Leakage- Resilient Cryptography: Recent Advances Research Exam May 20, 2010 Petros Mol 1

Leakage- Resilient Cryptography: Recent Advances

Research Exam

May 20, 2010

Petros Mol

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Cryptography in the ideal world KeyGen() sk $ SK pk f return pk

Encrypt(m) r c … return c

Decrypt($%^#) m f ($%^#) return m

• Primitives as mathematical objects

• Restricted interaction between primitive and user/attacker

• Secret information completely hidden from the adversary

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hidden

visible

• > 30 years of extensive research

• Primitives and notions for all tastes…– CPA/CCA encryption, UFCMA signature schemes – Collision resistant hash functions– NIZK protocols with honest but curious verifiers etc.– fully homomorphic encryption

• … and from many hardness assumptions– Number theoretic: Factoring, DLP, DDH,QR,...– Lattice based: GapSVP, uSVP,…

Cryptography in the ideal world

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• Provable Security: Any (efficient) adversary that “breaks” primitive Π, implies an efficient algorithm against a problem considered to be hard

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Ideal world: State of the art

primitives

(comp/nal) hardness

assumptions

reductions

Unless a major algorithmic breakthrough happens,

things look good

Cryptography in the real world

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• Protocols are executed in actual physical devices

• Adversary can attack the implementation of a protocol Interaction between adversary and primitive much less restricted

• secret information no longer perfectly hidden from the adversary

Real World situation • Systems vulnerable to such attacks

– Smart cards– General/specific purpose hardware– Software implementations

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• Primitives vulnerable to such attacks– All primitives actually implemented on a physical device

Side-Channel Attacks

definition: an attack based on extra information gained by the physical implementation of a protocol

…from Side Channel Attacks Database

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The decade 1999-2008• over 600 publications• over 50 patents• 19 PhD theses

Side-Channel Attacks (from computation)

• Timing Attacks: secret key information leaked as a result of

of measuring execution time

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• Power Analysis:

power consumption measurements reveal

sequence of instructions executed

• Fault Induction:

secret information revealed as a result of

execution of erroneous operation

Cold-boot (memory) attacks

•standard DRAM cells refresh interval: ~ms

•in practice cells retain content for much longer

•~ 30o C : complete loss only after tens of seconds

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At low temperatures (-50o C) contents might be retained for minutes or hours

[HSH+, USENIX Security 08]

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Memory (cold-boot) attacks- bits decay to known “ground states” following predictable patterns

5 sec 30 sec 5 min1 min

A (very partial) list of attacks

Type of Attack Protocol/software attacked

Timing RSA, El Gamal, DSA, AES, DES, RC5

Power Analysis RSA, DES, AES

Fault Induction RSA, ECC, DES, RC5, Blowfish, identification schemes

Memory Attacks AES, DES, RSA, FileVault, TrueCrypt, BitLocker

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HW/Implementation-level countermeasures

• “Obfuscate” computation by decreasing the Signal-to-Noise Ratio– Perform random operations– Randomize the execution sequence– Decorrelate input from intermediate values

• Consistency checks– Run (parts of the) algorithm twice and check if

results coincide

• Reduce information leaked about the key– Encrypt key in memory – Avoid storing redundant info about the key

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Timing & Power analysis

FaultInduction

Cold bootattacks

• Ad-hoc: target specific side channel attacks

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HW/Implementation-level countermeasures

• Expensive:

– hardware level masking very costly

– train programmers to write and maintain code that minimizes leakage

– incur significant slowdowns in the running time

• Insufficient: Can only decrease the efficiency of the attack but not completely eliminate it

Design process in practice

1st const.

Side-chan.attack 1

fix2nd const.

Side-chan.attack 2

fix

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. . . nth const.fix

Q: Can we construct primitives that are provable secure in the presence of side-channel information ?

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Idealizing the Real world

primitives assumptionsreductions

Rest of talk• Leakage Resilient Cryptography

o Modeling Leakage Formallyo Continuous Leakage

o Bounded (Memory) Leakage

o Auxiliary Input

o Security Definitions / Overview of techniqueso Example 1: LR stream cipher

o Example 2: LR password authentication

• Conclusions

o Has the picture changed ?

o Future Directions16

• Approach: It is impossible to– prevent side-channel attacks– predict all possible ways side-channel information can

be collected by an attacker

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Leakage-Resilient Cryptography (LRC)

• Face the truth: Devices are not black-boxes• No restriction on how the adversary obtains side-

channel information• only on what information he can obtain.

• Ultimate Goal: Develop Cryptography that is secure against wide classes of leakage

Modeling Side-Channel information

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[Micali, Reyzin 04: Physically Observable Cryptography]

separation of functionality and implementation

Physically implementedprimitive Π

Associated leakage

Abstract functionalit

y

Π= (A, L) A: implementation independentL: HW / implementation specific

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• Adversary has access to a leakage oracle that models all sources of side channel information

Modeling Side-Channel information

leakage oracle

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Micali- Reyzin model (somewhat relaxed)

at i-th execution round

Randomness (environmental noise)

Adversarially chosen (measurement)

Secret (internal) state of the protocol

Compactly (randomized f, Mi encoded in f)

** For stateless primitives Si might simply the secret key**

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Modeling Leakage for Cryptography

[MR04, Axiom 5]

Leaked information is efficiently computable

[MR04, Fundamental Physical Assumption]

physical implementations of primitives s.t. leakage does not reveal the entire secret state

Universal Requirement 1

Universal Requirement 2Given

it shouldn’t be trivial to recover Si

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(Partial) Taxonomy of Derived Models

• Type (family) of f• restricted• arbitrary This talk

• Domain (effective input) of f• entire secret state• active state [MR04, Axiom 1]

Only Computation leaks Information

• Range of f ( )

• bounded :

• unbounded :

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Model Domain Range Additional restrictions

Practical Scenario

continuousleakage

accessed state

unbound.bounded per invocation (round)

leakage from computation

Memory Leakage

entire sk boundedcold-boot attacks

Auxiliary Input

entire sk unbound.comp/ly hard to find sk, given f(sk)

both

(Partial) Taxonomy of Derived Models

Each model might or might not be appropriate depending on the actual practical setting

• continuous leakage– fails to capture memory attacks– proofs rely on the fact that parts of the key don’t participate

in the computation

• bounded (memory) leakage– captures scenarios where attacker has one-shot– are not very realistic in settings where many computations

take place

• auxiliary input– somehow combines best of both worlds– requires exponential hardness of the leakage function

hard to interpret what that means in practice24

Leakage Models: Comparison/Discussion

Outline• Leakage Resilient Cryptography



o Auxiliary Input



• Conclusions



• Security of PRFs:

Given pairs (for x’s of his choice) no

efficient adversary can tell which world he interacts with.

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New Security DefinitionsPseudorandom Functions (PRFs) (without leakage)

REAL

IDEAL

• Trivial Attack (single query, 1 bit of leakage)

Adversary encodes x in f

If output “REAL” else “IDEAL”

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New Security Definitions

Pseudorandom Functions (PRFs) (with leakage)

REAL

IDEAL

• Security of weak PRFs:

Given pairs (for random x’s) no efficient adversary can tell

which world he interacts with.

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Security in the presence of leakage

REAL

IDEAL

Weak PRFs

Q: Can we prove leakage resilience of wPRFs?

A: Yes (but security degrades exponentially in the leakage)

• Define

• F: ε-wPRF: for all PPT adversaries A

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[Pietrzak 09: A Leakage-Resilient Mode of Operation]

Weak PRFs are leakage resilient.

Theorem (informal) [Pietrzak 09]

Assume such that . If ε-wPRF for uniform keys, then δ-wPRF for keys K drawn according to , where

• Why ``only computation leaks information” ?

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Towards constructing a LR- stream cipher

• Π: stateful primitive

• State update (round i):

• Upper bound on |Si| : M

Attack (1 bit of leakage per round)

At round i: adversary queries leakage function

After M queries, adversary gets SM scheme completely broken!!

Leakage function takes the whole state as input

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Stream Cipher in the continuous leakage model

State

Computation

F: weak PRF

Update uses part of the state

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Leakage-resilient Stream Cipher

• K2i , K2i+1 evolve “independently”

• Intuition: If F is instantiated with a weak PRF and K, X have high min-entropy, then the output F(K,X) has high (pseudo-)entropy, even given the leakage f(K ,X)

Update rule:




o Auxiliary Input



• Conclusions



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Bounded leakage model (toy example)Password Authentication (w/o leakage)

secure channel clientserver

insecure storage

Problem: Client wants to authenticate itself to the server

(g, y=g(x)) (sk=x)

• Solution: Use One-Way Functions (OWF) . g is OWF- easy to compute - hard to invert

for all efficient A

secure storage

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Password Authentication (with leakage)


Problem: Attacker gets in addition l bits of leakage

(g, y=g(x)) (sk=x)

• Solution: Use l-Leakage-Resilient-OWF . g is l-LR-OWF

- … hard to invert. For all efficient A

insecure storagef(x)

leakage

• Observation: OWFs imply l-LR-OWFs with exponential (in l) loss in security

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Inverter

Input: (g,y=g(x)) Inverter (I) with Leakage

Advantage: ε

Proof Idea

x’

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(g, y=g(x)) (sk=x)

insecure storagef(x)

Using OWF g, the authentication protocol can resist l bits of leakage assuming g is - - hard to invert.

Q: Can we avoid this exponential loss in security?

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Workaround: Second Preimage Resistant Functions

is SPRF if:- easy to compute- given x, h hard to find such that h(x’)=h(x)

Known constructions of SPRFs from number-theoretic assumptions

• [Theorem (informal)]: If SPRF with then h is l-LR-OWF with

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AInput: (x, h) Inverter (I)

with Leakage

Advantage: ε

Proof Idea

x’Output: x’ if


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AInput: (x, h)

Inverter (I) with

Leakage

Advantage: εx’

Output: x’ if


Even given y= h(x) AND leakage = f(x) there exist on average preimages for y

But and

Security definitions for PKE

• Standard ind/lity under chosen plaintext attacks

(IND-CPA)

1.(sk,pk) KeyGen2.(m0,m1,state) A1(pk)

3.c* Enc(pk,mb)

4.b’ A2(c*,state)5.if (b’=b) return 1, else return 0

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• ind/lity under chosen plaintext attacks with leakage

(l-leakage-CPA)

1.(sk,pk) KeyGen2.(m0,m1,state) A1(pk,f(sk,pk))

3.c* Enc(pk,mb)

4.b’ A2(c*,state)5.if (b’=b) return 1, else return 0

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Adversary can query any (PT) leakage function f

No leakage allowed after the creation of the challenge c*

Security definitions for PKE

LRC: Primitives (overview)

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Con

tinuo

us

Leak

age

Bou

nded

Lea

kage

& A

uxili

ary

Inpu

t

2009 20102008I I I

LR- StreamCipher

SimplifiedLR- SC

LR st/fulsig/res

CPA-PKE,IBE

CPA / CCA-PKE,generic constructionsimproved leakage

CPA/CCA SKE

sign/resgen. const.

CPA PKE

LR PKE(gen. mod)




o Auxiliary Input



• Conclusions



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LRC in designing process

1st const.

Side-chan.attack 1

fix 2nd const.

Side-chan.attack 2

fix . . . nth const.fix

Before

Now

1. Consider classes of leakage functions that are as rich as possible (and for which there is a proof)

2. Construct primitives that are provably leakage resilient with respect to

3. Design hardware/ Write code whose leakage is faithfully modeled by

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LRC in designing process

1st const.

Side-chan.attack 1

fix 2nd const.

Side-chan.attack 2

fix . . . nth const.fix

Before

Now

1. Consider classes of leakage functions that are as rich as possible (and for which there is a proof)

2. Construct primitives that are provably leakage resilient with respect to

3. Design hardware/ Write code whose leakage is faithfully modeled by

How LRC changed the picture ?

Pros• Brought Side-Channel attacks (closer) to theoreticians’

attention and improved their understanding on the subject. • Set basis for formal treatment of a serious practical

problem• Can potentially bring Cryptography closer to hardware

designers : clearer engineering goals (though not easy ones)48

Cons• Modeling gaps (more on that in the next slide)

• constructions and models more tailored to proof techniques rather than to practical needs.

• some models unintuitive and hard to interpret in practice. • Performance implications:

• for same level of security size of secret keys might get much longer more storage, slower operations

More on Modeling Gaps

• What does it mean in practice for a smart card, to leak at most k bits per encryption?

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• Modeling leakage as an uninvertible function makes proofs go through but doesn’t make engineers’ life any easier.

• Adversary has no access to leakage oracle once given the challenge. Does this make sense?

(byproduct of considering arbitrary leakage functions)




o Auxiliary Input



• Conclusions



Look to the future

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• Construct more Leakage Resilient primitives- … that resist more leakage- … from more hardness assumptions

Flexible LR constructions: - Design independent of leakage- Security degrades with leakage but in the

absence of leakage primitive as secure as a leakage-free ones

[Goldwasser, Kalai, Peikert, Vaikuntanathan 10: Robustness of LWE

rob/ness of assumptions vs rob/ness of costructions

• Allowing (restricted) access to the leakage oracle after the creation of the challenge.

- How does this affect security guarantees?

Documents

Leakage- Resilient Cryptography: Recent Advances Research Exam May 20, 2010 Petros Mol 1