Full-Information Protocol Overview

• Full-information protocols are defined by the maximal disclosure of agents' entire history, ensuring that all past states are communicated in each round.
• They underpin diverse applications ranging from Byzantine agreement and coin-flipping in distributed systems to game synthesis and quantum no-forgetting protocols.
• These protocols critically influence resilience bounds and communication complexity, revealing trade-offs in adaptive corruption resistance and optimality in various models.

“Full-information protocol” is not a single formal object across the literature. In distributed computing, it usually denotes a protocol in which each agent’s local state records its entire history and, in each round, it re-sends its whole state to all other agents (Alpturer et al., 2023). Closely related work studies a full-information model in which all past broadcasts are public (Goldwasser et al., 2015), a game-theoretic communication primitive by which a player’s entire observation history is revealed to another player (Berwanger et al., 2023), and a quantum Byzantine setting in which the adversary knows the exact joint state of all unmeasured qubits (Li et al., 2024). In interactive quantum communication, the phrase is used differently again: a full-information or no-forgetting protocol is one with $CRIC(\Pi,\mu)=0$ (Lauriere et al., 2017). The common theme is maximal exposure or retention of protocol state, but the formal content depends on the problem domain.

1. Canonical meanings and formal scope

Across the cited literature, the phrase appears in several distinct but related roles. The table summarizes the main meanings that are formalized explicitly.

Context	Formal feature	Source
Eventual Byzantine Agreement	each agent records its entire history and rebroadcasts its whole local state	(Alpturer et al., 2023)
One-round coin-flipping	all past broadcasts, including adversarial ones, are public	(Goldwasser et al., 2015)
Imperfect-information games	communication action reveals the entire observation history of another player	(Berwanger et al., 2023)
Quantum communication	no-forgetting condition $CRIC(\Pi,\mu)=0$	(Lauriere et al., 2017)
Quantum Byzantine Agreement	adversary holds a purifying register of the entire pure global state	(Li et al., 2024)

In the runs-and-systems formulation used for eventual Byzantine agreement, an information-exchange protocol is a tuple

$E=\langle E_1,\dots,E_n\rangle,$

and a full-information protocol is obtained by taking local states of the form

$$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$

with $M_i=L_i$ and $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$, so that agent $i$ rebroadcasts its entire local state every round (Alpturer et al., 2023). This is the most literal protocol-level use of the term.

The game-synthesis literature generalizes the same intuition. A synchronous imperfect-information game with communication specifies observation functions

$O_i:S\times C\to \Sigma_i,$

and a communication graph $R_\sigma\subseteq P\times P$ such that when $(i,j)\in R_\sigma$, player $CRIC(\Pi,\mu)=0$0 “peeks” the full view of $CRIC(\Pi,\mu)=0$1 (Berwanger et al., 2023). Here, “full information” is not universal transparency; it is selective but history-complete disclosure.

A plausible implication is that “full-information” should be read as a family resemblance rather than a single definition. In each usage, the operative object is not merely a transcript, but a recursively enriched state that contains prior observations, communicated knowledge, or retained information.

2. Full-information coin-flipping and the adaptive-corruption barrier

Goldwasser, Kalai, and Park study one-round coin-flipping in the full-information model introduced by Ben-Or and Linial in 1985 (Goldwasser et al., 2015). In this model there are $CRIC(\Pi,\mu)=0$2 computationally unbounded players, communication takes place over a single reliable broadcast channel, each honest party has private fresh randomness, and all past broadcasts are public. The paper distinguishes two adversaries. A standard adaptive adversary can corrupt new parties based on messages seen so far, but cannot retroactively change honest broadcasts. A strong adaptive adversary first “peeks” at all honest messages that would be sent in a round, then decides which parties to corrupt and can replace or suppress those messages (Goldwasser et al., 2015).

The central upper bound is that any one-round full-information coin-flipping protocol secure against $CRIC(\Pi,\mu)=0$3 strong adaptive corruptions must satisfy

$CRIC(\Pi,\mu)=0$4

The same work proves that if a symmetric one-round protocol is secure against $CRIC(\Pi,\mu)=0$5 standard adaptive corruptions, then there exists a symmetric one-round protocol secure against $CRIC(\Pi,\mu)=0$6 strong adaptive corruptions; as a corollary, any symmetric one-round full-information coin-flipping protocol tolerates at most $CRIC(\Pi,\mu)=0$7 standard adaptive corruptions (Goldwasser et al., 2015).

The proof architecture is structurally important. The paper gives a robust-set characterization

$CRIC(\Pi,\mu)=0$8

and shows that strong-adaptive security is equivalent to lower bounds on the probability of landing in $CRIC(\Pi,\mu)=0$9 and $E=\langle E_1,\dots,E_n\rangle,$0. It then reduces arbitrary message length to $E=\langle E_1,\dots,E_n\rangle,$1 bits per player, then to single-bit players, and finally applies the Lichtenstein–Linial–Saks bound (Goldwasser et al., 2015).

One recurring misconception is that longer one-round messages might overcome the adaptive-corruption barrier. The paper rules this out: increased message length does not help in the strong-adaptive setting, and in the symmetric case it does not help against ordinary adaptive corruptions either (Goldwasser et al., 2015). The full-information constraint is therefore not merely a semantic choice; in one-round coin-flipping it induces a quantitative resilience barrier.

3. Eventual Byzantine agreement, knowledge tests, and optimality

In eventual Byzantine agreement with omission failures, Alpturer, Halpern, and van der Meyden separate a protocol into an information-exchange component $E=\langle E_1,\dots,E_n\rangle,$2 and an action protocol $E=\langle E_1,\dots,E_n\rangle,$3, allowing optimality to be studied relative to a fixed exchange discipline (Alpturer et al., 2023). Under a fixed $E=\langle E_1,\dots,E_n\rangle,$4, one says $E=\langle E_1,\dots,E_n\rangle,$5 if in every corresponding run with the same initial states and failure pattern, whenever $E=\langle E_1,\dots,E_n\rangle,$6 first decides in round $E=\langle E_1,\dots,E_n\rangle,$7, $E=\langle E_1,\dots,E_n\rangle,$8 does not decide at any earlier round. A protocol is optimal if no other EBA protocol strictly dominates it (Alpturer et al., 2023).

Their knowledge-based program $E=\langle E_1,\dots,E_n\rangle,$9 has the following structure: if an agent has already decided, it performs $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$0; if $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$1 or $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$2, it executes $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$3; if $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$4, it executes $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$5; otherwise it performs $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$6 (Alpturer et al., 2023). The intended epistemic reading is asymmetric: deciding $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$7 is triggered as soon as a $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$8-decision is known to exist, while deciding $$L_i=\{\langle t,init_i,decided_i,\mathit{hist}_i\rangle\},$$9 requires knowledge that no agent can be in the process of deciding $M_i=L_i$0.

Optimality is tied to a safety condition on the information-exchange protocol. Safety rules out “spurious” knowledge: if an agent has not received a $M_i=L_i$1-chain, then there must exist an indistinguishable point where all initial values are $M_i=L_i$2; and if an agent lacks knowledge that no one is about to decide $M_i=L_i$3, there must exist an indistinguishable point where some nonfaulty agent is indeed about to decide $M_i=L_i$4 (Alpturer et al., 2023). The theorem is then exact: if $M_i=L_i$5 is an EBA context and $M_i=L_i$6 is safe with respect to $M_i=L_i$7, every implementation of $M_i=L_i$8 is optimal in $M_i=L_i$9 (Alpturer et al., 2023).

The same paper gives two limited-exchange implementations. In $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$0, each agent sends exactly one non-null bit, namely its decision, to all $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$1 agents, for $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$2 bits total; if some agent starts with $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$3, all decide by round $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$4, while if all start with $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$5, they decide $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$6 at round $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$7 (Alpturer et al., 2023). In $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$8, undecided agents additionally send $$\mu_i(\langle t,\dots,\mathit{hist}\rangle,a)=\langle t,\dots,\mathit{hist}\rangle$$9, yielding $i$0 bits total and allowing all-1 failure-free runs to decide by round $i$1 (Alpturer et al., 2023).

The full-information exchange is stronger but not automatically optimal. The paper therefore adds common-knowledge tests and defines $i$2, whose leading rules are $i$3 for deciding $i$4 and $i$5 for deciding $i$6, before falling back to the $i$7 tests (Alpturer et al., 2023). Every FIP implementation of $i$8 is correct, and the added common-knowledge rules make it optimal among all FIP protocols. The usual graph-based full-information implementation costs $i$9 bits, but the action protocol can still be implemented in polynomial time by tracking a compact communication graph (Alpturer et al., 2023).

4. Full-information protocols as an overview target in games

The synthesis perspective treats full-information protocols as a communication discipline inside imperfect-information games. Berwanger, Mathew, and van den Bogaard model a synchronous game

$O_i:S\times C\to \Sigma_i,$0

with one active player, passive observers, explicit communication labels, deterministic transition function $O_i:S\times C\to \Sigma_i,$1, and perfect-recall observation functions $O_i:S\times C\to \Sigma_i,$2 (Berwanger et al., 2023). The key communication action is that player $O_i:S\times C\to \Sigma_i,$3 may receive the entire observation history of player $O_i:S\times C\to \Sigma_i,$4.

The reduction schema is classical in outline but technically specialized. One first forms the information tree $O_i:S\times C\to \Sigma_i,$5 of information sets $O_i:S\times C\to \Sigma_i,$6. To obtain a finite perfect-information game, the paper constructs a rectangular morphism $O_i:S\times C\to \Sigma_i,$7 of finite index satisfying rectangularity, morphism, and refinement conditions. In the full-information setting with $O_i:S\times C\to \Sigma_i,$8 observers, $O_i:S\times C\to \Sigma_i,$9 is built as a vector of knowledge-sets indexed by all coalitions $R_\sigma\subseteq P\times P$0, and the recursive update is driven by the synchronization operator $R_\sigma\subseteq P\times P$1 (Berwanger et al., 2023).

The resulting complexity is sharp. With $R_\sigma\subseteq P\times P$2 observers, full-information protocols can express indistinguishability relations not expressible with $R_\sigma\subseteq P\times P$3 observers, giving a strict hierarchy in expressiveness and complexity; the strategy-synthesis problem is $R_\sigma\subseteq P\times P$4-EXPTIME-complete (Berwanger et al., 2023). For fixed parity or $R_\sigma\subseteq P\times P$5-regular objectives, one can decide and synthesize a winning strategy in an FIP with $R_\sigma\subseteq P\times P$6 observers in time $R_\sigma\subseteq P\times P$7-EXPTIME, and reachability already yields the matching lower bound (Berwanger et al., 2023).

This suggests that full-information communication is not algorithmically trivial merely because it discloses entire views. The disclosure induces higher-order knowledge structures—knowledge of others’ views, of coalitional synchronization, and of recursively communicated histories—that are expensive to represent but still regular enough to admit finite quotients.

5. Quantum reinterpretations: no-forgetting protocols and full-information adversaries

In two-party interactive quantum communication, Laurière and Touchette use “full-information protocol” in a different technical sense: a protocol that never forgets information about the other party’s input (Lauriere et al., 2017). For classical inputs, they define the classical-input information cost $R_\sigma\subseteq P\times P$8, Holevo information cost $R_\sigma\subseteq P\times P$9, and reverse information cost $(i,j)\in R_\sigma$0, and prove the exact identities

$(i,j)\in R_\sigma$1

A protocol is full-information or no-forgetting precisely when

$(i,j)\in R_\sigma$2

(Lauriere et al., 2017).

The conceptual tool is the Information-Flow Lemma, which tracks the net change in conditional mutual information across an interactive process: $(i,j)\in R_\sigma$3 (Lauriere et al., 2017). Operationally, $(i,j)\in R_\sigma$4 decomposes into transmitted-input cost and forgotten-input cost. In classical reversible protocols, $(i,j)\in R_\sigma$5; in quantum protocols, the paper argues that forgetting is often necessary.

The lower-bound consequences are explicit. Any $(i,j)\in R_\sigma$6-round quantum protocol for $(i,j)\in R_\sigma$7 that never forgets information satisfies

$(i,j)\in R_\sigma$8

so the quadratic quantum speedup is lost (Lauriere et al., 2017). At zero error, the Inner Product function has

$(i,j)\in R_\sigma$9

and for a random Boolean function $CRIC(\Pi,\mu)=0$00, with overwhelming probability,

$CRIC(\Pi,\mu)=0$01

(Lauriere et al., 2017).

A second quantum usage concerns full-information adversaries rather than no-forgetting protocols. In the quantum Byzantine agreement model of (Li et al., 2024), players exchange qubits and the adversary holds a purifying register of the entire pure global state; equivalently, in round $CRIC(\Pi,\mu)=0$02 the adversary knows the exact joint state of all unmeasured qubits. The paper proves a black-box lifting theorem: any non-erasing classical private-channel BA protocol $CRIC(\Pi,\mu)=0$03 with resilience $CRIC(\Pi,\mu)=0$04, round complexity $CRIC(\Pi,\mu)=0$05, and communication cost $CRIC(\Pi,\mu)=0$06 yields a quantum full-information protocol $CRIC(\Pi,\mu)=0$07 with the same $CRIC(\Pi,\mu)=0$08, $CRIC(\Pi,\mu)=0$09, and $CRIC(\Pi,\mu)=0$10 (Li et al., 2024).

This lifting has concrete consequences. The quantum full-information model admits $CRIC(\Pi,\mu)=0$11-round agreement for $CRIC(\Pi,\mu)=0$12 against a fail-stop adversary and $CRIC(\Pi,\mu)=0$13-round agreement for $CRIC(\Pi,\mu)=0$14 against a Byzantine adversary, while the paper states that classically in the full-information model any BA with even a fail-stop adversary and $CRIC(\Pi,\mu)=0$15 requires $CRIC(\Pi,\mu)=0$16 rounds (Li et al., 2024). The term “full-information” therefore spans two distinct quantum notions: preservation of learned information inside a protocol, and maximal adversarial visibility of global state.

6. Adjacent cryptographic notions: full access, full opening, and information-theoretic security

Several neighboring literatures study settings that are not called full-information protocols but share the same design pressure: nothing essential should be hidden except what correctness or privacy requires. One cryptographic direction proposes “a truly (information theoretically) safe cryptographic communication system” that provides zero information to any passive adversary having full access to the channel, contingent on the existence of a suitable group action [0603107]. The available statement is only the abstract-level claim, but it illustrates an extreme version of the model: complete transcript exposure with information-theoretic secrecy.

A different information-theoretic primitive is the Information Checking Protocol in the full-information adversarial model. Patra and Pandu Rangan formalize an ICP with a dealer $CRIC(\Pi,\mu)=0$17, an intermediary $CRIC(\Pi,\mu)=0$18, $CRIC(\Pi,\mu)=0$19 verifiers, pairwise private channels, and a common broadcast channel, against a static, active, rushing, computationally unbounded $CRIC(\Pi,\mu)=0$20-threshold adversary (Patra et al., 2010). Their MVMS-ICP authenticates an $CRIC(\Pi,\mu)=0$21-vector $CRIC(\Pi,\mu)=0$22, achieves the $CRIC(\Pi,\mu)=0$23 round pattern for Gen/Ver/Reveal, has communication $CRIC(\Pi,\mu)=0$24 bits per phase, and satisfies a linearity property that supports batch composition into statistical VSS and MPC (Patra et al., 2010). Here the model is “full-information” on the adversarial side, not in the sense of agents rebroadcasting complete local state.

Card-based cryptography offers a further adjacent notion, full-open rather than full-information. In a single-shuffle full-open protocol, parties place face-down cards representing inputs, perform one shuffle, and then open all cards; privacy requires that for any two inputs with the same output, the distribution of revealed suits is identical (Eriguchi et al., 20 Oct 2025). The 2025 result that every Boolean function admits a single-shuffle full-open protocol establishes this for a model that completely reveals the final physical transcript yet leaks no more than $CRIC(\Pi,\mu)=0$25 (Eriguchi et al., 20 Oct 2025). Related single-cut full-open protocols use one random cut, then open all cards, with correctness and privacy obtained from disjoint rotation orbits $CRIC(\Pi,\mu)=0$26 and $CRIC(\Pi,\mu)=0$27 (Shinagawa et al., 4 Jul 2025).

This suggests an important distinction. Full-information protocols, full-information adversaries, and full-open protocols are not interchangeable terms. The first concerns how state or knowledge is communicated during execution; the second concerns what an adversary can observe; the third concerns how much of the final encoded object is revealed. What unifies them is the information-theoretic discipline that privacy or correctness must survive despite maximal disclosure of some designated view.