Once we fix the communication model (synchrony, asynchrony, or partial synchrony see here), and we fix a threshold adversary we still need to make important modeling decisions about the adversary power.

Here we will use the simplest model of a threshold adversary that can control up to $f$ parties given a static group of $n$ parties. We will later consider dynamic, permissionless, and bounded resource models.

In addition to the size of the threshold ($n>f$, $n>2f$, or $n>3f$), there are 4 more important parameters:

1. the type of corruption.
2. the computational power of the adversary.
3. the visibility of the adversary.

### 1. Type of corruption

The first fundamental aspect is what type of corruption the adversary can inflict on the $f$ parties is can corrupt. There are four classic adversaries: Passive, Crash, Omission, and Byzantine.

Passive: a passively corrupted party must follow the protocol just like an honest party, but it allows the adversary to learn information. A passive adversary (sometimes called Honest-But-Curious or Semi-Honest) does not deviate from the protocol but can learn all possible information from its view: i.e., the messages sent and received by parties it controls.

Crash: once the party is corrupted, it stops sending and receiving all messages.

Omission: once corrupted, the adversary can decide, for each message sent or received, to either drop or allow it to continue. Note that the party is not informed that it is corrupted.

Byzantine: this gives the adversary full power to control the party and take any (arbitrary) action on the corrupted party.

Note that each corruption type subsumes the previous. There are other types of corruption (most notable are variants of Covert adversaries) that we will cover later. Covert adversaries can be used to model rational behavior where there is fear (utility loss) from punishment.

### 2. Computational power

The computational power of the adversary is the next choice. There are two traditional variants and one newer one:

1. Unbounded: the adversary has unbounded computational power. This model often leads to notions of perfect security or statistical security.
2. Computationally bounded: the adversary is at most a polynomial advantage in computational power over the honest parties. Typically this means that the adversary cannot (except with negligible probability) break the cryptographic primitives being used. For example, typically assume the adversary cannot forge signatures of parties not in its control (see Goldreich’s chapter one for traditional CS formal definitions of polynomially bounded adversaries).
3. Fine-grained computationally bounded: there is some concrete measure of computational power and the adversary is limited in a concrete manner. This model is used in proof-of-work based protocols. For example, see Andrychowicz and Dziembowski for a way to model the hash rate.

### 3. Visibility

The visibility is the power of the adversary to see the messages and the states of the non-corrupted parties. Again, there are two basic variants:

1. Full information: here we assume the adversary sees the internal state of all parties and the content of all message sent. This often limits the protocol designer. See for example: Feige’s selection protocols, or Ben-Or et al’s Byzantine agreement.
2. Private channels: in this model, we assume the adversary cannot see the internal state of honest parties and cannot see the internal content of messages between honest parties. The adversary does know when a message is being sent and depending on the communication model can decide to delay it by any value that is allowed by the communication model.

For models that are round-based, another visibility distinction is the adversary’s ability to rush. When does the adversary see the messages sent to parties it controls? In the rushing adversary model, the adversary is allowed to see all the messages sent to parties it controls in round $i$ before it needs to decide what messages to send in its round $i$ messages. In the non-rushing adversary model, the adversary must commit to the round $i$ messages it sends before it receives any round $i$ messages from non-faulty parties.

Adaptivity is the ability of the adversary to corrupt dynamically based on information the adversary learns during the execution. There are three basic variants: static, adaptive, and mobile. The adaptive model has several sub-variant, we will cover here only the simplest one.

1. Static: the adversary has to decide which $f$ parties to corrupt in advance before the execution of the protocol.

2. Adaptive: the adversary can decide dynamically as the protocol progresses who to corrupt based on what the adversary learns over time. The main parameter that still needs to be decided is how long it takes between the adversary decision to corrupt a party and the event that the corruption occurs. One standard assumption is that this is instantaneous. Another is that it takes an additional round (for example here). We will later review several other options (for example, see here).

3. Mobile: the adversary can decide dynamically corrupt and un-corrupt parties. The total number of corrupted parties at any given time is at most $f$, but over time the set of corrupted parties may change. It is often required that there is a gap between the time the adversary corrupts one party and the time it is allowed to corrupt another. This model was introduced by Ostrovsky and Yung and exemplified by proactive secret sharing.

## Acknowledgments

Special thanks to Alin Tomescu for reviewing this post.