Module multi_degree 

Multi-degree sumcheck: runs multiple degree groups in lockstep with shared verifier randomness, producing a common evaluation point.

Protocol

Given G degree groups each with (degree_g, mles_g, comb_fn_g):

1. Absorb metadata: num_vars, num_groups, per-group degrees
2. For each round i = 1..num_vars:
   • Each group computes its round polynomial P_g (parallelizable)
   • Absorb all round messages in deterministic order
   • Sample ONE shared challenge r_i
   • All groups fix variable i at r_i
3. Each group produces a subclaim at the shared point r = (r_1, …, r_n)

Structs

MultiDegreeSubClaims

Sub-claims: shared evaluation point + per-group expected evaluation.

MultiDegreeSumcheck

MultiDegreeSumcheckGroup

A single degree group for the multi-degree sumcheck: (degree, mles, comb_fn).

MultiDegreeSumcheckProof

Proof for a multi-degree sumcheck.

Round1Output

Output of a Round1FastPath::round_1_message call. Carries the asserted sum and the round-1 polynomial tail (evaluations at 1, 2, ..., degree, omitting the constant term) that the framework wraps into a regular SumcheckProverMsg and absorbs into the transcript.

Traits

Round1FastPath

Opt-in hook on a MultiDegreeSumcheckGroup that bypasses SumcheckProverState::prove_round in the first round and replaces the full-size MLE fold by a closed-form half-size construction. Groups that don’t supply it are run as usual.