Public-Key Encryption in the - PowerPoint Presentation

Slide1

Public-Key Encryption in the Bounded-Retrieval Model

Joël Alwen, Yevgeniy Dodis, Moni Naor, Gil Segev, Shabsi Walfish, Daniel Wichs

Speaker: Daniel

Wichs

Eurocrypt

2010

Slide2

MotivationCryptographic security analyzed in formal “attack model”. Do our attack models capture reality?In

reality, extra information about secret-keys can leak. Side-channels attacks: timing, power, heat, EM radiation, acoustics...Cold-boot attack [HSH+ 08]VirusesLeakage-Resilient Crypto:Add key-leakage to the attack model.Build primitives that provably allow leakage

of secret key.

Slide3

B

ounded Retrieval Model [Dzi06,…,ADW09]:Grow secret-key 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.

Relative-Leakage Model[AGV09, DKL09,NS09,…]. Maximize ratio of L to |sk| (e.g. 90% of the key can leak).

sk

leak

[

Akavia-Goldwasser-Vaikuntanathan

09]

Slide4

Why 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 side-channel attacks:Leakage amount depends on the complexity of computation.Leakage-resilient schemes might be less secure:+ Leakage-resilience ) + Complexity ) + Leakage. BRM efficiency breaks the cycle.

Slide5

Prior Work on Leakage ResilienceMemory AttacksRelative-Leakage: Symmetric and Public-Key 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: Public-Key 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.

Slide6

Definition of PKE in BRMKey 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.AdversaryChallenger(pk,sk) Ã

KeyGen(1s )

pk

f : {0,1

}

*

!

{

0,1}

L

f(sk)

m0, m1bà {0,1} cÃEncrypt(mb,pk)cOutput b’, L

Pr[b’ = b]

·

½ +

negl

(s)

Slide7

A “high-level” template for constructing BRM schemes.“Identity Based Hash Proof System” (IB-HPS)

Overview of IB-HPS constructions and parameters.Outline of Talk

Slide8

Start 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 Parallel-Repetition)

Slide9

Template for BRM Schemes:1. Parallel-Repetition

EncryptionDecryption

sk1

sk2

sk

3

sk

n

…

SK=

PK=

pk

1

pk2pk3pkn…To encrypt under PK

.

Secret-share message

m into n shares

m1

,…,mn.

Encrypt

each share

m

i

separately under

pk

i

.

c

1

,

c

2

, …,

c

n

c

i

= Enc(m

i

,

pk

i

)

Slide10

Theorem (?): n-wise parallel repetition amplifies leakage-resilience 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 [Lewko-Waters 10]!Yes in special cases (“hash proof systems”). Stay tuned.Template for BRM Schemes:1. Security of Parallel-Repetition?

Slide11

Template for BRM Schemes:1. Efficiency of Parallel-Repetition?

EncryptionDecryption

sk1

sk2

sk

3

sk

n

…

SK=

PK=

pk

1

pk2pk3pkn…Problem

1: Ciphertext-size, computation proportional to

n.

Problem 2: Public-key size proportional to n.

c

1

,

c

2

, …,

c

n

c

i

= Enc(m

i

,

pk

i

)

Slide12

Template for BRM Schemes:2. Small random subsets.

EncryptionDecryption

sk

1sk2

sk

3

sk

n

…

SK=

PK=

pk

1

pk2pk3pkn…Encryptor chooses small random subset of t <<

n indices.

Encrypts t shares under the corresponding

t public-keys.

Hope:

to break scheme, need to have leaked

L’

bits on almost all indices (all of the ones that are later chosen).

(idx

1

, c

1

)…,(

idx

t

,

c

t

)

c

i

= Enc(m

i

,

pk

idx

i

)

Slide13

Template for BRM Schemes:3. Adding a Master Public Key.

EncryptionDecryption

sk

1sk2

sk

3

sk

n

…

SK=

PK=

Use Identity-Based Encryption (IBE)

PK

is master-public-key of IBE. SK consists of keys ski for identities i=1,…,n.(idx1, c

1

)…,(idxt,

ct)

c

i = Enc(m

i, idx

i

)

MPK

Slide14

Template for BRM Schemes:3. Adding a Master Public Key.

EncryptionDecryption

sk

1sk2

sk

3

sk

n

…

SK=

PK=

Scheme meets

efficiency

requirements of the BRM.Security?Does not amplify leakage-resilience in general.Rest of talk: make it work with special IBE.

(idx

1, c

1)…,(idx

t

, ct)

c

i

= Enc(m

i

,

idx

i

)

MPK

Slide15

A “high-level” template for constructing BRM schemes.“Identity Based Hash Proof System” (IB-HPS) IB-HPS constructions and parameters.Outline of Talk

Slide16

A KEM can be used to encrypt a random message m. (pk, sk

)ÃKeyGen(1s)(c, m)ÃEncap(pk)m à Dec(c, sk)Key Encapsulation Mechanism (KEM)

Slide17

Hash Proof System (HPS): A Special KEMFor each pk, many possible sk. KeyGen outputs skÃ

SKpk .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, SKpk)

Dec(

c*

,

SK

pk

)

Slide18

HPS and Leakage Resilient KEMTheorem [Naor-Segev 09]: A HPS is a Leakage-Resilient 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

)

Slide19

Parallel-Repetition of HPSTheorem: Parallel repetition of a HPS amplifies leakage-resilience.Leakage of HPS is L ¼ log(|

SKpk |) n-wise parallel repetition results in new HPS with SK’pk = SKpk x SKpk x … x

SKpk

Can show that “random subset selection” also works.

n

times

Slide20

Identity-Based Hash Proof System (IB-HPS)Global ‘master’ parameters: (MPK, MSK).For each identity, the secret-key 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

…

Slide21

Applications of IB-HPSDirectly gives leakage-resilient IBE in relative-leakage model. Can be used to instantiate our framework. Leakage-amplification works! )

Get PKE/IBE in the Bounded Retrieval Model.

Slide22

A “high-level” template for constructing BRM schemes.“Identity Based Hash Proof System” (IB-HPS) IB-HPS constructions and parameters.Outline of Talk

Slide23

ConstructionsSchemeAssumption

RelativeLeakageBilinear Groups[Gen06]ABDHEStandard Model1/2Quadratic Residuosity[BGH07]QR RO Model1/O(s)Lattices

[GPV08]LWERO Model

(1-²)

Slide24

Thank You!Questions?

Slide25

ConstructionsThree constructions of IB-HPS based on prior IBE schemes.[Gentry 06]: Based on a “bilinear groups” assumptions (TABDHE) in standard model.Gives relative leakage ½.

[Boneh-Gentry-Hamburg 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 leakage-resilient IBE. [AGV09]Gives relative leakage (1- ²) for any

²>0.