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

From: Lloyd Fournier <[email-masked]>
Date: Wed, 25 Mar 2020 00:00:45 +1100

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.

There was no citation for this claim, so I did some digging and found two
papers that seemed like they might be the origin of the idea [3,4] (I had
no idea about these attacks before). A relatively easy to understand
explanation of DPA attacks against is in [3]:

The fundamental principle behind all DPA attacks is that at some point in
> an algorithm’s execution, a function f exists that combines a fixed secret
> value with a variable which an attacker knows. An attacker can form
> hypotheses about the fixed secret value, and compute the corresponding
> output values of f by using an appropriate leakage model, such as the
> Hamming Distance model. The attacker can then use the acquired power
> consumption traces to verify her hypotheses, by partitioning the
> acquisitions or using Pearson’s correlation coefficient. These side-channel
> analysis attacks are aided by knowledge of details of the implementation
> under attack. Moreover, these attacks can be used to validate hypotheses
> about implementation details. In subsequent sections, these side-channel
> analysis attacks are referred to as DPA attacks.

For example, in the original BIP-340 proposal the nonce derivation was
vulnerable to DPA attacks as it was derived simply by doing
H_derive(sec_key || message). Since, the message is known to the attacker
and variable (even if it is not controller by her), the SHA256 compression
function run on (sec_key || message) may leak information about sec_key. It
is crucial to understand that just hashing sec_key before passing it into
the H_derive does *not* fix the problem. Although the attacker would be
unable to find sec_key directly, they could learn H(sec_key) and with that
know all the inputs into H_derive and therefore get the value of the secret
nonce k and from there extract the secret key from any signature made with
this nonce derivation algorithm.

The key thing I want to argue with this post is that there is no advantage
of (1) over (2) against DPA attacks, at least not given my understanding of
these papers. The way the attack in [3] works is by assuming that
operations in the compression function leak the "hamming distance" [5] (HD)
between the static secret thing that is being combined with the variable
public thing. In practice the attack involves many particulars about SHA256
but that is, at a high level, the right way to simplify it I think. The way
the paper suggests to fix the problem is to mask the secret data with
secret randomness before each sensitive operation and then strip off the
secret randomness afterwards. This seems to be the inspiration for the
structure of updated BIP-340 (1), however I don't believe that it provides
any extra protection over (2). My argument is as follows:

Claim A: If the randomness used during signing is kept secret from the
attacker then (2) is secure against DPA.

Since SHA256 has 64-byte blocks the hash H_derive(sec_key || rand ||
pub_key_x || message) will be split up into two 64 byte blocks, one
containing secret data (sec_key || rand) and the other containing data
known to the attacker (pub_key_x || message). The compression function will
run on (sec_key || rand) but DPA will be useless here because the
HD(sec_key, rand) will contain no information about sec_key since rand is
also secret. The output of the compression function on the first block will
be secret but *variable* so the intermediate hash state will not reveal
useful information when compressed with the second block.

Then I thought perhaps (1) is more robust in the case where the randomness
is known by the attacker (maybe the attacker can physically modify the
chipset to control the rng). We'd have to assume that the sec_key ^
H_aux(rand) isn't vulnerable to DPA (the LHS is under the control of the
attacker) to be true. Even under this assumption it turned out not to be
the case:

Claim B: If the randomness used during signing is known to the attacker,
then (1) is not secure against DPA.

In (1)  there are 96 bytes to be hashed and therefore two SHA256 blocks:
(H_aux(sec_key) ^ rand || pub_key_x) and (message). During the first
compression function call the attacker gets the HD of:
HD( sec_key ^ H_aux(rand),  pub_key_x)
which is equal to the following as applying the same XOR to both sides does
not change the HD.
HD(sec_key, H_aux(rand) ^ pub_key_x)
Since the LHS is secret and static, and the RHS is variable and known to
the adversary we have a successful DPA attack -- the attacker will learn
sec_key after enough runs.

Maybe it's just a general rule if you can't produce randomness hidden to
the attacker then no defence is possible against DPA but I wanted to check
this anyway.

My conclusion from this is that (2) is preferable to (1) because it is
simpler and more efficient (it has one less SHA256 compression run) and no
less secure against DPA (in this model). This is not really my area so
perhaps there is a justification for (1) over (2) that I don't understand
yet. If so, someone needs to write it down! If not then I think changing
the proposal to (2) is preferable.

Cheers,

LL

[1] https://github.com/bitcoin/bips/pull/893
[2]
https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-February/017639.html
[3] http://www.oocities.org/mike.tunstall/papers/MTMM07.pdf
[4] https://www.cryptoexperts.com/sbelaid/articleHMAC.pdf
[5] https://en.wikipedia.org/wiki/Hamming_distance