# Keys & Key Derivation

### Key Derivation

The user's spending key, which we will refer to as $\text{sk}$, is a uniformly-sampled 32-byte string.

In Nocturne's MetaMask Snap, we derive the $\text{sk}$ using the

`snap_getBip44Entropy`

method at derivation path`m / 44' / 6789'`

.

Let $G$denote the generator of Baby Jubjub's prime-order subgroup. The user's spending public key, which we will refer to as $\text{PK}$, is an element of Baby Jubjub defined as $\text{PK} = \text{SHA512}(\text{sk})[0:32] \times G$ ($[0:32]$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 $\mathbb{F}_r$ defined as $vk = H(\text{PK.X}\ ||\ \text{PK.Y}\ ||\ \text{vkNonce})$, where $\text{PK.X}, \text{PK.Y} \in \mathbb{F}_p$are the x and y coordinates of $\text{PK}$ respectively, $\text{vkNonce} \in \mathbb{F}_p$, and $\text{vkNonce}$ must be chosen such that the output of the hash is an element of $\mathbb{F}_r$.

That last provision is needed because $H$ returns an element of $\mathbb{F}_p$, but we need an element of $\mathbb{F}_r$. A reduction modulo $r$ would bias the key generation, and using Poseidon over $\mathbb{F}_r$ 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 $\mathbb{F}_r$.

In theory, rejection sampling like this comes small performance cost. ~91% of the possible $\text{vkNonce} > r$ , 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.

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