Leakage-Resilient Storage Francesco Davì Stefan Dziembowski Daniele Venturi SCN /09/2010 Sapienza University of Rome

Plan 1. 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 /09/2010

How to construct secure cryptographic devices? CRYPTO cryptographic device very secure Security based on well-defined mathematical problems not secure! Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010

The problem hard to attack easy to attack CRYPTO cryptographic device Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010

Information leakage cryptographic device Side channel information: power consumption, electromagnetic radiation, timing information, … Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010

Leakage-Resilient Cryptography Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 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:The Models Leakage-Resilient Cryptography: The Models Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 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 leaks Computationally hard to recover the secret from the leakage

Bounded memory-leakage model Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 The adversary is allowed to learn (adaptively) the values of t leakage functions (chosen by her) on the internal data used by the cryptographic scheme The adversary is allowed to learn (adaptively) the values of t leakage functions (chosen by her) on the internal data used by the cryptographic scheme

Leakage functions Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 very restricted class (read-off wires) 0110 f f(x) general leakage (any input-shrinking function) x chooses retrieves chooses

Plan 1. 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 /09/2010

Leakage-Resilient Storage Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 Enc(m) Enc Dec Note: no secret key m m g 1,…,g t m m chooses (adaptively) t functions g i : {0,1} |Enc(m)| → {0,1} c i є Γ retrieves c i bits computationally unbounded total leakage < C very realistic Decode є Γ input-shrinking C < |Enc(m)| All-Or-Nothing Transform it should be hard to reconstruct a message if not all the bits of its encoding are known

Security definition Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 A scheme (Enc, Dec) is secure if for every m 0, m 1 no adversary can distinguish Enc(m 0 ) from Enc(m 1 ) A scheme (Enc, Dec) is secure if for every m 0, m 1 no adversary can distinguish Enc(m 0 ) from Enc(m 1 ) we will require that m 0, m 1 are chosen by the adversary Enc(m 0 ) Enc(m 1 )

Security definition adversaryoracle chooses m 0,m 1 є {0,1} α m 0,m chooses a random b = 0, calculates τ := Enc(m b ) outputs b’ (Enc,Dec) is ( Γ, C, t, ε )-secure if no adversary wins the game with probability greater than 1/2 + ε Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 Enc : {0,1} α → {0,1} β Dec : {0,1} β → {0,1} α for i = 1,...,t chooses g i : {0,1} β → {0,1} c i є Γ calculates g i ( τ ) gi(τ)gi(τ) gigi wins if b’ = b advantage

Problem Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 each leakage function can depend only on some restricted part of the memory each leakage function can depend only on some restricted part of the memory the cardinality of Γ is restricted randomness extractors -wise independent hash functions For a fixed family Γ how to construct secure (Enc,Dec)?

A weaker adversary Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 Enc(m):=(Rand, f(Rand) m) Enc m m gigi g i (Rand, f(Rand) m) Enc(m) g i (Enc(m)) g’ i g’ i (Rand) adversaryweak adversary

Lemma Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 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 ˙ 2 α )-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 ˙ 2 α )-secure for α is the length of the message

Proof Idea Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 wins with advantage δ can simulate replacing f(Rand) m with a random string z є {0,1} α Consider Construct wins with advantage ε = δ ˙ 2 -α = ε ˙2α = ε ˙2α

Two-source Extractor Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 source 1 source 2 Two-Source Extractor extracted 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: construction Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 R0R0 R0R0 R1R1 R1R1 Ext Ext(R 0,R 1 ) Enc(m):=(,, m) R0R0 R0R0 R1R1 R1R1 Ext(R 0,R 1 ) Dec(,, m*):= m*. R0R0 R0R0 R1R1 R1R1 Ext(R 0,R 1 ) M0M0 M1M1 each leakage function can depend only on some restricted part of the memory each leakage function can depend only on some restricted part of the memory remind

Memory divided into 2 parts: contribution Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 R0R0 R0R0 R1R1 R1R1 Ext Ext(R 0,R 1 ) Enc(m):=(,, m) R0R0 R0R0 R1R1 R1R1 Ext(R 0,R 1 ) Dec(,, m*):= m*. R0R0 R0R0 R1R1 R1R1 Ext(R 0,R 1 ) M0M0 M1M1 each leakage function can depend only on some restricted part of the memory each leakage function can depend only on some restricted part of the memory remind IfExtis a two-source extractor then is secureEnc Dec (), against an adversary such that

Proof Idea Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 It suffices to show that (Enc,Dec) is secure against every One can prove that even given One can prove that even given g’ 1 (, ),…, g’ t (, ) R0R0 R0R0 R1R1 R1R1 Enc(m):=(,, m) R0R0 R0R0 R1R1 R1R1 Ext(R 0,R 1 ) R0R0 R0R0 R1R1 R1R1 R0R0 R0R0 R1R1 R1R1 are still independent have high min-entropy (with high probability) remind and

Problem Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 each leakage function can depend only on some restricted part of the memory each leakage function can depend only on some restricted part of the memory the 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 Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 H={h s :X→Y} sє I is -wise independent if uniformly random S є I X Y { x 1,…,x } hShS {h S (x 1 ),…, h S (x ) } uniform over Y

Boolean circuits of small size: construction Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 the cardinality of Γ is restricted remind the set of functions computable by Boolean circuits of a fixed size Enc s (m):=(R, h S (R) m) Dec s (R, m*):=( h S (R) m*) H={h s :X→Y} sє I is -wise independent R є X is random

Plan 1. 1.Leakage-Resilient Cryptography - Motivation - Leakage models 2. Our contribution: Leakage-Resilient Storage - Definition and Properties - Construction 3. Conclusion Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010

Conclusion and Future work Davì, Dziembowski, Venturi – Leakage-Resilient StorageSCN /09/2010 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