Non interactive anti-exfil

anti-exfil-nonce-generation.md

Signing protocol:

x: private key
X: public key
m: message to sign
n: nonce extra

H: cryptographically secure hash committment

1. signing device

q = H(x,m,n)
Q = q·G
k = q + H(Q,m,n)
R = k·G
e = H(R,X,m)

Schnorr signature

s = k + x·e

ECDSA signature

r = R.x
s = k⁻¹·(m + r·x)

2. return to wallet app

Q, s

3. wallet app calculates

R = Q + H(Q,m,n)·G
R, s

4. verify

Schnorr verify

e = H(R,X,m)

s·G ?= R + e·X

ECDSA verify

r = R.x

s⁻¹·m·G + s⁻¹·r·X ?= R

Q&A

Where does the SD get n from?

Typically an opensource "Companion App" or view only wallet software would provide the n nonce extra.

The PSBT standard can either be extended with a new field, or n may be included as an invalid but recognizable "signature" that the SD replaces with (Q,s).

What is the function of H, why not simply hash the concatenated values?

m1 = "hello ", n1 = "world!"
m2 = "hello", n2 = " world!"
q = hash(x|"hello world!")
Q = q·G
k = q + hash(Q|"hello world!")

Would generate the exact same nonce, but the SD would sign a different message ("hello " and "hello"), allowing malicious software to extract the private key from the device.

s1 = k + x·e1
s2 = k + x·e2
s1 - s2 = (k - k) + x(e1 - e2)

With k eliminated, the private key x is revealed to the malicious software.

What function is appropriate as H?

With fixed sized fields/values hash(v1|v2|..|vn) can be safely used.

With variable size binary strings:

• hash(serialize(v1, v2, .., vn))
• hash(hash(v1)|hash(v2)|..|hash(vn))
• merkle_root(v1, v2, .., vn)

Any reversible serialization may also be used.

For example: hash(6|"hello "|6|"world!") != hash(5|"hello"|7|" world!")

What is the reason for committing to n in generation of q?

Let's take a look at what happens, when we sign the same message twice, with different nonce extras!

q = H(x,m)
Q = q·G
s1 = q + H(Q,m,n1) x·e1
s2 = q + H(Q,m,n2) x·e2
s1 - s2 = (q - q) + H(Q,m,n1) - H(Q,m,n2) + x(e1 - e2)

With q eliminated, the private key x is revealed to the malicious software.

Are low bandwidth attacks still possible?

Yes, theoretically the SD with malicious firmware could churn the final R point to leak information using for example FEC codes (Pieter Wuille on delving) with 2ⁿ rounds of churning SD can leak the seed phrase by creating bits(seed)/n signatures. For example if the attacker chooses to leak 4 bits each time it takes 32 signatures to leak a 128 bit (12 word) seed phrase. Due to the public transaction graph and plainly observable wallet characteristics the attacker can make good guesses at which signatures could belong to the same seed.

• The attacker can just generate a random q, normally this can not be detected
• The verifier needs to know the private key to verify generation of q (same as RFC6979)
• The attacker can encrypt and obfuscate his own channel
• The attacker decides his channel bandwidth making it impossible to estimate "seed health"
• Transactions may have multiple inputs making signing them not deterministic in time
• Said attack is likely to be "always on" and would be caught by verifying generation of q
• Evil maid scenario is still problematic, factory and user acceptance tests are already passed
• Tamper evident storage of signing devices is still heavily recommended
• This is not a huge concern for cold storage scenarios with very infrequent signing
• This protocol offers protection from immediate catastrophic leaks via chosen nonce
• 24 words are better in this scheme than 12 words

References

1. https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-March/017667.html (Pieter Wuille 2020)
2. https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-February/017649.html
3. https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2020-February/017655.html
4. https://blog.eternitywall.com/2018/04/13/sign-to-contract/
5. https://bitcointalk.org/index.php?topic=893898.msg9861102#msg9861102
6. https://bitcointalk.org/index.php?topic=893898.msg9841502#msg9841502
7. https://medium.com/cryptoadvance/hardware-wallets-can-be-hacked-but-this-is-fine-a6156bbd199
8. https://youtu.be/j9Wvz7zI_Ac?t=640
9. https://diyhpl.us/wiki/transcripts/sf-bitcoin-meetup/2019-02-04-threshold-signatures-and-accountability/
10. bitcoin-core/secp256k1#590
11. bitcoin-core/secp256k1#669
12. https://eprint.iacr.org/2017/985
13. https://moderncrypto.org/mail-archive/curves/2017/000925.html

Where R is not a valid point for the scheme. At least my understanding is, that can happen. Also with BIP-340, because you need an even R (0x02).

as we use secp256k1_schnorrsig_sign_custom and only provide it with custom noncefp my understanding is that this is done for us in secp256k1_schnorrsig_sign_internal https://github.com/bitcoin-core/secp256k1/blob/1988855079fa8161521b86515e77965120fdc734/src/modules/schnorrsig/main_impl.h#L165-L180