10/25/04 security of ad hoc and sensor networks (sasn) 1/22 an attack on the proactive rsa signature...
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10/25/04 Security of Ad Hoc and Sensor Networks Security of Ad Hoc and Sensor Networks (SASN)(SASN)
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An Attack on the Proactive RSA Signature Scheme in the URSA Ad Hoc Network
Access Control Protocol
Stanislaw Jarecki, Stanislaw Jarecki, Nitesh SaxenaNitesh Saxena, Jeong Hyun Yi, Jeong Hyun Yi
School of Information and Computer ScienceSchool of Information and Computer Science
University of California, IrvineUniversity of California, Irvine
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Outline
Introduction: Access control in ad hoc groups
Threshold cryptography Proactive signatures URSA proactive RSA scheme Our attack: efficient key recovery Discussion: Insecurity of URSA Open issues
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Access Control in Ad Hoc Groups
Access control is required to prevent unauthorized entities from joining the group bootstrap other security services, e.g., secure routing remove misbehaving members in general, make group decisions
However, ad hoc group has no infrastructure no trusted group authority dynamic membership
Challenge:How to provide secure access control in a such a decentralized and dynamic environment?
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Zhou and Haas [IEEE Comm. Mag’99] (t+1,n) secret sharing of group secret;
Shamir [ACM COMM.’79]
Threshold signatures any set of t+1 members can sign messages on behalf of the
group tolerate up to t corruptions in the lifetime of the system
Proactive Signatures threshold signatures with increased resilience, lifetime is divided into intervals secret shares are updated tolerate up to t corruptions in every interval
Distribution of Trust using Threshold
Cryptography
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Access Control using Proactive Signatures
Step 1: Certification request
Step 2: Join commit (Signed Vote)
Step 3: Certificate acquisitionMnewMnew
New member (Mnew) wants to join the group If a quorum of t+1 current members approve, Mnew is
issued a signed certificate via proactive signing protocol If no quorum found, membership is denied
Vote1Vote2
Vote2Vote2
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Provably Secure Proactive Signatures
RSA based Frankel, et al. [FOCS’97] [Crypto’97],
Rabin [Crypto’98] DSA based;
Gennaro, et al. [EC’96] [IANDC’01] Schnorr based
Gennaro, et al. [RSA Security’03] BLS based
Boldyreva [PKC’03]
None applicable for access
control in ad hoc groups
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Recent Access Control Schemes URSAURSA: Ubiquitous and Robust Access Control
Luo, et al. [ICNP’01, ISCC’02, WCMC’02, ToN’04]
Proposes a new proactive RSA scheme
Others Based on proactive DSA; Narasimha, et al.
[ICNP’03], Saxena, et al. [SASN’03] Based on proactive BLS; Saxena, et al.
[ICISC’04]
Under scrutiny in this work
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URSA Proactive RSA Scheme (1/3) SetupSetup
Dealer generates RSA private key d and public key (e, N) Randomly picks polynomial f(x) of degree t
Member Mj is issued a secret share:
f(x) = d + a1x + a2x2 + … + atxt (mod N)
Signature generationSignature generation (signing group G, |G|=t+1) Polynomial interpolation:
, , where partial key:
Mj outputs partial signature: )N(modms jd
j
ssj = f(j) (mod N)
)N(modddGj
j
)Nmod( lssd jjj
Recall: RSA signature
s = md (mod N)
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URSA Proactive RSA Scheme (2/3)Signature reconstructionSignature reconstruction:
Since
Try all (t+1) values of α , s.t. se = m (mod N)
Ndeach and )N(moddd jGj
j
}t,...,0{ somefor integers),(over NddGj
j
]t,...,0[ somefor ),N(modm)s(ms N
Gjj
d
Note: α is revealed
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Problems with URSA Proactive RSA Robustness; Narasimha, et al. [ICNP’03]
Shares are computed mod N Regular verifiability mechanisms fail No verifiability No robustness
Fix Share secret d modulo a large prime q Use special purpose zero-knowledge proofs;
Boudot [EC’00] & Camenisch and Michels [Crypto’99]
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Problems with URSA Proactive RSA
Is this scheme (modified with the robustness fix) secure in the presence of
a coalition of t corrupt members?
The answer is: negative
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Our Attack (example): Binary Search
t=1, n=2 Players M1, M2 , Signing group G={1,2} Adversary A corrupts M1
Recall: d = d1 + d2 – αN Signing protocol reveals α
If α = 0, d = d1 + d2 d ≥ d1 o/w if α = 1, d = d1 + (d2 - N) d < d1
During proactive updates, A can choose ss1 s.t.
With every update round, the search interval is halved Binary search recovers d in log2(N) rounds
0 d1 N
Recall d1 = ss1l1 (mod N)
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Our Attack: (t+1)-ary Search
Adversary A corrupts M1, M2, …,Mt (w.l.o.g) Signing group Gp={1,2,…,t, p}, where p > t A learns if d ≥ Dp or d < Dp, where
During proactive updates, A can choose ss1,
ss2,…, sst s.t.
Every round reveals log2(t+1) MSBs of d (t+1)-ary search recovers d in rounds
pj,Gj
)G(jjp
p
p )N(modlssD
0 Dp1 Dp2 Dpt N
)1t(log
|N|
2
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Optimal Choice of New Shares
Solve following set of deterministic equations for ss1, ss2, …, sst
)N(modDlss...lsslss
......
)N(modDlss...lsslss
)N(modDlss...lsslss
ttptptp
22p2p2p
11p1p1p
p)G(
tt)G(
22)G(
11
p)G(
tt)G(
22)G(
11
p)G(
tt)G(
22)G(
11
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URSA Proactive Update
Simplified Classic protocol; Herzberg et al. [Crypto’95] Update the shares but keep the same group secret d A set of at least t+1 members update the
polynomials Each M i chooses random poly. δi(z) of degree t
s.t. δi(0) = 0 Mj gives δj(i) to Mi
Mi’s new share becomes ssi (old share was ssi‘)
ssi’ is deleted
)N(mod )i('ssssn
1jjii
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Adversarial Behavior in Share Update
B : t members corrupted by A Mb B : member who “speaks last ” Update polynomial New shares are computed as
Mb waits until it receives all other shares and chooses its polynomial δb(z) s.t.
This sets A’s share to be ss1, ss2,…,sst
)N)(modz()z()z( b}M\{jj
b
)N(mod )i('ssss ii
)N(mod)i('ssss)i(}M\{jjiib
b
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Speeding-up the Attack Attack requires r = rounds Recover last 40-bits of d by brute-force given
RSA public key (e,N) r = Apply known results on RSA partial key
exposure; Boneh, et al. [AC’01], Blomer-May [Crypto’03],
Thm1: log2(e) MSBs of d determine 512-MSBs
r = e.g., for t = 7, |N|=1024, e = 65537 r = 163
e = 3 r = 158
)1t(log
|N|
2
)1t(log
40|N|
2
)1t(log
402/|N|)e(log
2
2
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Speeding-up the Attack
Number of proactive update rounds required for a given logN(e) value, for t=7 & |N|=1024
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Attack Assumptions
1. Adversary corrupts t members of the update group Ω, one of whom “speaks last ”
2. In every round, t runs of the signing protocol are executed, the signing groups consisting of all bad and one (distinct) good player.
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Insecurity of URSA
For a modest threshold t=7, |N|=1024 and e=65537, the attack requires 163 proactive update rounds and a total of 1148 runs of the signing protocol
The leakage is very fast e.g. in just 34 rounds, 600 MSBs of d are revealed
Other faster attacks are possible with signing group consisting of less than t bad players
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Positive Result in a Related Work
Jarecki and Saxena [in submission] URSA proactive RSA scheme (plus robustness
fix) with additive-secret sharing is provably secure
2-4 times faster than the state-of-the-art Rabin’s proactive RSA [Crypto’98]
However, not applicable for access control in ad hoc groups
Open Problem: to design a provably secure proactive RSA scheme that yields an efficient access control mechanism for ad hoc groups!!
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Thank You!
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Speeding-up the Attack
Thm2: For prime e ε [2m, 2m+1], with m ε [|N|/4,|N|/2], m MSBs of d determine d
Thm3: For e ε [2m, 2m+1] and product of
at most r primes, with m ε [|N|/4,|N|/2], m
MSBs determine d given factorization of e
Thm4: For e ε [N0.5, N0.25],
MSBs of d determine d, where α = logN(e)
)1t(log2
|N|r
)1t(log4
|N|
22
151236238
|N| 2
15123623)1t(log8
|N|r 2
2
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Our Attack: (t+1)-ary search
Adversary A corrupts M1, M2, …,Mt (w.l.o.g) Signing group Gp={1,2,…,t, p}, where p ε [t+1,..2t] Recall
Signing protocol reveals α(Gp)
Compute
If Sp ≥ α(Gp)N , A learns d ≥ Dp o/w if Sp < α(Gp)N , A learns d < Dp
During proactive updates, A chooses ss1, ss2,…, sst such that
Every round reveals log2(t+1) MSBs of d (t+1)-ary search recovers d in rounds
pj,Gj
)G()G(p
)(Gj
p
ppp Nddd
)N(modSD ,integers) over(dS pppj,Gj
)G(jp
p
p
0 Dt+1 Dt+2 D2t N-1
)1t(log
|N|
2