Vela and Carina: Fast Pairing-Based Multilinear Polynomial Commitments from Reciprocal Polynomials

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Pairing-based multilinear polynomial commitments offer succinct verification for sum-check-based proof systems, but existing schemes trade prover group work against proof size and verifier cost. Building on Mercury's representation of a multilinear evaluation as the constant coefficient of a Laurent polynomial, we construct two schemes with different performance trade-offs. Vela uses inversion symmetry to derive a half-length auxiliary polynomial and opens two polynomials at $z$ and $z^{-1}$ w