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Keys & Key Derivation

Key Derivation

The user's spending key, which we will refer to as
sk\text{sk}
, is a uniformly-sampled 32-byte string.
In Nocturne's MetaMask Snap, we derive the
sk \text{sk}
using the snap_getBip44Entropy method at derivation path m / 44' / 6789'.
Let
GG
denote the generator of Baby Jubjub's prime-order subgroup. The user's spending public key, which we will refer to as
PK\text{PK}
, is an element of Baby Jubjub defined as
PK=SHA512(sk)[0:32]×G\text{PK} = \text{SHA512}(\text{sk})[0:32] \times G
(
[0:32][0:32]
means "0th through 32nd byte"). This is only used to verify signatures in-circuit. It never appears on-chain or leaves the client.
What we refer to as the "generator" is often called the "base point" in order to differentiate between generator of Baby Jubjub's curve group and the generator of Baby Jubjub's prime-order subgroup. Since all operations are performed in the prime-order subgroup, we're ignoring this distinction and using the word "generator" to refer to the generator of the prime-order subgroup.
The user's viewing key is an element of
Fr\mathbb{F}_r
defined as
vk=H(PK.X  PK.Y  vkNonce)vk = H(\text{PK.X}\ ||\ \text{PK.Y}\ ||\ \text{vkNonce})
, where
PK.X,PK.YFp\text{PK.X}, \text{PK.Y} \in \mathbb{F}_p
are the x and y coordinates of
PK\text{PK}
respectively,
vkNonceFp\text{vkNonce} \in \mathbb{F}_p
, and
vkNonce\text{vkNonce}
must be chosen such that the output of the hash is an element of
Fr\mathbb{F}_r
.
That last provision is needed because
HH
returns an element of
Fp\mathbb{F}_p
, but we need an element of
Fr\mathbb{F}_r
. A reduction modulo
rr
would bias the key generation, and using Poseidon over
Fr\mathbb{F}_r
would be prohibitively expensive in-circuit. But this approach suffers from neither issue - during key generation, we can increment vkNonce and try again if the output of the hash is not an element of
Fr\mathbb{F}_r
.
In theory, rejection sampling like this comes small performance cost. ~91% of the possible
vkNonce>r\text{vkNonce} > r
, so we expect that, on average, it will take 10-11 tries to find a "good" nonce. In practice, the cost is negligible - 11 attempts takes ~30ms with a very naive implementation.