Leakage-Resilient Storage

Leakage-Resilient StorageLeakage-Resilient Storage

Francesco DavìStefan DziembowskiDaniele Venturi

SCN 2010 13/09/2010

Sapienza University of Rome

PlanPlan

1.Leakage-Resilient Cryptography- Motivation- Leakage models

2. Our contribution: Leakage-Resilient Storage- Definition and Properties- Constructions

3. Conclusion

Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN 2010 13/09/2010

How to construct secure cryptographic How to construct secure cryptographic devices?devices?

CRYPTO

cryptographic device

very secure

Security based on well-defined mathematical problems

not secure!


The problemThe problem

hard to attack

easy to attack

CRYPTO

cryptographic device


Information leakageInformation leakage

cryptographic deviceSide channel information:

• power consumption, • electromagnetic radiation, • timing information,

…


Leakage-Resilient CryptographyLeakage-Resilient Cryptography


Design cryptographic protocols that are secure

even

on the machines that leak information

Design cryptographic protocols that are secure

even

on the machines that leak information

Leakage-Resilient Cryptography:Leakage-Resilient Cryptography: The ModelsThe Models


• Continual leakage(MR04, DP08, Pie09, FKPR10, FRRTV10, GR10, JV10)

• Bounded memory-leakage(ISW03, IPSW06, AGV09, ADW09, KV09, NS09, DHLW10)

• Auxiliary input(DKL09, DGKPV10)

• Continual memory-leakage(BKKV10, DHLW10)

• Continual leakage(MR04, DP08, Pie09, FKPR10, FRRTV10, GR10, JV10)

• Bounded memory-leakage(ISW03, IPSW06, AGV09, ADW09, KV09, NS09, DHLW10)

• Auxiliary input(DKL09, DGKPV10)

• Continual memory-leakage(BKKV10, DHLW10)

Only computation leaks

Total leakage unbounded

All the memory leaks

Total leakage bounded

All the memory leaks

Total leakage unbounded

All the memory leaksComputationally hard to recover

the secret from the leakage

Bounded memory-leakage modelBounded memory-leakage model


The adversary is allowed to learn (adaptively)

the values of t leakage functions(chosen by her)

on the internal data used bythe cryptographic scheme

The adversary is allowed to learn (adaptively)

the values of t leakage functions(chosen by her)

on the internal data used bythe cryptographic scheme

Leakage functionsLeakage functions


very restricted class (read-off wires)very restricted class (read-off wires)

0 1 1 0

f

f(x)

general leakage (any input-shrinking function)general leakage (any input-shrinking function)

0 0 1 0 1 1 0 1

x

chooses

retrieves

retrieves

chooses

PlanPlan


2. Our contribution: Leakage-Resilient Storage- Definition and Properties- Constructions

3. Conclusion


Leakage-Resilient StorageLeakage-Resilient Storage


Enc(m)Enc(m)Enc Dec

Note:no secret key

mm

g1,…,gt

mm

chooses (adaptively) t functions

gi : {0,1}|Enc(m)| → {0,1}ci є Γ

retrieves ci bitscomputationally

unbounded

total leakage < C • very realistic

• Decode є Γ

• input-shrinking

C < |Enc(m)|

All-Or-Nothing TransformAll-Or-Nothing Transformit should be hard to reconstruct a messageif not all the bits of its encoding are known

Security definition Security definition


A scheme (Enc, Dec) is secure if for every m0, m1

no adversary can distinguish Enc(m0) from Enc(m1)A scheme (Enc, Dec) is secure if for every m0, m1

no adversary can distinguish Enc(m0) from Enc(m1)

we will require that m0, m1 are chosen by the adversary

Enc(m0)Enc(m0) Enc(m1)Enc(m1)

Security definitionSecurity definition

adversary oracle

chooses m0,m1 є {0,1}α m0,m11. chooses a random b = 0,12. calculates τ := Enc(mb)

outputs b’

(Enc,Dec) is (Γ, C, t, ε)-secureif no adversary wins the game

with probability greater than 1/2 + ε


Enc : {0,1}α → {0,1}β

Dec : {0,1}β → {0,1}α

for i = 1,...,t

chooses gi : {0,1}β → {0,1}ci є Γ calculates gi(τ)gi(τ)

gi

wins if b’ = b

advantage

ProblemProblem


each leakage function can depend only on some restricted part

of the memory


of the memorythe cardinality of Γ is restrictedthe cardinality of Γ is restricted

randomness extractors

-wise independent hash

functions

For a fixed family Γ

how to construct secure (Enc,Dec)?

A weaker adversaryA weaker adversary


Enc(m):=(Rand, f(Rand) m)Enc(m):=(Rand, f(Rand) m)Encmm

gi gi(Rand, f(Rand) m)

Enc(m)Enc(m)

gi(Enc(m))g’i g’i(Rand)

adversaryweak adversary

LemmaLemma


For any Γ, c, t and ε,

if an encoding scheme is (Γ, c, t, ε )-secure for

then it is also (Γ, c, t, ε˙2α )-secure for

For any Γ, c, t and ε,

if an encoding scheme is (Γ, c, t, ε )-secure for

then it is also (Γ, c, t, ε˙2α )-secure for

α is the length of the message

Proof IdeaProof Idea


wins with advantage δ

can simulate

replacing f(Rand) m with a random string z є {0,1}α

ConsiderConsider

ConstructConstruct

wins with advantage ε= δ˙2-α

= ε ˙2α

Two-source ExtractorTwo-source Extractor


source1source1

source2source2

Two-SourceExtractor

extracted stringextracted string

Example:

inner product modulo 2

deterministic

Independent

Random

Far from uniform

A lot of min-entropy

Almost uniformly random

Memory divided into 2 parts: constructionMemory divided into 2 parts: construction


R0R0

R1R1

Ext Ext(R0,R1)Ext(R0,R1)

Enc(m):=( , , m)R0R0 R1R1 Ext(R0,R1)Ext(R0,R1)

Dec( , , m*):= m* .R0R0 R1R1 Ext(R0,R1)Ext(R0,R1)

M0 M1each leakage function can depend

only on some restricted partof the memory


of the memory

remind

Memory divided into 2 parts: contributionMemory divided into 2 parts: contribution


R0R0

R1R1

Ext Ext(R0,R1)Ext(R0,R1)


Dec( , , m*):= m* .R0R0 R1R1 Ext(R0,R1)Ext(R0,R1)

M0 M1each leakage function can depend

only on some restricted partof the memory


of the memory

remind

If Extis a two-source extractorthen

is secureEnc

Dec

( ),

against an adversary such that

Proof IdeaProof Idea


It suffices to show that It suffices to show that (Enc,Dec)(Enc,Dec) is secure against every is secure against every

One can prove that even given One can prove that even given g’1( , ),…, g’t( , )

R0R0 R1R1


R0R0 R1R1 R0R0 R1R1

• are still independent

• have high min-entropy (with high probability)

remind

andand

ProblemProblem



of the memory


of the memorythe cardinality of Γ is restrictedthe cardinality of Γ is restricted

randomness extractors

-wise independent hash

functions

For a fixed family Γ

how to construct secure (Enc,Dec)?

-wise independent hash functions-wise independent hash functions


H={hs:X→Y}sєIis -wise independent if

uniformly random S є I

X Y

{x1,…,x} hS {hS(x1),…,hS(x)}

uniform over Y

Boolean circuits of small size: constructionBoolean circuits of small size: construction


the cardinality of Γ is restrictedthe cardinality of Γ is restricted

remind

the set of functions computable by Boolean circuits of a fixed size

Encs(m):=(R, hS(R) m)

Decs(R , m*):=(hS(R) m*)

H={hs:X→Y}sєIis -wise independent

R є X is random

PlanPlan


2. Our contribution: Leakage-Resilient Storage- Definition and Properties- Construction

3. Conclusion


Conclusion and Future workConclusion and Future work


Achieved:• We have defined a primitive to securely store

information in hardware that may leak information• We have given constructions of such a scheme in two

relevant scenarios

Open:• Refreshing of the storage• From storage to computation: compute with encoded

data• Find more applications

Achieved:• We have defined a primitive to securely store

information in hardware that may leak information• We have given constructions of such a scheme in two

relevant scenarios

Open:• Refreshing of the storage• From storage to computation: compute with encoded

data• Find more applications

Leakage-Resilient Storage

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