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PublicKey Encryption in the BoundedRetrieval Model
Joël Alwen, Yevgeniy Dodis, Moni Naor, Gil Segev, Shabsi Walfish, Daniel Wichs
Speaker: Daniel
Wichs
Eurocrypt
2010
Slide2Motivation
Cryptographic security analyzed in formal
“attack model”
.
Do our attack models capture
reality
?
In
reality
, extra information about secretkeys can
leak
.
Sidechannels attacks:
timing, power, heat, EM radiation, acoustics...
Coldboot attack
[HSH+ 08]
Viruses
LeakageResilient Crypto:
Add
keyleakage
to the
attack model
.
Build primitives that
provably
allow
leakage
of secret key.
Slide3Bounded Retrieval Model [Dzi06,…,ADW09]:Grow secretkey to allow for more leakage. Even many Gigabytes.Efficiency does not degrade as sk grows. {Public key, ciphertext, computation time}
f
(sk)
Model of Leakage: Memory Attacks
Adversary can learn
any efficiently computable function f : {0,1}* {0,1}L of the secret key. L = Leakage Bound.
RelativeLeakage Model[AGV09, DKL09,NS09,…]. Maximize ratio of L to sk (e.g. 90% of the key can leak).
sk
leak
[
AkaviaGoldwasserVaikuntanathan
09]
Slide4Why design schemes for the BRM?
Security against Viruses:
Upper bound how much attacker can download (e.g. 10 GB)
.
Bandwidth too low, cost too high, system security may detect.
OK
if secret key is large.
Not OK
if efficiency degrades.
Security against sidechannel attacks:
Leakage amount depends on the
complexity of computation
.
Leakageresilient schemes might be less secure:
+ Leakageresilience
)
+ Complexity
)
+ Leakage.
BRM efficiency breaks the cycle.
Slide5Prior Work on Leakage Resilience
Memory Attacks
RelativeLeakage
:
Symmetric
and PublicKey
Encryption
and
Authentication/Signatures.
[AGV09,DKL09,ADW09,
KV09,NS09,…].
Bounded Retrieval Model:
Symmetric and Public Key
“Authenticated key Agreement.”
Requires
interaction
.
[Dzi06,CDD
+
07, ADW09]
.
This work:
PublicKey Encryption
in the Bounded Retrieval Model.
Restricted types of leakage functions
.
[CDH+00, DSS01,KZ03, ISW03 , MR04, DP08, Pie09, FKPR10, GR10, FRR+10, JV10]
Does not seem applicable to e.g. virus attacks.
Slide6Definition of PKE in BRM
Key generation gets L as input. Adversary learns L bit leakage.Efficiency: pk size, ciphertext size, encryption/decryption times are all bounded by some fixed polynomials, independent of L.
Adversary
Challenger
(
pk,sk
) Ã KeyGen(1s )
pk
f : {0,1
}
* ! {0,1}L
f(
sk)
m0, m1
bÃ {0,1}
cÃEncrypt(mb,pk)
c
Output b’
, L
Pr[b’ = b]
·
½ +
negl
(s)
Slide7A “highlevel” template for constructing BRM schemes.
“Identity Based Hash Proof System” (IBHPS)Overview of IBHPS constructions and parameters.
Outline of Talk
Slide8Start with: Scheme resilient to L’ bits of leakage.Construct: Scheme resilient to L >> L’ bits of leakage.Idea: Leakage Amplification via Parallel Repetition.
Template for BRM Schemes:
1. Leakage Amplification (via ParallelRepetition)
Slide9Template for BRM Schemes:1. ParallelRepetition
Encryption
Decryption
sk
1
sk
2
sk
3
sk
n
…
SK=
PK=
pk
1
pk
2
pk
3
pkn
…
To encrypt under PK.Secretshare message m into n shares m1,…,mn.Encrypt each share mi separately under pki.
c
1, c2, …, cn
c
i
= Enc(m
i
,
pk
i
)
Slide10Theorem (?): nwise parallel repetition amplifies leakageresilience by a factor of n. Hope: Need to leak L’ bits on each of n keys to break the ‘repetition scheme’. … but maybe not a different L’ bits on each key.So is the theorem true?Not in general. Recent counterexample by [LewkoWaters 10]!Yes in special cases (“hash proof systems”). Stay tuned.
Template for BRM Schemes:
1. Security of ParallelRepetition?
Slide11Template for BRM Schemes:1. Efficiency of ParallelRepetition?
Encryption
Decryption
sk
1
sk
2
sk
3
sk
n
…
SK=
PK=
pk
1
pk
2
pk
3
pkn
…
Problem 1: Ciphertextsize, computation proportional to n. Problem 2: Publickey size proportional to n.
c
1, c2, …, cn
c
i
= Enc(m
i
,
pk
i
)
Slide12Template for BRM Schemes:2. Small random subsets.
Encryption
Decryption
sk
1
sk
2
sk
3
sk
n
…
SK=
PK=
pk
1
pk
2
pk
3
pkn
…
Encryptor chooses small random subset of t << n indices.Encrypts t shares under the corresponding t publickeys.Hope: to break scheme, need to have leaked L’ bits on almost all indices (all of the ones that are later chosen).
(idx
1, c1)…,(idxt, ct)
c
i
= Enc(m
i
,
pk
idx
i
)
Slide13Template for BRM Schemes:3. Adding a Master Public Key.
Encryption
Decryption
sk
1
sk
2
sk
3
sk
n
…
SK=
PK=
Use IdentityBased Encryption (IBE)
PK
is masterpublickey of IBE. SK consists of keys ski for identities i=1,…,n.
(idx
1, c1)…,(idxt, ct)
ci = Enc(mi, idxi)
MPK
Slide14Template for BRM Schemes:3. Adding a Master Public Key.
Encryption
Decryption
sk
1
sk
2
sk
3
sk
n
…
SK=
PK=
Scheme meets
efficiency
requirements of the BRM.Security?Does not amplify leakageresilience in general.Rest of talk: make it work with special IBE.
(idx
1, c1)…,(idxt, ct)
ci = Enc(mi, idxi)
MPK
Slide15A “highlevel” template for constructing BRM schemes.
“Identity Based Hash Proof System” (IBHPS) IBHPS constructions and parameters.
Outline of Talk
Slide16A KEM can be used to encrypt a random message m. (pk, sk)ÃKeyGen(1s)(c, m)ÃEncap(pk)m Ã Dec(c, sk)
Key Encapsulation Mechanism (KEM)
Slide17Hash Proof System (HPS): A Special KEM
For each pk, many possible sk. KeyGen outputs skÃSKpk .Correctness: if (c, m)ÃEncap(pk) then Dec(c, sk) = m for all sk.Bad Encapsulation: c* Ã Encap*(pk).Dec(c*, sk) is different for each sk. Can’t distinguish c* from c (even given sk).
SKpk
Dec(c,
SK
pk
)
Dec(
c*
,
SK
pk
)
Slide18HPS and Leakage Resilient KEM
Theorem [NaorSegev 09]: A HPS is a LeakageResilient KEM. L ¼ log(SKpk ).Proof:
sk
Ã
SK
pk
Dec(c,
sk
)
Show: Looks random
Can’t distinguish
‘bad’
ciphertext
m
still has entropy given view of adv.
Use extractors.
If leakage
< log(
SK
pk
)
adv still has uncertainty about sk.
Dec (
c*
,
sk
)
Slide19ParallelRepetition of HPS
Theorem: Parallel repetition of a HPS amplifies leakageresilience.Leakage of HPS is L ¼ log(SKpk ) nwise parallel repetition results in new HPS with SK’pk = SKpk x SKpk x … x SKpkCan show that “random subset selection” also works.
n
times
Slide20IdentityBased Hash Proof System (IBHPS)
Global ‘master’ parameters: (MPK, MSK).For each identity, the secretkey skID comes from a large set.Can efficiently sample from any SKID only if given MSK.Encapsulation targets a specific identity:Good (c, m) Ã Encap(ID, MPK) Bad c* Ã Encap*(ID, MPK).
SKID1
SK
ID2
…
Slide21Applications of IBHPS
Directly gives leakageresilient
IBE
in relativeleakage model.
Can be used to instantiate
our framework. Leakageamplification works!
)
Get
PKE/IBE
in the
B
ounded
R
etrieval
M
odel
.
Slide22A “highlevel” template for constructing BRM schemes.
“Identity Based Hash Proof System” (IBHPS) IBHPS constructions and parameters.
Outline of Talk
Slide23Constructions
Scheme
Assumption
Relative
Leakage
Bilinear
Groups
[Gen06]
ABDHE
Standard Model
1/2
Quadratic
Residuosity
[BGH07]
QR
RO Model
1/O(s)
Lattices
[GPV08]
LWE
RO Model
(1
²
)
Slide24Thank You!
Questions?
Slide25Constructions
Three constructions of IBHPS based on prior IBE schemes.
[Gentry 06]:
Based on a “bilinear groups” assumptions (TABDHE) in standard model.
Gives relative leakage
½
.
[
Boneh
GentryHamburg 07]:
Based on “quadratic
residuosity
” in Random Oracle model.
Gives relative leakage
1/s
(
s
= security parameter).
[Gentry
Peikert

Vaikuntanathan
08]:
Based on lattices and the LWE problem in Random Oracle model.
Already used to get leakageresilient IBE.
[AGV09]
Gives relative leakage
(1
²
)
for any
²
>0
.
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