Introduction to the Bounded-Retrieval Model

Stefan Dziembowski

University of Rome La Sapienza

Warsaw University

The main ideaBounded-Retrieval Model:

Construct cryptographic protocols where the secrets are so large that they cannot be efficiently stolen.

D. Dagon, W. Lee, R. J. LiptonProtecting Secret Data from Insider Attacks. Financial Cryptography 2005

G. Di Crescenzo, R. Lipton and S. Walfish Perfectly Secure Password Protocols in the Bounded Retrieval ModelTCC 2006

S. Dziembowski Intrusion-Resilience via the Bounded-Storage ModelTCC 2006

Perfectly Secure Password Protocols in the Bounded Retrieval ModelD. Cash, Y. Z. Ding, Y. Dodis, W. Lee, R. Lipton and S. Walfish Intrusion-Resilient Authenticated Key Exchange in the Bounded Retrieval Model without Random OraclesTCC 2007

S. DziembowskiOn Forward-Secure StorageCRYPTO 2006

Plan

Introduction to the Bounded Retrieval Model

Motivation An entity-authentication protocol Connections to the BSM

Forward-Secure Storage

The problem

Computers can be infected by mallware!

installs a virus

The virus can: take control over the machine, steal some secrets stored on the machine.

Can we run any crypto on such machines?

retrieves some data

Is there any remedy?

If

the virus can download all the data stored on the machine

then

Assume that he cannot do it!

the situation looks hopeless.

Idea:

The general model

installs a virus

retrieves some data

installs a virus

retrieves some data

no virus

no virus

no virus

The total amount of retrieved data is bounded!

Our goal

Try to preserve as much security as possible (assuming the scenario from the previous slide).

Of courseas long as the virus is controlling the machine

nothing can be done.

Therefore

we care about the periods when the machine is free of viruses.

Two variants

How does the virus decide what the retrieve?

Variant 2 [CLW06,…]He can only access some individual bits on the

victim’s machine (“slow memory”)

Variant 1 [D06a,D06b,CDDLLW07]He can compute whatever he wants on the

victim’s machine.

Practicality?

An example: entity authentication

the bank

How can the bank verify the authenticity of the user?

We solve the following problem:

the user

example of f:Y={y1,…,ym} is a set of indices in R f(Y,(R1,…,Rt)) = (Ry1,…,Rym)

Entity authentication – the solution

random Y

key R 00011010011101001001101011100111011111101001110101010101001001010011110000100111111110001010

X = f(Y,R)verifies

00011010011101001001101011100111011111101001110101010101001001010011110000100111111110001010

y1 y2 ym

11 . . . 0

…

Security of the authentication protocol

Theorem [D06,CDDLLW07]

The adversary that “retrieved” a constant fraction of R does is not able to impersonate the user.

(This of course holds in the periods when the virus is not on the machine.)

A related concept: the Bounded Storage Model

This is related to the Bounded Storage Model (BSM) [Maurer 1992]

In the BSM the security of the protocols is based on the assumption that one can broadcast more bits than the adversary can store.

In the BSM the computing power of the adversary may be unlimited.

The Bounded-Storage Model (BSM) –an introduction

can perform any computationon R, but the result U=h(R) has to be much smaller than R

shortinitialkey K

X = f(K,R)

000110100111010010011010111001110111111010011101010101010010010100111100001001111111100010101001000101010010001010010100101011010101001010010101

randomizer R:

knows:U=h(R)

randomizer disappears

X ?

Eve shouldn’t be able to distinguish X from random

s

How is BSM related to our model?

Seems that the assumptions are oposite:

transmission storage

BSM cheap expensive

LCM expensive cheap

BSM vs. BRMBounded-Storage Model:

Bounded-Retrieval Model

R comes from a satellite

stored value U

R is stored on a computer

retrieved value U

Consider again the authentication protocol

Observation

In the authentication protocol one could use a BSM-secure function f.

random Y

X = f(Y,R)verifies

Overview of the results

An entity authentication protocol

A session-key exchange protocol in the Random Oracle Model [D06a] in the plain model [CDDLLW07]

Forward Secure Storage [D06b] – “an encryption scheme secure in the BRM”

Plan

Forward-Secure Storage

IT-secure computationally-secure a scheme with a conjectured hybrid security

Connections with the theory of Harnik and Naor

Forward Secure Storage (FSS) - the motivation

key K

message M

C = E(K,M)

Cinstalls a virus

retrieves C

One of the following happens: 

• The key K leaks to the adversary or

• The adversary breaks the scheme

The adversary can compute M

The idea

Design an encryption scheme such that the ciphertext C is so large that the

adversary cannot retrieve it completely

message M

ciphertext C=Encr(K,M)

Forward-Secure Storage – a more detailed view

The adversary to compute an arbitrary function h of C.

ciphertext C=Encr(K,M)

function h

retrieved value U=h(C)

length t

length s << t

K M ?

Computational power of the adversary

We consider the following variants:

computational: the adversary is limited to poly-time

information-theoretic: the adversary is infinitely-powerful

hybrid: the adversary gains infinite power after he computed the function h.

This models the fact that the in the future the current cryptosystems may be broken!

Information-theoretic solution – a wrong idea

K R

X

M

Y

f( ),

=message

key

ciphertextin the BSMencryption

f – secure in the BSM

xor

ciphertext(R,Y)

Shannon theorem this cannot work!

What exactly goes wrong?

Suppose the adversary has some information about M.

He can see(R, f(K,R) xor M ).

So, he can solve (for K) the equation W = f(K,R) xor M.

If he has enough information about M, and K is short, he will succed!

Idea: “Blind” the message M!

denote it W

A better idea

K R

X

M

Y

f( ),

=

message

key is a pair (K,Z)

ciphertext(R,Y)

Z

xor

Why does it work?

IntuitionThe adversary can compute any function h of: 

Y is of no use for him, since it is xor-ed with a random string Z! 

So if this FSS scheme can be broken then also the BSM function f can be broken 

(by an adversary that uses the same amount of memory).

R Y = f(K,R) xor M xor Z

Problem with the information-theoretic scheme

The secret key needs to be larger than the message!

What if we want the key to be shorter?

We need to switch to the computational setting...

Computational FSS (with a short key)

(Encr,Decr) – an IT-secure FSS(E,D) – a standard encryption scheme

Encr1(

Encr(

E(

)

)

)=

,

,

,

K

K K’

K’

M

K’ is a random key for the standard encryption scheme

M

Intuition: when the adversary learns K he has no idea about K’ and therefore no idea about M.

large

small

Hybrid security

What about the hybrid security?

Recall the scenario:

ciphertext C=Encr(K,M)

h

retrieved value

U=h(C)

M ?

Is this scheme secure in the hybrid model?

The adversary retrives only the second part! 

Later, when she gets infinite computing  power, she can recover the message M! 

Thus, the scheme is not secure in the  hybrid model!

Encr(

E(

)

)

,

,

K K’

K’ M

A scheme (Encr2,Decr2)

Does there exist an FSS scheme with hybrid security (and a short key)?

Idea: Generate K pseudorandomly!

(Encr,Decr) – an IT-secure FSSG – a cryptographic PRG

Encr2( )=,K M

Encr( ),G(K) M

Is the scheme from the previous slide secure?It cannot be IT-secure, but is it

computationally-secure? secure in the hybrid model? We leave it as an open problem. Looks secure...

We can show the following:

Very informally,

it is secure if one-way functions cannot be used to construct Oblivious Transfer.

Computational security of Encr2 (1/2)

there exists an adversary Athat breaks the (Encr2,Decr2) scheme

We show that if

then

one can construct an Oblivious Transfer protocol with:

an unconditional privacy of the Sender privacy of the Receiver based on the security of the

PRG G.

Computational security of Encr2 (2/2)

Simplification: assume that |M| = 1 and the adversary can guess it with probability 1.

We construct an honest-but-curious Rabin OT.

receiver

Encr(X,M)K

M

U - memory of the adversary

A computationally-limited sendercannot distinguish these cases!

If X is random then the receiver learns nothing about M (this follows from the IT-security of Encr)!

If then the adversary outputs M.

if if then thenX := G(K) X random

senderinput: M

How to interpret this result?

Which PRGs G are safe to use in this protocol?

In some sense: “those that cannot be used to construct OT”.

But maybe there exist “wrong” PRGs...

(see: S. Dziembowski and U. MaurerOn Generating the Initial Key in the Bounded-

Storage Model, EUROCRYPT '04)

Hybrid security of Encr2

The argument for the hybrid security is slightly weaker.

We can construct only an OT-protocol with a computationally-unbounded algorithm for the Receiver...

This is because the receiver has to simulate an unbounded adversary.

receiver

Summary

ITsecurity

hybrid security

comp. security

the first scheme secure secure secure

the second scheme

notsecure

notsecure secure

the third scheme

notsecure

maybesecure

maybesecure

A complexity-theoretic view

Suppose the adversary wants to know if a given C is a ciphertext of some message M.

NP-language:L = {C : there exists K such that C = Encr(K,M)}.

standard encryption FSS

is C in L?Can we compress C to some U, s.t. |U| << |C| so that later we can decide if C is in L basing on U, and using infinite computing power?

The theory of Harnik and Naor

This question was recently studied in:Danny Harnik, Moni Naor On the Compressibility of NP Instances andCryptographic Applications FOCS 2006

See also:Bella Dubrov, Yuval Ishai On the Randomness Complexity of Efficient SamplingSTOC 2006

Compressibility of NP Instances

Informally, an NP language L is compressible if there exists an efficient algorithm that

compresses every string X to a shorter string U,

in such a way that an infinitely-powerful solver can decideif X is in L basing only on U.

Proving that some language is incompressible(from standard assumptions)

is an open problem..

This is why showing an FSS scheme provably-secure in the hybrid model may be hard!

Thanks!