PSO Security: Protocols, Privacy & Optimization

• PSO Security is a set of techniques defining cryptographic protocols for private set operations, enabling secure multi-party computations.
• Modern PSO protocols leverage simulation-based proofs, zero-sharing, and composable methods to achieve efficient and robust privacy guarantees.
• In optimization, PSO methods integrate penalty functions and discrete controls to meet operational constraints in power systems while balancing computational costs.

PSO Security refers to a family of security notions and techniques relating to private set operations (PSO), predicate singling-out (PSO) security in privacy-preserving data release, and Particle Swarm Optimization (PSO) in the context of enforcing operational security constraints. The term thus encompasses methodology from cryptographic protocol design, privacy bounds in generative synthetic data, and optimization under security constraints, each with specific, rigorous formulations. The following sections provide a technical synthesis spanning multi-party cryptographic frameworks, advanced notions of privacy in data generation, and security guarantees in network optimization.

1. Foundations: Private Set Operations and Security Models

Private Set Operations (PSO) protocols enable two or more parties, each holding a private set, to compute set-theoretic functions (e.g., intersection, union) without revealing any further information except what is deducible from the output. The security model generally adopted is simulation-based semi-honest security: for any coalition of parties, there exists a PPT simulator whose output is indistinguishable from the coalition’s joint real transcript and outputs, given only their private inputs and legitimate outputs. Formal definitions require that, for PSO protocols $\Pi$ computing $f$ on inputs $x=(x_1,\dots,x_m)$, there exists a simulator $\mathrm{Sim}$ such that

$$\left\{\mathrm{Sim}(A, x_A, f_A(x), f(x))\right\}_x \approx_c \left\{\mathsf{View}_A^\Pi(x), \mathsf{Output}_A^\Pi(x)\right\}_x$$

for all coalitions $A$ (Dong et al., 10 Apr 2025).

Security notions further refine the output granularity: classic PSO protocols allow all participants to obtain the function output, while external decider models yield the output exclusively to a designated decider party, with formal definitions ensuring that all other parties—including the decider—learn nothing beyond their entitled outputs (Ramezanian et al., 2021).

2. Modern PSO Protocols: Composability, Flexibility, and Efficiency

Recent work achieves general frameworks for Multi-Party Private Set Operations (MPSO), extending functionality beyond intersection and union to arbitrary secure set formulas, including difference and complex predicates. The core primitive of (Dong et al., 10 Apr 2025) is predicative zero-sharing (PZS): for parties with inputs $x_i$, a Boolean predicate $Q(x_1,\dots,x_m)$ is evaluated such that each party receives a field element $s_i$ satisfying

$$\sum_{i=1}^m s_i = \begin{cases} 0, & \text{if } Q(x)=1\ \text{uniform in } \mathbb{F}, & \text{if } Q(x)=0 \end{cases}$$

Compound predicates (arbitrary Boolean formulas in literals) are handled by composing “simple” PZS instances with standard secret-sharing operations (additions for AND, Beaver triples for OR).

The protocol composes

• Hashing-to-bins (e.g., Cuckoo hashing) for input randomization,
• Batched membership zero-sharing for parallelized set-operations,
• Robust secret-shared shuffles to prevent information leakage via ordering,
• Explicit reconstruction of shared set-memberships or (optionally) summary statistics such as cardinalities.

These protocols, instantiated with symmetric-key cryptographic primitives, achieve optimal $O(mn)$ (where $m$ is the number of parties, $n$ is per-party set size) computational and communication complexity for MPSI, MPSI-card, and related variants, matching the insecure baseline and surpassing prior solutions in both efficiency and composability.

Empirically and theoretically, such MPSO frameworks circumvent the need for generic MPC and outperform public-key-based solutions in both leader and client cost (Dong et al., 10 Apr 2025).

3. Predicate Singling-Out (PSO) Security in Synthetic Data

A distinct notion, Predicate Singling-Out (PSO) security, addresses privacy in synthetic data generation. Introduced to formalize GDPR requirements, PSO security demands that no adversary (even computationally unbounded) upon observing a released dataset $S$ can construct any predicate $p$ that (a) holds for some real individual $x$ and (b) uniquely identifies an element of $S$ as stemming from $x$ (i.e., $p(x)=1$ and $p(x')=0$ for all $x'\neq x$) (Dombrowski et al., 22 Jun 2025).

Formally, for a generator $M$ mapping datasets $D$ to synthetic sets $S=M(D)$, and adversary $\mathcal{A}$ outputting $(p, i)$ after observing $S$, the PSO advantage is

$$\mathsf{Adv}_{\mathrm{PSO}^M(\mathcal{A})} = \Pr \left[ \exists i: p(D_i)=1\ \land\ |\{s \in S : p(s)=1\}| = 1 \right]$$

and $M$ is $\delta$–PSO secure if this probability is bounded by $\delta$ for all adversaries.

Operationalizing this in synthetic medical data, (Dombrowski et al., 22 Jun 2025) enforces diversity-aware sampling to prevent mode collapse, uses pseudo-conditional feature extraction, and filters samples via a re-identification model $\mathcal{P}(x,x')$ to empirically limit the PSO advantage. Image Retrieval Score (IRS) quantifies the diversity and thus the expected PSO security, while empirical evaluations report downstream model AUC gaps under 1 percentage point compared to real data, and re-identification AUC of 0.96, indicating practical PSO resistance.

4. Security Analysis and Proofs: Simulation Paradigm

Security proofs for PSO and MPSO protocols are generally simulation-based. For the case of external decider protocols (Ramezanian et al., 2021), proof strategies distinguish between corruptions of a subset of parties $T \subseteq \{P_1, ..., P_n, D\}$:

• If $T$ contains only parties $P_i$, the simulator generates random ciphertexts for unknown elements, plants zeros only where prescribed by the output, and leverages semantic security of encryptions.
• If $T$ consists of the decider $D$, the simulator produces fresh, indistinguishable encryptions of zeros and non-zeros based solely on the function output—guaranteeing that nothing is leaked except the legitimate result.

For MPSO based on predicative zero-sharing (Dong et al., 10 Apr 2025), hybrid arguments demonstrate that replacement of actual protocol invocations by their ideal functionalities does not alter adversary view distributions except with negligible probability. Privacy, correctness, and independence properties of compound PZS suffice to guarantee UC (Universally Composable) security with information-theoretic or computational guarantees depending on the primitives used.

5. Computational and Communication Complexity

The complexity characteristics of PSO protocols critically depend on the protocol family and functionality supported:

Protocol Variant	Computation per Client	Communication per Client	Leader Cost
MPSI, MPSI-card (Dong et al., 10 Apr 2025)	$O(n)$	$O(n)$	$O(mn)$
MPSI-card-sum	$O(n)$	$O(n)$	$O(mn)$
MPSU, MPSU-card	$O(mn)$	$O(mn)$	$O(m^2n)$
HE-based PSO (Ramezanian et al., 2021)	$O(u)$	$O(u \cdot \lambda)$	$O(u)$ decryptions

Here $n$ is set size per client, $m$ number of parties, $u$ universe size, $\lambda$ the security parameter. Notably, batched symmetric-key instantiations avoid costly public-key operations and achieve communication matching the secret-sharing lower bound.

Limitations arise in universality (hash-based schemes often do not reveal element identity) and, for public-key protocols, feasibility when $u$ becomes large. For fully malicious security, zero-knowledge proofs and repetition-based consistency checks are necessary, but at substantially increased cost (Ramezanian et al., 2021).

6. PSO in Security-Constrained Optimization

In a separate context, Particle Swarm Optimization (PSO) has been applied to Security Constrained Optimal Power Flow (SCOPF) problems. Here, "security" pertains to operational feasibility under system constraints such as voltage, current, and dispatch bounds, not cryptographic privacy (Sarstedt et al., 2021). PSO enables direct handling of in-phase and quadrature transformer tap changes, incorporates discrete and continuous control spaces, and enforces constraints via penalty functions added to the fitness objective: $$f(\mathbf{x}_i) = c_{\rm loss} P_{\rm loss}(\underline{\mathbf v}_N) + \mathbf{c}_N^{T}\Delta \mathbf{p}_N + g(\mathbf{x}_i)$$ with

$$g(\mathbf{x}) = \sum_k \alpha_V \max\{0, |v_k| - v_{\max}\} + \sum_\ell \alpha_I \max\{0, |i_\ell| - i_{\max}\}$$

Adapting PSO with discrete rounding for tap steps, boundary "bounce back," velocity clamping, random mutation, and parallel independent runs, the method demonstrates robust search and constraint satisfaction. However, guaranteeing N-1 security (contingency-resilient feasibility) requires explicit inclusion of all scenario penalties in the fitness function. PSO achieves competitive loss minimization with deterministic SQCQP solvers, with the advantage of computation time scaling less dramatically with added discrete variables. Nevertheless, stochastic scatter in results and difficulty in ensuring hard-constraint satisfaction under all contingencies are noted limitations.

7. Limitations, Applications, and Distinctions

Key distinctions and open challenges across PSO security domains are as follows:

• In cryptographic MPSO, universality and flexibility have been attained with theoretically optimal efficiency under standard semi-honest corruption. However, universes of arbitrary size and elements of high entropy remain computationally intensive or limit outputs to cardinality/emptiness.
• PSO security for synthetic data addresses a core privacy concern unaddressed by classical differential privacy, yet its empirical and heuristic nature (IRS-based) relies on quality of re-identification filters and may not fully eliminate predicate-based adversaries unless underlying model risks are fully accounted for.
• In the context of constrained optimization, "security" is unrelated to privacy or adversarial information leakage, but instead addresses resilience and guaranteed operational bounds in complex, hybrid-integer systems.

Applications span privacy-preserving statistical analysis in healthcare, privacy-compliant data release under regulatory regimes, secure multi-party collaborations, privacy-filtered machine learning on synthetic corpora, and resilient power systems optimization. Limitations include communication blowup in large-universe protocols, the need for advanced zero-knowledge techniques for malicious-security upgrades, and practical challenges in setting empirically meaningful privacy parameters in synthetic data scenarios.

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References:

• "Multi-Party Private Set Operations from Predicative Zero-Sharing" (Dong et al., 10 Apr 2025)
• "Multi-party Private Set Operations with an External Decider" (Ramezanian et al., 2021)
• "Enabling PSO-Secure Synthetic Data Sharing Using Diversity-Aware Diffusion Models" (Dombrowski et al., 22 Jun 2025)
• "Comparison of Convexificated SQCQP and PSO for the Optimal Transmission System Operation based on Incremental In-Phase and Quadrature Voltage Controlled Transformers" (Sarstedt et al., 2021)