[bitcoin-dev] Mitigating Differential Power Analysis in BIP-340

[Pieter Wuille]

On Tuesday, March 24, 2020 6:00 AM, Lloyd Fournier via bitcoin-dev <bitcoin-dev at lists.linuxfoundation.org> wrote:

> Hi List,
>
> I felt this topic deserved it's own thread but it follows on from the mailing list post [2] announcing a new PR [1] to change BIP-340 in several ways, including adding random auxiliary data into the nonce derivation function. Rather than hashing the randomness with the secret key and message etc, the randomness is hashed then XOR'd (^) with the secret key and the result is hashed like so to determine the secret nonce k:
>
> (1) k = H_derive( sec_key ^ H_aux(rand) || pub_key_x || message)
>
> The claim made in the mailing list post is that this is more secure against "differential power analysis" (DPA) attacks than just doing the simpler and more efficient:
>
> (2) k = H_derive(sec_key || rand || pub_key_x || message)
>
> The TL;DR here is that I don't think this is the case.

Hi Lloyd,

Thank you for looking into this. I very much agree we haven't given nearly enough justification for the use of a non-standard scheme here.

I'll try to summarize the discussion we had that led to this choice, but most of it is on https://github.com/sipa/bips/issues/195 if you want the details.

Let me first try to address what I think you're overlooking: in a BIP32/Taproot like scenario, the private key that goes into the signing algorithm functions as *both* secret and known to the attacker. That is to say, there is a master secret s, and signing key x that goes into the hash is x=s+a (mod n) for some value a that the attacker knows, and can modify (he cannot control it directly, but he may be able to grind it to have a structure he likes). I believe that means that feeding x to a hash directly itself is already a problem, regardless of what else goes into the hash - interactions between bits inside the hash operation that all come from x itself can leak bit-level information of x.  XORing (or any other simple mix operation that does not expose bit-level information) into the private key before giving it to a hash function seems like it would address this.

That said, all these DPA issues are very hard to reason about, as it's hard to find a realistic attack model that both (a) leaks some information but (b) doesn't obviously leak the entire key immediately. In the reasoning above I assumed an attacker who can observe word-level Hamming weight aggregates of all variables in the algorithm (which seems to match what one of the papers observed), but not bit level information. It also assumes that somehow the computation of x itself is immune from leaks (something you pointed out in a previous e-mail, I noticed).

So really, all of this is trying to choose one alternative among a set of (when ignoring DPA) nearly equally good constructions. Note that randomness is useful for protection against fault attacks, but for that purpose it doesn't matter where in the hash it goes, or even that it's particularly strong randomness (a counter would also work). There are a number of other concerns we discussed in the linked thread:
* Efficiency (how many SHA256 transformations, including the ability for some to be precomputed)
* The risk that the randomness added is correlated with the private key in a way that cancels things out when they're naively XORed together.
* The risk of having a midstate in the hash function leak (without leaking the actual private key, but enough to predict nonces).
* The issue with public keys that are input to the signing algorithm which come directly from an attacker (which is the reason why pubkey goes into the nonce function too).

The solution we came up with (H(priv XOR H(aux) || pub || msg)) is the only that ticks most of the boxes - but a different prioritization may certainly lead to a different conclusion.

I'm happy for any input you may have here. In particular, the recent discussions around the interactions between anti-covert channel protection, randomness, and the ability to spot check hardware devices may mean we should revise the advice to consider not adding randomness unless such a anti-covert channel scheme is used.

Cheers,

--
Pieter