> ## Documentation Index > Fetch the complete documentation index at: https://docs.stoffelmpc.com/llms.txt > Use this file to discover all available pages before exploring further. # Protocol definitions > Definitions for phases, adversaries, corruption timing, and asynchronous protocol guarantees. Use these definitions when reading the security, cost, and protocol-reference pages. They separate properties that are often collapsed into a single claim such as “asynchronous” or “Byzantine secure.” ## Execution phases | Term | Meaning | | ----------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Offline phase / preprocessing | Work performed before application values are computed. It creates correlated secret material such as random shares, multiplication triples, random bits, and fixed-point masks. | | Online phase | Work that accepts application inputs, computes over secret values, and reconstructs or sends authorized outputs. It may consume material created offline. | | Preloaded material | Valid, unused preprocessing material already available when online execution begins. Preloading can separate online latency from preprocessing latency, but it does not change the material's security assumptions. | An online phase cannot repair invalid preprocessing. A preprocessing outage may leave online work unaffected until the reserve of valid material is exhausted. ## Adversary behavior | Term | Meaning | | --------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Passive / semi-honest | Corrupted parties follow the protocol but expose their internal state to the adversary. This model does not cover deliberately malformed messages or incorrect local computation. | | Active / malicious | Corrupted parties may deviate arbitrarily, including sending inconsistent values, withholding messages, replaying material, or computing the wrong relation. A protocol may detect some active behavior without providing end-to-end active security. | | Byzantine fault | Arbitrary faulty behavior, whether caused by compromise, malicious operation, or software failure. The exact tolerated behavior still depends on the protocol's checks and progress assumptions. | ## Corruption timing | Term | Meaning | | ------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | Static corruption | The corrupted party set is fixed before the protocol execution being analyzed. | | Adaptive corruption | The adversary chooses additional parties to corrupt as it observes the execution. | | Mobile corruption | The adversary may move between party hosts over time. Even if no more than `t` parties are corrupted at once, the accumulated exposure can exceed a static model. | | Proactive security | Parties periodically refresh or reshape secret state so prior compromises do not accumulate indefinitely. This requires an explicit refresh protocol; restarting a phase does not provide it. | When the same parties perform preprocessing and online execution, a static corruption budget spans both phases and the stored material between them. It does not reset when online execution begins. ## Asynchronous guarantees | Property | What it guarantees | What it does not guarantee | | -------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------ | | Asynchronous safety | Message delay alone cannot make honest parties accept incompatible or incorrect results or cross the stated privacy boundary. | Completion. A safe execution may wait or stop. | | Asynchronous liveness | Honest parties eventually complete despite the allowed faulty parties, assuming the protocol's delivery and scheduling conditions hold. | Completion during a permanent partition, without required client input, or after valid preprocessing is exhausted. | | Conditional liveness | Progress is available only while named preconditions hold, such as valid preloaded material, enough responsive parties, and eventual honest-to-honest delivery. | A blanket guarantee that every operation or replenishment attempt terminates. | | Expected-round termination | A randomized protocol has a bounded expected number of rounds under its stated scheduler and delivery model. | A fixed deadline for every execution. | Randomized asynchronous agreement may have an expected-round bound without a deterministic completion deadline. FLP rules out deterministic consensus termination in every fully asynchronous execution with even one fault.\[14] ## Sharing and reconstruction | Term | Meaning | | --------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | `n` | Number of computing parties. Input and output clients are not included unless a protocol explicitly assigns them a computing-party role. | | `t` | Maximum corrupted-party count covered by the stated guarantee. It is one total budget, not separate allowances for malicious, crashed, and offline parties. | | Degree-`t` sharing | A secret-sharing polynomial of degree `t`. Any `t+1` correct evaluations determine the polynomial, but Byzantine correction or distributed progress can require more parties. | | Verified share | A share checked against a common commitment or proof. Verification can establish consistency with that commitment without proving every higher-level relation the application needs. | | Robust reconstruction | Reconstruction that identifies or corrects inconsistent evaluations within a stated error bound rather than assuming every supplied share is correct. | ## Sources \[14] Fischer, Lynch and Paterson. Impossibility of Distributed Consensus with One Faulty Process (1985). [https://groups.csail.mit.edu/tds/papers/Lynch/jacm85.pdf](https://groups.csail.mit.edu/tds/papers/Lynch/jacm85.pdf)