Definition 4 specifies Share, Redistribute, and Open with termination, correctness, and secrecy under a phase-by-phase mobile corruption threshold τ(n(i)) while party set and threshold change.

1. threat model

   Mobile adversary

   defined

   The adversary may corrupt changing parties across phases but not more than the active threshold in any phase; perfect and statistical variants use different parameter regimes.

   SS, PSS, and DPSS definitions · Perfect/statistical parameter and group-change constraints
2. assumption

   Dynamic-change constraints

   explicit

   Perfect security limits each population change to a factor of two and requires old honest shares to interpolate new polynomials while old corrupt shares cannot; the statistical base protocol constrains threshold changes to a factor of two.

   Perfect/statistical parameter and group-change constraints

Four Threshold Change protocols handle increasing/decreasing thresholds with or without batch-size changes; Refresh Recovery then masks, checks, and transfers the refreshed packed shares.

1. component

   Threshold and batch transformation

   specified

   Hyper-invertible matrices distribute transformed sharings among old parties, who emulate trusted threshold conversion; Berlekamp-Welch reconstruction tolerates corrupted contributions.

   Threshold Change and Refresh Recovery protocols
2. component

   Masked refresh and share transfer

   specified

   Old parties create U sharings of existing shares and random/zero V sharings; each new party receives a linear combination equal to its share of H+Q without the old group learning Q or the new group learning H.

   Threshold Change and Refresh Recovery protocols
3. component

   Statistical threshold amplification

   described

   Committee-based party virtualization raises the low-threshold statistical base construction toward one-half corruption while preserving constant-round redistribution.

   Statistical party virtualization

The paper reports first dynamic proactive schemes with O(1) amortized communication per secret, information-theoretic security, near-optimal thresholds, and a DPMPC application.

1. claim

   Amortized communication optimality

   analytically supported

   Batching O(n) secrets and moving O(n)-scale data yields O(1) amortized field elements per secret; the paper explicitly does not claim optimal non-amortized complexity.

   Problem, techniques, contributions, and comparison · Perfect/statistical parameter and group-change constraints
2. claim

   Near-optimal corruption thresholds

   proved conditional

   The reported perfect variant supports any threshold fraction below one third under chosen constants; party virtualization raises the statistical variant toward one half.

   Problem, techniques, contributions, and comparison · Perfect/statistical parameter and group-change constraints · Statistical party virtualization
3. claim

   Dynamic proactive MPC

   construction sketch

   The application section proposes executing Redistribute between circuit layers of an existing proactive MPC to let the computing party set and threshold evolve.

   Dynamic proactive MPC application

The source provides precise functionality definitions, protocol pseudocode, parameter inequalities, complexity reasoning, and UC ideal/simulator material.

1. evidence

   Security argument

   paper proof not machine checked

   Appendix B writes an ideal functionality and simulator for Threshold Change2 and states analogous functionality for the other threshold-change and refresh components; this audit did not mechanically verify the extrapolation or all imported subprotocol theorems.

   UC ideal functionality and representative threshold-change simulator

The result assumes synchrony, threshold-bounded phases, admissible population changes, large batches for amortization, and an external mechanism that agrees on new membership.

1. limitation

   Membership governance is external

   out of scope

   Voting, a trusted administrator, or a predetermined schedule must decide additions and removals; the paper explicitly leaves that mechanism outside the protocol.

   Threshold Change and Refresh Recovery protocols
2. limitation

   No implementation evidence

   no empirical evaluation

   The source supplies no code, benchmarks, failure experiments, concrete field sizes, or distributed-system deployment.

   Dynamic proactive MPC application