Combinatorial Privacy: Private Multi-Party Bitstream Grand Sum by Hiding in Birkhoff Polytopes

A recent line of work initiated by Chiesa and Gur and further developed by Herman and Rothblum investigates the sample and communication complexity of verifying properties of distributions with the assistance of a powerful, knowledgeable, but untrusted prover. In this work, we initiate the study of differentially private distribution property verification. After all, if we do not trust the prover to help us with verification, why should we trust it with our sensitive sample? We map a landscape of differentially private verification of properties of distributions. In the non-private case it is known that one-round private-coin protocols can have substantially lower complexity than public-coin (AM) protocols. In contrast, the possibility for improvement in differentially private interactive proofs depends on the privacy parameter regime and model. Drawing on connections between privacy and replicability and privacy amplification techniques in the literature we show: 1. There exists a reduction from any one-round $(\varepsilon,δ)$-differentially private private-coin protocol to a differentially private AM protocol for the parameter regime $\varepsilon = O(1/\sqrt{s})$ and $δ= O(1/s^{5/2})$ with the same privacy and sample and communication complexities. In the local model, this is relaxed to $\varepsilon = O(1/\sqrt{\log s})$ 2. However, when the privacy guarantee is very relaxed ($\varepsilon \in Ω(\log s)$), private coins indeed reduce sample and communication complexities. We also obtain a computationally efficient Merlin-Arthur proof for privately testing whether samples are drawn from a product distribution and prove that its sample complexity is optimal up to a $polylog N$ factor by reducing uniformity testing to independence testing with Boolean attributes and appealing to known lower bounds on sample complexity for private uniformity testing.

In decentralized learning, clients train a machine learning model while keeping their data locally and share their model parameters or gradients with a set of neighbors, making it vulnerable to both privacy attacks and model poisoning attacks. Jointly addressing privacy and Byzantine behavior in decentralized learning is challenging. Indeed, Differential Privacy (DP) applied on the client side is known to increase the variance of honest model updates, which weakens Byzantine-robust aggregation against poisoning attacks. Distributed DP can mitigate this issue by adding noise after aggregation, but implementing it without a trusted curator requires both confidential and Byzantine-resilient computation. Recent work addresses this challenge using secure multi-party computation (MPC) to implement robust aggregation while preserving input confidentiality. However, these solutions scale badly: they either require all-to-all communication between participants or delegate the entire computation to a small subset of parties whose computational and communication cost grows with network size. In this paper, we present Giskard, a scalable protocol for private and Byzantine-robust decentralized aggregation. Giskard organizes $n$ parties into a tree of committees of size $O(\log n)$ and evaluates a differentially private coordinate-wise approximate median via a committee-adapted secure quantile search over the value domain, using MPC within each committee. We assess Giskard both theoretically by proving its security and privacy properties and empirically through extensive experiments involving up to one million participants. Compared to its closest competitors, Giskard reduces per-party communication complexity by close to three orders of magnitude over the closest competitor while exhibiting comparable robustness under up to $n/4$ Byzantine parties.

Privacy-preserving aggregation is a cornerstone for AI systems that learn from distributed data without exposing individual records, especially in federated learning and telemetry. Existing two-server protocols (e.g., Prio and successors) set a practical baseline by validating inputs while preventing any single party from learning users' values, but they impose symmetric costs on both servers and communication that scales with the per-client input dimension $L$. Modern learning tasks routinely involve dimensionalities $L$ in the tens to hundreds of millions of model parameters. We present TAPAS, a two-server asymmetric private aggregation scheme that addresses these limitations along four dimensions: (i) no trusted setup or preprocessing, (ii) server-side communication that is independent of $L$ (iii) post-quantum security based solely on standard lattice assumptions (LWE, SIS), and (iv) stronger robustness with identifiable abort and full malicious security for the servers. A key design choice is intentional asymmetry: one server bears the $O(L)$ aggregation and verification work, while the other operates as a lightweight facilitator with computation independent of $L$. This reduces total cost, enables the secondary server to run on commodity hardware, and strengthens the non-collusion assumption of the servers. One of our main contributions is a suite of new and efficient lattice-based zero-knowledge proofs; to our knowledge, we are the first to establish privacy and correctness with identifiable abort in the two-server setting.