A computationally bounded adversary adaptively chooses passive and active corruptions between predefined refresh phases and may eventually visit every party, while the within-phase multi-threshold remains satisfied.

1. assumption

   Network, reset, and cryptographic assumptions

   assumed

   Parties use synchronous pairwise secure channels and authenticated broadcast, reset parties return to a pristine state, old shares are not retained, and proofs rely on DDH-based homomorphic commitments plus Paillier and zero-knowledge multiplication components.

   Mobile mixed adversary and cryptographic assumptions · Resharing, masking, product, verification, and Mult

Inputs are batched and shared, then each arithmetic-circuit layer permutes operands, applies Add or Mult, redistributes shares for planned departures and arrivals, optionally refreshes, and finally reconstructs designated outputs.

1. protocol

   Proactive and dynamic maintenance

   reused and adapted

   Refresh rerandomizes current batches, Recover restores reset parties, Redistribute changes membership, and gradual Reconstruct releases outputs fairly within a separate threshold region.

   Eight-subprotocol dynamic PMPC · Updated PSS proofs, multiplication proofs, permutation, and full composition
2. protocol

   Multi-dealer batched Share

   specified

   Multiple input owners first Shamir-share their values and the parties combine those univariate sharings into a common bivariate batch, removing the earlier requirement that every batched secret originate from one dealer.

   Multi-dealer batched proactive sharing
3. protocol Bivariate batched multiplication specified

   The protocol splits each batch into roughly square-root-sized univariate groups, creates bivariate zero masks, computes blinded cross-products with pairwise zero-knowledge multiplication, verifies random evaluations, and recombines product sharings.

   Bivariate multiplication overview · Resharing, masking, product, verification, and Mult

   1. mechanism

      Commitment and random-evaluation checks

      cryptographically checked

      Homomorphic commitments check local algebra, Paillier-based ZK-Mult protects pairwise products, and a jointly sampled random evaluation detects malformed bivariate commitments except with the theorem's bounded probability.

      Mobile mixed adversary and cryptographic assumptions · Resharing, masking, product, verification, and Mult · Updated PSS proofs, multiplication proofs, permutation, and full composition

Theorems separately state multiplication security, full PMPC secrecy and correctness, conditional fairness, and amortized communication; the thresholds and ideal functionalities must be read together.

1. claim

   Quadratic amortized communication

   asymptotic theorem

   For batches of size n minus 2, the complete PMPC reports O(n squared) amortized communication per secret; the new multiplication and permutation procedures report O(n times square root of n) amortized communication.

   Problem, contributions, and headline thresholds · Multiplication security and communication · Updated PSS proofs, multiplication proofs, permutation, and full composition
2. claim

   Dishonest-majority proactive security

   proved under model

   Under DDH and the formal model, Theorem 1 gives a secrecy and correctness region with a square-root batch-size loss from n minus 3, plus a distinct mixed active/passive threshold for fair reconstruction.

   Full PMPC security and fairness theorem · Updated PSS proofs, multiplication proofs, permutation, and full composition
3. claim

   No-precomputation multiplication

   proved under model

   Theorem 2 realizes the multiplication ideal functionality for batches within the degree bound and a square-root-loss corruption threshold, without a preprocessing phase.

   Multiplication security and communication · Updated PSS proofs, multiplication proofs, permutation, and full composition

Membership changes are assumed planned before execution and occur between circuit layers; corruption sets change only during refresh; fairness holds only in its narrower region; the protocol assumes synchrony, secure channels, reset and erasure; and evidence is formal rather than implemented or benchmarked.

1. limitation

   Refresh frequency is a policy input

   deployment tradeoff

   More frequent refresh permits the adversary less time within a corruption set but costs more communication; the protocol accepts a set of refresh layers rather than deriving an operational schedule.

   Eight-subprotocol dynamic PMPC · Scheduling, planned membership, fairness, and evidence boundaries