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Plan Introduction to the Bounded Retrieval Model Motivation An entity-authentication protocol Connections to the BSM Forward-Secure Storage
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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!
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