CPPDD: Consensus Privacy in Data Distribution

• Consensus-Based Privacy-Preserving Data Distribution (CPPDD) is a framework that secures data aggregation by combining distributed masking with consensus protocols to protect individual inputs.
• It employs a two-phase process where agents first mask their inputs by exchanging random values and then use consensus algorithms to recover precise aggregate statistics.
• CPPDD provides exact aggregation with strong information-theoretic privacy guarantees and scalability, making it applicable to sensor networks, federated analytics, and collaborative learning.

Consensus-Based Privacy-Preserving Data Distribution (CPPDD) describes a class of distributed protocols for securely aggregating, distributing, or agreeing on data across a network of agents while ensuring strong privacy of individual inputs—even against coalitions of passive adversaries. CPPDD frameworks decompose into two essential components: a privacy-enforcing data masking mechanism and a consensus protocol for aggregate computation. They operate in arbitrary network topologies and support a spectrum of privacy notions from information-theoretic (statistical) privacy to differential privacy, with correctness guarantees to recover exact aggregates when desired.

1. Network Model, Adversarial Threats, and Privacy Definitions

CPPDD frameworks model agents as nodes in an undirected, connected graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ (Gupta et al., 2018). Each agent $i \in \mathcal{V}$ privately holds a scalar or vector input $x_i$. Communication between agents occurs over private, authenticated channels specified by the edges $\{i,j\} \in \mathcal{E}$.

Adversaries are considered passive (honest-but-curious): they follow protocol steps but collude, share all messages, and have complete knowledge of network structure and algorithms. Information-theoretic privacy is formalized as follows: CPPDD mechanisms are $C$-private for an adversarial set $C \subset \mathcal{V}$ if, for any two input vectors $x, x'$ with $x_C = x'_C$ and $\sum_{i \in H} x_i = \sum_{i \in H} x'_i$ (where $H = \mathcal{V} \setminus C$), the adversary's view of protocol execution is identical between $x$ and $x'$ (Gupta et al., 2018). Thus, adversaries learn nothing about honest nodes' individual inputs beyond what is implied by global aggregation.

Privacy is guaranteed against collusion up to size $t$ if the underlying graph is $(t+1)$-connected; that is, no coalition of $t$ nodes forms a vertex cut disconnecting $\mathcal{G}$ (Gupta et al., 2018).

2. Protocol Architecture and Mechanism Composition

A canonical CPPDD protocol comprises two phases:

Phase 1: Distributed Masking—Privacy Subroutine

• Each agent samples and exchanges random field elements with its neighbors, forming pairwise edge masks.
• The agent's "masked input" $\tilde{x}_i$ is computed as its true input plus the sum of signed incoming and outgoing edge masks modulo a large prime $p$.
• Summation-consistency holds: $\sum_{i} \tilde{x}_i = \sum_{i} x_i$ (modulo $p$), and for appropriate parameterization, the aggregate is exact.

Phase 2: Consensus Aggregation—Black-box Consensus

• Agents run any exact, distributed consensus algorithm (e.g., average consensus, gossip, push-sum) using the masked inputs.
• Every agent decodes the aggregate (sum or average) from the output and recovers the desired statistic after modulus reduction and division (Gupta et al., 2018, Liu et al., 2018).

This architecture is modular: the masking phase is agnostic to the consensus routine, enabling scalability and extensibility to diverse aggregation functions beyond simple averaging, including additively separable statistics and vector/matrix data (Gupta et al., 2018).

3. Privacy Guarantees and Connectivity Requirements

The information-theoretic privacy of CPPDD is characterized by the network's vertex connectivity. If adversarial colluders do not form a vertex cut, then their joint view is statistically independent of honest agents' individual inputs, subject only to the sum or average revealed. For a coalition of size $t$, privacy is unconditional provided the graph is $(t+1)$-connected (Gupta et al., 2018).

The core mathematical property is as follows: the distributed masking ensures that the combined random variables observable to any adversarial coalition lie in a high-dimensional subset where honest inputs are statistically indistinguishable, except for their aggregate sum. Consequently, adversaries cannot infer individual data, even with unlimited computational resources (Gupta et al., 2018).

4. Extensions: Additive Functions, Vector Data, and Generalization of CPPDD

CPPDD mechanisms generalize readily:

• Additively separable functions: Agents may apply arbitrary function $f(x_i)$ locally, then mask $\tilde{y}_i = f(x_i) + a_i$ and run consensus on $\{\tilde{y}_i\}$. The protocol reveals only the aggregate $\sum_i f(x_i)$ (Gupta et al., 2018).
• Vector and high-dimensional data: Pairwise random vectors are exchanged and aggregated, and privacy/correctness arguments apply coordinate-wise, provided statistical independence of masks (Gupta et al., 2018).
• Data Distribution: By encoding dataset indicators or histograms as input vectors, CPPDD enables privacy-preserving distribution and aggregation of complex statistics.

5. Correctness, Overhead, and Scalability

CPPDD achieves exact consensus: agents recover the aggregate or average with zero error, in contrast to differential privacy-based mechanisms which inherently trade-off accuracy for privacy (Gupta et al., 2018). The masking phase requires one round of neighbor exchange per edge for random field elements, with total communication overhead scaling linearly with network size/type. The consensus phase's overhead matches the chosen routine, ensuring practical deployment over large, sparse networks.

Because the protocol applies masking only once before consensus, scalability with respect to computation, communication, and memory is maintained, and the method supports integration with any scalable consensus algorithm (Gupta et al., 2018).

6. Comparative Analysis and Trade-offs

CPPDD mechanisms offer statistical (information-theoretic) privacy without degrading result accuracy, assuming sufficient graph connectivity. In differential privacy schemes, accuracy is sacrificed due to additive noise, whereas CPPDD ensures perfect accuracy at the cost of stricter topological requirements.

Scalability is achieved as the masking routine is a local operation and consensus is run over masked values. The overhead is proportional to the number of edges and agents, and practical for large, sparse graphs. The trade-off is strict: privacy is lost if network connectivity drops below threshold (colluders form a vertex cut), but within the guarantees, privacy is unconditional (Gupta et al., 2018).

7. Generality and Applicability Across Distributed Systems

CPPDD frameworks underpin secure aggregation and distribution in sensor networks, federated analytics, distributed optimization, secure voting, and collaborative learning scenarios. The abstraction of privacy via anonymous masking coupled with consensus allows deployment in diverse application environments without the need for trusted third parties or external aggregators.

Mechanisms are easily generalizable to sum-consistent gossip, push-sum, event-triggered quantized consensus, and cryptographic (homomorphic encryption, secret-sharing) settings for both average and more general aggregate consensus objectives (Gupta et al., 2018, Liu et al., 2018, Gao et al., 2018, Ruan et al., 2017, Zhang et al., 2021, Zhang et al., 2020).

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For a full analysis and construction of the protocol, proofs of privacy, and extensions to arbitrary aggregation, see "Information-Theoretic Privacy in Distributed Average Consensus" (Gupta et al., 2018) and related formalizations in consensus and gossip settings (Liu et al., 2018).