> For the complete documentation index, see [llms.txt](https://nocturne-xyz.gitbook.io/nocturne/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nocturne-xyz.gitbook.io/nocturne/protocol-details/algebraic-primitives-and-notation.md).

# Algebraic Primitives & Notation

### Algebraic Primitives

Unless otherwise specified, all values are assumed to be elements of the Scalar field of BN254 as defined in [EIP-197](https://eips.ethereum.org/EIPS/eip-197). We will refer to this field as the "BN254 Scalar field" or $$\mathbb{F}\_p$$ for short. We will use $$p$$ to refer to the order of the BN254 scalar field.

For all elliptic curve cryptography, including our addressing and signature schemes, we use the Baby Jubjub curve, the curve whose *base* field is $$\mathbb{F\_p}$$ as defined in [EIP-2494](https://eips.ethereum.org/EIPS/eip-2494). All operations are performed in Baby Jubjub's prime-order subgroup, which we will refer to as "Baby Jubjub", or $$\mathbb{G}$$ (blackboard "G" as opposed to regular "G") for short.

We will use $$r$$ to refer to the order of Baby Jubjub's scalar field (again, the prime-order subgroup, not the curve group), and we will refer that scalar field as "Baby Jubjub Scalar field", or $$\mathbb{F}\_r$$ for short.

We will use additive notation. More specifically, we use:

* "$$+$$" to denote curve addition or field addition depending on whether we are adding curve elements or field elements
* "$$\cdot$$" or juxtaposition to denote field multiplication
* "$$\times$$" to denote scalar multiplication, in order to clearly distinguish it from field multiplication

Additionally, for byte arrays, we will use "slice" notation to specify "sub-arrays" - $$\text{bytes}\[K:N]$$ means "from the $$K$$th byte up to, but not including the $$N$$th byte".

Furthermore, we will use all-capital letters for curve elements and all-lowercase lettrs for field elements.

### Hashing

We use the Poseidon hash function defined over the $$\mathbb{F}\_p$$, the BN254 scalar field, instantiated using the [parameters set in `circomlib`'s implementation](https://github.com/iden3/circomlib/blob/master/circuits/poseidon_constants.circom). We will denote this function $$H$$.