1. Leakage-resilient Signatures
Vinod Vaikuntanathan
(IBM)
Jonathan Katz
(IBM & Univ. of Maryland)

2. Leakage-resilient Crypto
Crypto Device
Secret-Memory
Secret-Key
L(SK)
L(SM)
=SK+…
L: any polynomial-size circuit
[MR’03,DP’08,P’09,AGV’09,…]
What leaks?
How much?
L: smaller class of circuits
[Riv’97,B’99,CDH+’00,ISW’03,FRT’09,RV’09] 2

3. Models of Leakage What leaks? How much? Bounded Continual
Memory Leakage
[HSH+’08, AGV’09]
“All secret memory leaks”
Computational
Leakage [MR’03]
“Only computation
leaks information”
How much?
Bounded
Continual
Total leakage
< α(|secret|)
Leakage in any
time-period < α(|secret|) 3

4. Models of Leakage Bounded Continual Memory Leakage [HSH+’08, AGV’09]
[AGV’09, NS’09,
ADW’09]
This Work
Computational
Leakage [MR’03]
[MR’03, DP’08,
P’09,FKPR’09]
Bounded
Continual 4

5. Leakage-Resilient Signatures
GMR-security
against bounded α(.)-memory attacks
For every PPT Adv, if |L(SK)| ≤ α(|SK|),
Pr[Adv wins] is negligible. PK L
Adv
L(SK) m
Sign(m)
(m*,σ*) 5

6. Leakage-Resilient Signatures
[ADW’09]
Bounded (1/2-ε)n memory leakage, in random oracle model
[FKPR’09]
Continual α(n) comp. leakage, assuming 2α(n)-hardness
Memory
Leakage
[ADW’09]
Comp.
Leakage
[FKPR’09]
Bounded
Continual 6

7. Our Results Setting: bounded, memory leakage A New Scheme GMR-secure
(1-ε) fraction leakage,∀ε>0
Assumption: Semantically secure enc. + NIZK
More generally, I would like to come up with a Recipe
An Old Scheme (+ tweaks)
one-time signature (generally, t-time)
≈ 1/4 fraction leakage
Assumption: One-way functions
(and more…) 7

8. Our Results Theorem [FKPR’09]
Bounded α(n) leakage ⇒ Continual α(n)/3 comp. leakage
(3-time sig)
(fully-secure sig)
Computational
Leakage
Memory
Bounded
Continual
This Work
Theorem of FKPR
This Work +
[FKPR’09] 8

9. Leakage-resilient One-way Functions
Definition:
Hard to invert f given L(x), for any L s.t. |L(x)| ≤ α(n).
Lemma:
Any UOWHF is a leakage-resilient OWF.
“Proof”: (for CRHFs)
h:{0,1}n → {0,1}n/2 is a CRHF
L:{0,1}n → {0,1}n/2-1 is any leakage function
x has min-entropy ≥ 1 given h(x) and L(x)
x has min-entropy n/2 given h(x)
Given h(x) and L(x), an inverter returns x'≠x w.p ≥ 1/2 9

10. Fully-secure Signature
UOWHF+
Public-key Encryption+
Simulation-sound NIZK [BFM,Sahai]
Assumptions:
x є {0,1}n
SK:
PK:
(h, h(x), PKenc, CRSnizk)
C = Enc(PKenc,(x,m))
Π = Proof in SS-NIZK that “∃x s.t PK contains h(x) and C is the enc. of (x,m)”
Sign(m):
Output (C, Π). 10

11. Proof of Security Three Ideas:
Signature contains no (computational) info. on SK
- NIZK proof Π is simulatable
- Enc(x,m) ≈c Enc(0,m)
PK=(h,h(x),…)
L(x)
Adv m
σ=(Enc(0,m),Π)
σ=(Enc(x,m),Π)
(m*,σ*) 11

12. Proof of Security Three Ideas:
Signature contains no (computational) info. on SK
Forgery ⇒ extract a secret-key.
- simulation-soundness
PK=(h,h(x),…)
L(x)
Adv
σ* contains Enc(x*,m*)
where h(x*)=h(x)
(m*,σ*) 12

13. Proof of Security Three Ideas:
Signature contains no (computational) info. on SK
Forgery ⇒ extract a secret-key.
- simulation-soundness
PK=(h,h(x),…)
L(x)
Adv
x* s.t. h(x*)=h(x) 13

14. Proof of Security Three Ideas:
Signature contains no (computational) info. on SK
Forgery ⇒ extract a secret-key.
UOWHF = Leakage-resilient OWF.
Contradiction.
PK=(h,h(x),…)
L(x)
Adv
x* s.t. h(x*)=h(x) 14

15. A Recipe? Given signature scheme s.t.
H∞[SK given Adv’s view] is non-zero
Leakage-resilient
Signature
Forgery ⇒ extract a secret-key
Finding two SK’s for a PK is an “attack” 15

16. One-time Signature xn,0 x1,0 x2,0 x1,1 … xn,1 x2,1 y1,0 y1,1 yn,0 yn,1
(based on Lamport’78)
xn,0
x1,0
x2,0
Assumption: OWF f
SK:
PK:
x1,1 …
xn,1
x2,1
y1,0
y1,1
yn,0
yn,1
y2,0
y2,1
(where yi,j = f(xi,j))
(xi,j unif. random)
Sign(m1…mn) = (x1,0 x2,1 … xn,0)
=01…0
Q: Is Lamport leakage-resilient? 16

17. ! One-time Signature x1,0 … xn,0 y1,0 y2,0 … yn,0 x2,0 x1,1 x2,1 …
(based on Lamport’78)
Assumption: OWF f
x1,0 …
xn,0
y1,0
y2,0 …
yn,0
x2,0
SK:
PK:
x1,1
x2,1 …
xn,1
y1,1
y2,1 …
yn,1
Leakage
Sign(01…0) + !
Sign(11…0) 17

18. One-time Signature xn,0 x1,0 x2,0 x1,1 … xn,1 x2,1 y1,0 y1,1 yn,0 yn,1
(based on Lamport’78)
xn,0
x1,0
x2,0
Assumption: OWF f
SK:
PK:
x1,1 …
xn,1
x2,1
y1,0
y1,1
yn,0
yn,1
y2,0
y2,1
Sign'(m) =
Sign(ECC(m)) 18

19. One-time Signature xn,0 x1,0 x2,0 x1,1 … xn,1 x2,1 y1,0 y1,1 yn,0 yn,1
(based on Lamport’78)
xn,0
x1,0
x2,0
Assumption: OWF f
SK:
PK:
x1,1 …
xn,1
x2,1
y1,0
y1,1
yn,0
yn,1
y2,0
y2,1
Sign'(m) =
Sign(ECC(m))
Still insecure:
Consider f(x) that ignores 99% of x; outputs OWF(1% of x).
Solution: Let f be a leakage-resilient OWF (=UOWHF) 19

20. One-time Signature xn,0 x1,0 x2,0 x1,1 … xn,1 x2,1 y1,0 y1,1 yn,0 yn,1
(based on Lamport’78)
xn,0
x1,0
x2,0
Assumption: UOWHF h (=OWF [NY,R])
SK:
PK:
x1,1 …
xn,1
x2,1
y1,0
y1,1
yn,0
yn,1
y2,0
y2,1
Sign'(m) =
Sign(ECC(m)) 20

21. ? An Open Question This Work: Bounded, memory leakage +FKPR’09:
Continual, computational leakage
Best of both worlds? ?
Memory
Leakage
This Work
Computational
Leakage
This Work +
[FKPR’09]
Bounded
Continual 21

22. Thanks! 22