[Feature] Add additional features to the "group" element type

[matthewdgreen]

🚀 Feature

Currently in Leo (and SnarkVM) there is no way to implement the following three features on "group" elements:

1. Obtaining the X-coordinate of an elliptic curve group element (as a field element)
2. Hashing group elements through functions like Poseidon (this can be implemented using the feature above)
3. Obtaining the generator point of the elliptic curve

Motivation

These three features are necessary to build Elgamal/ECIES (crypto_box) style encryption, which in a simplified version will look something like this.

Let "random_field" be a random field element. Let "generator_group" be the generator of the elliptic curve group. Let "message_field" be some message to encrypt. Let "PK_group" be the public key to encrypt to, as a group element.

C0 = random_field * generator_group,
C1 = Poseidon(random_field * PK_group) + message_field

Implementation

It seems likely that this will require additional opcodes in SnarkVM as well as a new command in Leo. Ideally this would be very simple:

• g.x would provide the x coordinate of group element "g".
• group.generator() might obtain the standard generator point (G) of the group.

Thank you!