the clients jointly run a secure co
Page 305 and 306:
to compute F from scratch. The work
Page 307 and 308:
I. Pre-processing phase. In this ph
Page 309 and 310:
Online Phase: 1. Each client D i ge
Page 311 and 312:
2n ciphertexts in order to be sure
Page 313 and 314:
[KMR11] [KMR12] [Lip12] [LTV12] Sen
Page 315 and 316:
For any set of adversarial parties
Page 317 and 318:
C.5 Secure Computation We make use
Page 319 and 320:
Experiment H 4 . This experiment is
Page 321 and 322:
of A only in the case when A guesse
Page 323 and 324:
leading to concrete implementations
Page 325 and 326:
y modeling each block by an interac
Page 327 and 328:
z i P 2 Q 1 y i P 1 s 1 r 1 x i w i
Page 329 and 330:
2 Distributed Systems, Composition,
Page 331 and 332:
[G | H](y) = (w, x): z = y w = y x[
Page 333 and 334:
infinite chains, such as (M ∗ ;
Page 335 and 336:
2.3 Multi-party computation, securi
Page 337 and 338:
BCast(x) = (y 1 , . . . , y n ): y
Page 339 and 340:
Sim(y A , s A ) = (y ′ A , r A):
Page 341 and 342:
WCast(x ′ , w) = (y 1 ′ , . . .
Page 343 and 344:
s ′ 0 s ′ A r ′ A y ′ H s
Page 345 and 346:
p r ′ A Net’ s ′ r ′ H r A
Page 347 and 348:
Notice that we have already underta
Page 349 and 350:
simpler protocol description. For t
Page 351 and 352:
[29] Andrew Chi-Chih Yao. Protocols
Page 353 and 354:
2 Mihir Bellare, Stefano Tessaro, a
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
10 Mihir Bellare, Stefano Tessaro,
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Multi-Instance Security and its App
Page 373 and 374:
Multi-instance Security and Passwor
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Page 389 and 390:
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 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
show all

a590003

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?