Module algebra 

Algebraic structure traits shared by DHT identifiers and elliptic-curve groups.

This module names the algebraic structure carried by a domain type. It is deliberately about carriers and operations, not about object hierarchies: the implementing type is the carrier set, and each trait states which operations and laws are part of that type’s public model.

Model Boundary

Rust trait bounds can require operation shapes such as Add<Output = Self>, Neg<Output = Self>, or Mul<Output = Self>. They cannot prove associativity, commutativity, distributivity, identity, or inverse laws. Implementing one of these traits is therefore a proof obligation: the implementation asserts the law, and law tests witness the assertion on representative samples.

The public surface is intentionally small:

• Zero and One name additive and multiplicative identities together with the operations whose identities they are.
• AbelianGroup is the additive structure used by Chord identifiers and elliptic-curve points.
• CommutativeRing combines an additive abelian group with commutative multiplication and a multiplicative identity.
• Field is a commutative ring whose non-zero elements have inverses.
• Module is a right scalar action of a commutative ring on an abelian group.
• JoinSemilattice is the CRDT merge structure used by replicated DHT state.

Rings DHT

crate::dht::Did is the carrier for Chord identifier arithmetic. It is an AbelianGroup under addition in Z / 2^160, which is exactly the operation used for clockwise offsets, biased ordering, finger targets, and affine replica placement. It intentionally does not implement CommutativeRing: Chord does not use identifier multiplication as a protocol operation, so the public model should not expose it.

Elliptic Curves

Curve points are additive abelian groups. Curve scalars are finite fields. A point group with scalar multiplication is modeled as Point<C>: Module<Scalar<C>>. The module action is a right action because Rust’s operator implementation in this crate is Point<C> * Scalar<C>. This keeps cryptographic algorithms phrased in algebraic terms while curve libraries remain adapters behind crate::ecc::group::CurveGroup.

Law Witnesses

The assert_*_laws functions are test helpers, not proofs in the type system. They are useful because every implementation can be checked through the same vocabulary, but they remain finite-sample witnesses. A new implementation must still explain why its carrier and operations satisfy the stated laws for all values, usually by delegating to a native finite-field or group implementation with documented semantics.

Identities are functions rather than associated constants because several curve adapters obtain identity values through their native libraries.

Traits

AbelianGroup

Abelian group under addition.

CommutativeRing

Unital commutative ring.

Field

Field.

JoinSemilattice

Join-semilattice for state-based CRDT merge.

Module

Right scalar action of a commutative ring on an abelian group.

One

Multiplicative identity for a multiplicative carrier.

Zero

Additive identity for an additive carrier.