WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Longer Randomly Blinded RSA Keys may be Weaker than Shorter Ones Colin D. Walter

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 2/16 Outline Aims History Key Blinding Counter-Measure Side Channel Leakage Model Best Fit Metric Phases 1 & 2 Computational Feasibility Conclusion

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 3/16 Aim The aim is to investigate the implementation strength of public key cryptography assuming: –standard counter-measures –imperfect side-channel leakage Do counter-measures interact to weaken a system? Are published attacks impossible in real life? Are longer keys more secure?

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 4/16 History Refs to side channel leakage in patents ( USPTO 1978): see Abstract to US Patent Kocher et al (C RYPTO 1996, 1999): Timing and Power Attacks on smart cards – the concepts. Coron (CHES 1999): Lists three standard randomising counter-measures for ECC. Fouque et al (CHES 2006): Attack on Blinded RSA keys. Here (WISA 2007): Extension of Fouque to an imperfect side channel.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 5/16 4-ary Exponentiation Inputs: key D = (d n-1 d n-2 …d 1 d 0 ) 4 ; modulus M; ciphertext C. Precompute C d mod M for each digit value d. P  1 ; For i  0 to n  1 do Begin P  (P 2 ) 2 mod M ; if d i ≠0 then P  C d i ×P mod M ; End ; Output: plaintext P = C D mod M The side channel may distinguish squares from multiplications but not mult ns by different values of d.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 6/16 The Leakage Model Standard counter-measures are in place. There is a (weak) side channel which gives a probability that a square or multiply occurs. I/O of the exponentiation is unknown. The adversary knows the algorithms. The public parameters M and E are available. The correctness of secret key D can be checked. Only a realistic number of side channel traces are allowed.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 7/16 Assumed Counter-Measures An m-ary or sliding windows algorithm is used to prevent attackers from distinguishing exponent digits. Input text is blinded to prevent attacker from modelling the identical leakage. The secret key D is blinded on each re-use to prevent the adversary improving the signal to noise ratio with repeated use of the same D. This means D is replaced by D i = D+r i φ(N) for 20- to 32-bit random r i.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 8/16 Initial Calculations The top half bits of φ(N) and N are identical, so known. DE = 1+kφ(N) where k < E, D < φ(N). D i = (1+(k+r i E))φ(N) / E ≈ (1+(k+r i E))N / E k+r i E is typically a 32- to 48- unknown number. Use the leakage from the first half of trace for D i to guess k+r i E. Information theoretically, a leakage of 1 bit per 32 key bits means we need 32×32 to 32×48 bits in the top half to guess k+r i E successfully, i.e. keys of 2048 to 3072 bits – or longer keys.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 9/16 Best Fit Metric & Phase 1 Let p j = prob that j th operation of trace is a squaring. Let D' be a guess at the key used for the trace tr. Putd j = +1 if j th operation of D' is a squaring, d j = –1 if j th operation is a multiplication Let μ m (tr,D') = Σ 0≤j<m d j (p j –½) This measures how well D' matches the leakage tr over the first m operations (for the top half of N). For each side channel trace tr, choose k+rE, and hence D', to maximise this.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 10/16 Does it Work? Is the best guess at k+rE the correct one? The leakage is weak, so there are better incorrect guesses. The correct guess lies in the top fraction of best guesses. The fraction containing the correct guess is (almost) independent of how many bits need guessing. The fraction containing the correct guess gets rapidly smaller as key length increases, so the search space is smaller. k+rE becomes known if the key is long enough.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 11/16 Phase 2: Recovering φ(N) We now assume k+r i E is known for D i = (1+(k+r i E))φ(N) / E Phase 2: Choose bits of φ(N) to maximise the metric μ m = Σ j μ m (tr j,D j ) Bits are chosen one by one from most to least significant. (m picked to measure only contributions of chosen bits.) Use several bits lookahead to allow for the influence of carries and bit recoding in the exponentiation algorithm. Are the bits choices correct? What influences their correctness?

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 12/16 Bit Accuracy Accuracy of each bit determination depends on: –Number of available traces –Level of leakage –Number of lookahead bits The algorithm is self-correcting – bit errors are isolated. For 2048-bit key, 10 lookahead bits, 100 traces,… the prob of correct bit was This is 1 error in 2000, so half of a set of keys will be recovered correctly.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 13/16 Computational Feasibility Phase 1 – recovering the k+r i E : O( REt log(RE) ) leaked bit operations to process whereR = # choices for r (2 16 to 2 32 ) E = public exponent (e.g ) t = # traces needed in phase 2. It is highly parallelisable, with low space requirements. Phase 2 – recovering φ(N) : much less work than phase 1. The attack is computationally feasible if RE is not too large

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 14/16 Improved Counter-Measures Bits are determined more accurately for longer keys. Other work suggests fewer bit errors for longer keys. Different exponentiation & modular multiplication algorithms only affect the level of leakage per key bit. Computational feasibility decreases with more blinding: it is essentially proportional to this. So: If possible, choose less leaky HW & algorithms Choose enough blinding & large enough public key to make it computationally infeasible to check every value of k+rE.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 15/16 Conclusions Many essential and first-class SW counter- measures can be inadequate on their own. Information theoretic expectations should be treated as realisable. Imprecise leakage is useful to an adversary. Longer keys tend to be weaker for a fixed level of randomisation counter-measures. Randomisation needs to be scaled up for longer keys.

WISA 2007 Jeju Island, Korea, 27th – 29th Aug 2007 Colin Walter, Comodo 16/16