On Forward-Secure Storage

Stefan Dziembowski

Warsaw University

and

University of Rome La Sapienza

The main idea

Limited Communication Model:

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

D.Intrusion-Resilience via the Bounded-Storage ModelTCC 2006

D. Cash, Y. Z. Ding, Y. Dodis, W. Lee, R. Lipton and S. Walfish Intrusion-Resilient Authentication in the Limited Communication

Model

(There it was used to construct intrusion-resilient protocols for authentication and session-key generation.)

The problem that we consider

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

Our 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)

We call it a

Forward-Secure Storage (FSS)

Practicality?

Forward-Secure Storage

We allow 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!

Our Contribution

Formal definition of FSS

Constructions of FSS schemes:

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

Connections with the theory of Harnik and Naor

A tool: the Bounded Storage Model

It turns out that 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.

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

BSM – previous results

Several key-expansion functions f were proven secure [DR02, DM04b, Lu04, Vad04].

Of course their security depends on the bound on the memory of the adversary.

We call a function s-secure if it is secure against an adversary that has memory of a size s.

How is BSM related to our model?

Seems that the assumptions are oposite:

transmission storage

BSM cheap expensive

LCM expensive cheap

BSM vs. LCM

Bounded-Storage Model:

Limitted-Communication Model:

R comes from a satellite

stored value U

C is stored on a computer

retrieved value U

Information-theoretic solution – a wrong idea

K R

X

M

Y

f( ),

=message

key

ciphertextin the BSMencryption

f – s-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 using 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 settings...

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 senderinput: M

X = G(K) with prob. 0.5X is random with prob. 0.5

Encr(X,M)K

If X = G(K) then the adversary outputs M.

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)!

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!

Questions?

?