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

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 primitives are black-box Interaction depends on the security notion hidden visible

Cryptography in the ideal world > 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,…

Ideal world: State of the art (comp/nal) hardness assumptions primitives reductions Provable Security: Any (efficient) adversary that “breaks” primitive Π, implies an efficient algorithm against a problem considered to be hard Central Concept: Provable security Unless a major algorithmic breakthrough happens, things look good

Cryptography in the real world 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 Primitives vulnerable to such attacks All primitives actually implemented on a physical device Present motivation in a better way

Side-Channel Attacks …from Side Channel Attacks Database definition: an attack based on extra information gained by the physical implementation of a protocol …from Side Channel Attacks Database 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 Power Analysis: power consumption measurements reveal sequence of instructions executed power analysis: If the execution path depends on processed data (secret key etc) Common feature: side-channel info is leaked as a result of computation Fault Induction: secret information revealed as a result of execution of erroneous operation

Cold-boot (memory) attacks [HSH+, USENIX Security 08] standard DRAM cells refresh interval: ~ms in practice cells retain content for much longer ~ 30o C : complete loss only after tens of seconds At low temperatures (-50o C) contents might be retained for minutes or hours

Memory (cold-boot) attacks - bits decay to known “ground states” following predictable patterns 5 sec 30 sec 5 min 1 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 RC5 is a block cipher Other types of SC attacks; Electromagnetic radiation, heat, sound, cache

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 Timing & Power analysis Fault Induction Cold boot attacks

HW/Implementation-level countermeasures Ad-hoc: target specific side channel attacks 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. 2nd const. nth const. fix fix . . . fix Side-chan. attack 2 Side-chan. attack 1

Idealizing the Real world assumptions primitives reductions Central Concept: Provable security Q: Can we construct primitives that are provable secure in the presence of side-channel information ?

Rest of talk Leakage Resilient Cryptography Modeling Leakage Formally Continuous Leakage Bounded (Memory) Leakage Auxiliary Input Security Definitions / Overview of techniques Example 1: LR stream cipher Example 2: LR password authentication  Conclusions Has the picture changed ? Future Directions

Leakage-Resilient Cryptography (LRC) Approach: It is impossible to prevent side-channel attacks predict all possible ways side-channel information can be collected by an attacker 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 [Micali, Reyzin 04: Physically Observable Cryptography] separation of functionality and implementation Physically implemented primitive Π Abstract functionality Associated leakage We don’t care about how the information is collected (we don’t care about the specific side-channel attack) This allows to model any side-channel information abstractly (the leakage) Here we present a relaxation of the Micali Reyzin model Leakage is hardware/implementation specific A: implementation independent L: HW / implementation specific Π= (A, L)

Modeling Side-Channel information leakage oracle Adversary has access to a leakage oracle that models all sources of side channel information

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**

Modeling Leakage for Cryptography Universal Requirement 1 [MR04, Axiom 5] Leaked information is efficiently computable Universal Requirement 2 Given it shouldn’t be trivial to recover Si [MR04, Fundamental Physical Assumption] physical implementations of primitives s.t. leakage does not reveal the entire secret state

(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 :

(Partial) Taxonomy of Derived Models Domain Range Additional restrictions Practical Scenario continuous leakage accessed state unbound. bounded per invocation (round) leakage from computation Memory Leakage entire sk bounded cold-boot attacks Auxiliary Input comp/ly hard to find sk, given f(sk) both Each model might or might not be appropriate depending on the actual practical setting

Leakage Models: Comparison/Discussion 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 practice

Outline Leakage Resilient Cryptography Modeling Leakage Formally Continuous Leakage Bounded (Memory) Leakage Auxiliary Input Security Definitions / Overview of techniques Example 1: LR stream cipher Example 2: LR password authentication  Conclusions Has the picture changed ? Future Directions

New Security Definitions Pseudorandom Functions (PRFs) (without leakage) REAL IDEAL PRFs are central primitive in cryptography building block for more advanced primitives Security of PRFs: Given pairs (for x’s of his choice) no efficient adversary can tell which world he interacts with.

New Security Definitions Pseudorandom Functions (PRFs) (with leakage) REAL IDEAL PRFs are central primitive in cryptography building block for more advanced primitives Trivial Attack (single query, 1 bit of leakage) Adversary encodes x in f If output “REAL” else “IDEAL”

Security in the presence of leakage Weak PRFs REAL IDEAL Security of weak PRFs: Given pairs (for random x’s) no efficient adversary can tell which world he interacts with. PRFs are central primitive in cryptography building block for more advanced primitives Q: Can we prove leakage resilience of wPRFs? A: Yes (but security degrades exponentially in the leakage)

Weak PRFs are leakage resilient. [Pietrzak 09: A Leakage-Resilient Mode of Operation] Define F: ε-wPRF: for all PPT adversaries A Theorem (informal) [Pietrzak 09] Assume such that . If ε-wPRF for uniform keys, then δ-wPRF for keys K drawn according to , where

Towards constructing a LR- stream cipher Why ``only computation leaks information” ? Π: stateful primitive State update (round i): Upper bound on |Si| : M Leakage function takes the whole state as input Attack (1 bit of leakage per round) At round i: adversary queries leakage function After M queries, adversary gets SM scheme completely broken!!

Stream Cipher in the continuous leakage model F: weak PRF Init ; ; Update uses part of the state State Computation

Stream Cipher in the continuous leakage model F: weak PRF State Update uses part of the state Computation

Leakage-resilient Stream Cipher Update rule: 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)

Outline Leakage Resilient Cryptography Modeling Leakage Formally Continuous Leakage Bounded (Memory) Leakage Auxiliary Input Security Definitions / Overview of techniques Example 1: LR stream cipher Example 2: LR password authentication  Conclusions Has the picture changed ? Future Directions

Bounded leakage model (toy example) Password Authentication (w/o leakage) secure channel client server (g, y=g(x)) (sk=x) insecure storage secure storage Problem: Client wants to authenticate itself to the server Solution: Use One-Way Functions (OWF) . g is OWF easy to compute hard to invert for all efficient A

Password Authentication (with leakage) server secure channel client (g, y=g(x)) (sk=x) f(x) insecure storage Problem: Attacker gets in addition l bits of leakage Solution: Use l-Leakage-Resilient-OWF . g is l-LR-OWF - … hard to invert. For all efficient A leakage

Password Authentication (with leakage) Observation: OWFs imply l-LR-OWFs with exponential (in l) loss in security Proof Idea Inverter (I) with Leakage Advantage: ε Inverter Input: (g,y=g(x)) x’

Password Authentication (with leakage) server secure channel client (g, y=g(x)) (sk=x) f(x) insecure storage 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?

Password Authentication (with leakage) 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

Password Authentication (with leakage) [Theorem (informal)]: If SPRF with then h is l-LR-OWF with Proof Idea Inverter (I) with Leakage Advantage: ε A Input: (x, h) Output: x’ if x’

Password Authentication (with leakage) Inverter (I) with Leakage Advantage: ε A Input: (x, h) Output: x’ if x’ But and Even given y= h(x) AND leakage = f(x) there exist on average preimages for y

Security definitions for PKE Standard ind/lity under chosen plaintext attacks (IND-CPA) 1.(sk,pk)  KeyGen 2.(m0,m1,state)  A1(pk) 3.c* Enc(pk,mb) 4.b’ A2(c*,state) 5.if (b’=b) return 1, else return 0

Security definitions for PKE ind/lity under chosen plaintext attacks with leakage (l-leakage-CPA) 1.(sk,pk)  KeyGen 2.(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 Adversary can query any (PT) leakage function f No leakage allowed after the creation of the challenge c*

LRC: Primitives (overview) LR PKE (gen. mod) LR- Stream Cipher Simplified LR- SC Continuous Leakage LR st/ful sig/res 2008 2009 2010 I I I CPA/CCA SKE sign/res gen. const. CPA-PKE, IBE Bounded Leakage & Auxiliary Input CPA PKE CPA / CCA-PKE, generic constructions improved leakage

Outline Leakage Resilient Cryptography Modeling Leakage Formally Continuous Leakage Bounded (Memory) Leakage Auxiliary Input Security Definitions / Overview of techniques Example 1: LR stream cipher Example 2: LR password authentication  Conclusions Has the picture changed ? Future Directions

LRC in designing process 1st const. Side-chan. attack 1 fix 2nd const. attack 2 . . . nth const. Before Consider classes of leakage functions that are as rich as possible (and for which there is a proof) Construct primitives that are provably leakage resilient with respect to Design hardware/ Write code whose leakage is faithfully modeled by Now

LRC in designing process 1st const. Side-chan. attack 1 fix 2nd const. attack 2 . . . nth const. Before Consider classes of leakage functions that are as rich as possible (and for which there is a proof) Construct primitives that are provably leakage resilient with respect to Design hardware/ Write code whose leakage is faithfully modeled by Now

How LRC changed the picture ? 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 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)

More on Modeling Gaps What does it mean in practice for a smart card, to leak at most k bits per encryption? 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)

Outline Leakage Resilient Cryptography Modeling Leakage Formally Continuous Leakage Bounded (Memory) Leakage Auxiliary Input Security Definitions / Overview of techniques Example 1: LR stream cipher Example 2: LR password authentication  Conclusions Has the picture changed ? Future Directions

Look to the future 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 rob/ness of assumptions vs rob/ness of costructions [Goldwasser, Kalai, Peikert, Vaikuntanathan 10: Robustness of LWE Allowing (restricted) access to the leakage oracle after the creation of the challenge. - How does this affect security guarantees?