Protecting Circuits from Leakage the computationally bounded and noisy cases Sebastian Faust Eurocrypt 2010, Nice Joint work with KU Leuven Tal Rabin Leo Reyzin Eran Tromer Vinod Vaikuntanathan IBM Research Boston University MIT IBM Research

2 Theory vs. Reality K XY Standard security analysis: Controls inputs/outputs, e.g. CPA Computation completely unknown K XY Attacking the implementation: input key output Adversary obtains leakage Use physical observations: e.g. power consumption, timing,… Completely break crypto schemes! implement

3 Countermeasures? Hot topic: ISW03, MR04, DP08, P09, AGV09, ADW09, KV09, DKL09,… Many more citations in the paper We may try to defeat specific attacks, e.g. power analysis, timing attacks,… Or we can try to go for a broad class! Most other work: Security of specific scheme This work: How to securely implement any scheme?

4 How to extend the standard model? K Modeled by a leakage function f Adversary obtains leakage f(state) Real-life leakages don’t leak complete key Power consumption: e.g. f(st) ≈ Hamming weight of wires in circuit Arbitrary leakage function? No…  e.g.: f(st) = K means no security Some restrictions are necessary XY Probing: f(st) = some bits of state

5 Restrictions: Bounded leakage Bounded total leakage K … f(st) K K e.g. used to model cold boot attacks Continuous leakage Amount of leakage << length of key K Bounded per observation, but: total leakage >> |K|

6 Restrictions: Bounded leakage Bounded total leakage K … f(st) K1K1 f(st 1 ) KnKn f(st n ) Bounded per observation, but: total leakage >> |K| e.g. power analysis Continuous leakage requires refreshing of key: K  K i e.g. used to model cold boot attacks Amount of leakage << length of key K

7 Restrictions: Local vs. Global Local leakage Global leakage e.g. probing: leakage is oblivious to most of the computation e.g. power analysis: power consumption depends on all computation

8 Restrictions: Weak/Noisy vs. PPT (requires bounded leakage) Weak or Noisy leakage K f є L = {computationally weak functions} Leakage can be described by “simple” aggregated function Is this reasonable? Yes! E.g. probing, power consumption… f(st) weak

9 Weak or Noisy leakage K f(st) K f є L = {Noisy functions}: Leakage is a noisy function of the secret key Restrictions: Weak/Noisy vs. PPT (requires bounded leakage) weak noisy

10 Weak or Noisy leakage K f(st) K Powerful! Restrictions: Weak/Noisy vs. PPT (requires bounded leakage) weak noisy

11 Weak or Noisy leakage K f(st) K Polynomial-time leakage K f(st) f є L = {PPT functions} Leakage is arbitrary PPT function Restrictions: Weak/Noisy vs. PPT (requires bounded leakage) Powerful! weak noisy PPT Probably stronger than leakage in reality

12 Q: Is there computation that can be protected against global, continuous, but weak or noisy leakage? A challenge… A: Any Computation! If we have a simple leak-free component Reduce some complex computation to very simple shielded component [MR04]

13 Earlier work: Ishai, Sahai, Wagner ‘03 Main drawback: No proof of security for global functions, e.g. Hamming Weight Q: Is there computation that can be protected against global, continuous, but weak or noisy leakage? A: Any Computation! local probing

14 1.Circuit Compilers 2.Our Result Rest of this talk…

15 Circuit compiler: C‘ with K‘ has same functionality as C with K K XY C YX K’K’ C’C’ Circuit compilers  Is resistant to continuous leakages from some large function class L (Security Definition by Simulation) Input: description of arbitrary circuit C and key K Functionality preserving:  Uses same gates as C Transformed circuit C‘: + leak-free gate (later more) Output: description of transformed circuit C‘ and key K‘

16 Our Result Theorem 1: A compiler that makes any circuit resilient to computationally weak leakages. Set of leakage functions L can be large, but they cannot compute a certain linear function One example: AC 0 = Const depth and poly size circuits of Λ or V gates. What does this mean? L = AC 0  L cannot compute linear function parity!

17 Our Result Theorem 2: A compiler that makes any circuit resilient to noisy leakages. What does this mean? Leakages are {wire i + noise ƞ i }  ƞ i = 0, with probability 1-p  ƞ i = 1, with probability p Both compilers assume leak-free gates in transformed circuit!

18 Leak-free gates  Leak-free processor: oblivious RAM (1) Many previous usages in leakage-resilience:  Leak-free memory: “only computation leaks”, one-time programs (2) Our leak-free gate is: Small & simple: Much smaller than size of Stateless: No secrets are stored Computation independent: No inputs For Theorem 1: random t-bit string (b 1,…,b t ) with parity 0 (1) [G89,GoldOstr95], (2) [MicRey04], [DziPie08], [GoldKalRoth08] For Theorem 2: above properties, but a bit more complicated

19 Compiler: high-level C M ● + ● ● + C ● M Circuit topology is preserved 1. Memory:Encoded memory Bit b e.g. “Parity” encoding”: uniform t-bit string (b 1 …b t ) with parity b

20 Compiler: high-level C M ● + ● ● + C ● M 2. Each wire w Wire bundle that carries the encoding of w, e.g. a t-bit string with parity w

21 Two key properties of our encoding Let (a 1,…a t ) and (b 1,…b t ) be bit strings with parity 0 and 1 (resp.) f(a 1,…a t ) or f(b 1,…b t ) 2. Noise indistinguishable [XOR Lemma] (a 1 + ƞ 1,+…a t + ƞ t ) or (b 1 + ƞ 1,…b t + ƞ t ) ?? in AC 0 Flip each bit with prob. p 1. L=AC 0 indistinguishable [Has86,DubrovIshai06] ??

22 Compiler: high-level C M ● + ● ● + C ● M 3. Gates Gadgets: built from normal gates and leak-free gates and operate on encodings Properties of the encoding do not suffice for security!

23 Conclusion Two circuit compilers …. global leakages : i.e. leakage can depend on all the computation, all intermediate results,… continuous leakage : the amount of leakage over time is unbounded  eliminate leak-free gates compile any circuit Open problems:  For security parameter t: blow-up ≈ t 2

24 Thank you!

25 Simulation: Real: indistinguishable L-Security: Simulation [ISW03] Intuition: Adversary learns no more than by input/output access X1f1 ∈LX1f1 ∈L Y 1 f 1 (wires 1 ) Simulation: K1K1 X1X1 Y1Y1 … K’1K’1 Xnfn ∈LXnfn ∈L Y n f n (wires n ) K’nK’n … refresh key Can e.g. be some low complexity function class