Learning-Augmented PSMM Protocol

• The paper presents the Learning-Augmented PSMM protocol that integrates information-theoretic privacy via polynomial masking with computational efficiency gained from low-rank tensor decompositions.
• It guarantees perfect privacy under a semi-honest model with up to t-1 colluding agents and achieves optimal recovery thresholds through careful algebraic encoding and polynomial interpolation.
• By leveraging learned tensor decompositions, the protocol significantly reduces local computation costs—demonstrating up to an 80% speedup for large matrices—while maintaining rigorous security guarantees.

Learning-Augmented PSMM (Perfectly Secure Collaborative Matrix Multiplication) is a protocol designed for secure multiparty computation (MPC) of matrix products, specifically $A^\top B$ over finite fields $\F$, under strict storage and privacy constraints. The core innovation integrates information-theoretic secrecy based on polynomial-masking techniques with computational speedup through learning-based, low-rank tensor decompositions, yielding substantial improvements in local computation while maintaining perfect security guarantees (He et al., 14 Jan 2026).

1. Problem Framework and Security Model

Given $N$ semi-honest agents, connected to a controller and able to store at most a $1/k$ fraction of each input matrix (i.e., one $m \times (m/k)$ block of $A$ and $B$), the protocol considers the scenario where at most $t-1$ agents may collude. The objective is for a controller to compute $A^\top B$ exactly, ensuring:

• Information-theoretic privacy: Any coalition of up to $t-1$ agents obtains no information about the inputs.
• Local storage constraint: Each agent holds exactly one block of each matrix plus masking randomness.
• Optimal recovery threshold: The number of agents $N$ achieves matching lower bounds for polynomial-sharing-based secure matrix multiplication.

This setting adheres to the standard security definitions in MPC and coded computing, with explicit attention to storage and collusion bounds, and assumes a trusted source and private authenticated channels (He et al., 14 Jan 2026).

2. Algebraic Structure: Polynomial Masking and Coefficient Alignment

Input matrices $A, B \in \F^{m \times m}$ are partitioned into $k$ column blocks:

$A = [A_1~\cdots~A_k],~~B = [B_1~\cdots~B_k],~~A_i, B_j \in \F^{m \times (m/k)}.$

Each block is encoded as a sparse masking polynomial; for example: \begin{align*} g_A(x) &= \sum_{i=1}^k A_i x^{i-1} + \sum_{\ell=1}^{t-1} R^{(A)}_\ell x^{k^2+\ell-1}, \ g_B(x) &= \sum_{j=1}^k B_j x^{k(j-1)} + \sum_{\ell=1}^{t-1} R^{(B)}_\ell x^{k^2+\ell-1}. \end{align*} The “signal support” term contains input blocks; the “masking tail” terms ($R^{(A)}_\ell$, $R^{(B)}_\ell$ drawn uniformly and independently over $\F^{m \times (m/k)}$) ensure information-theoretic security.

For each publicly chosen $\alpha_n \in \F$, agent $n$ receives $g_A(\alpha_n), g_B(\alpha_n)$. Local computation yields

$M(\alpha_n) = g_A(\alpha_n)^\top g_B(\alpha_n),$

which, as a polynomial in $x$, decomposes into coefficients $M_\nu$ such that the indices $\{i-1 + k(j-1)\}$ directly recover the $k^2$ products $A_i^\top B_j$, while all other coefficients are linear combinations involving random masks.

The scheme is reminiscent of Beaver-style MPC but is realized via algebraic encoding: the masking tail effectively plays the role of classical Beaver triples, ensuring that any set of $t-1$ evaluations (i.e., views of colluding agents) reveals zero information about the signals, as formalized by the polynomial masking lemma (He et al., 14 Jan 2026).

3. Recovery Thresholds and Information-Theoretic Privacy

The number of nonzero coefficients in $M(x)$ determines the minimal number of agents required for recovery:

$N^\star(k,t) = \min\{2k^2+2t-3,~k^2+kt+t-2\}.$

By assigning $N \ge N^\star(k,t)$ agents and choosing $\alpha_n$ randomly, the protocol constructs a block-Vandermonde interpolation system of full rank with high probability. This approach guarantees both perfect privacy against up to $t-1$ colluders and optimal recovery, matching known information-theoretic lower bounds for polynomial-sharing protocols, under the given storage and privacy parameters (He et al., 14 Jan 2026).

4. Learning-Augmented Protocol: Tensorization and Low-Rank Methods

The learning-augmented extension—LA-PSMM—replaces each agent’s dense multiplication by a lower-rank, learned tensor decomposition. Conventional local operations require $\mathcal{O}((m/k)^3)$ finite-field operations, which becomes prohibitive for large $m$. Instead, matrices are multiplied in bilinear tensorized form:

$$\operatorname{vec}(C) = \sum_{r=1}^T \langle u_r,\operatorname{vec}(A) \rangle \langle v_r, \operatorname{vec}(B) \rangle w_r,$$

where $T$ is the rank of the decomposition; for Strassen’s method, $T=7$ for $2\times2$ matrix multiplication, while learned decompositions (e.g., via AlphaTensor) achieve larger $T_l \ll (m/k)^3$, enabling scalable reductions in local computation.

The local step thus becomes:

$\hat{M}(\alpha_n) = \sum_{r=1}^{T_l} \langle u_r, \operatorname{vec}(g_A(\alpha_n)) \rangle \langle v_r, \operatorname{vec}(g_B(\alpha_n)) \rangle \mat(w_r),$

with learned $(u_r, v_r, w_r)$ from tensor decomposition, and $T_l$ the learned rank (He et al., 14 Jan 2026).

Operator-invariance: If the local bilinear mapping is exactly equivalent to $g_A(\alpha_n)^\top g_B(\alpha_n)$ for all $\alpha_n$, it preserves the distribution of signal and masking coefficients in $M(x)$, thus does not compromise privacy or the recovery threshold.

5. Protocol Workflow and Computational Complexity

Protocol Workflow

The LA-PSMM protocol consists of the following steps:

Step	Actor	Operation
Partition	Source plant	Split $A,B$ into $k$ blocks
Masking	Source plant	Sample $R^{(A)}_\ell, R^{(B)}_\ell$
Polynomial encoding	Source plant	Form $g_A(x), g_B(x)$
Point selection	Source plant	Choose $\alpha_1,\ldots,\alpha_N$
Share distribution	Source plant	Send $g_A(\alpha_n), g_B(\alpha_n)$
Local multiplication	Agent $n$	Compute $\hat M(\alpha_n)$ via learned expansion
Upload results	Agent $n$	Send $\hat M(\alpha_n)$ to controller
Interpolation	Controller	Reconstruct $A_i^\top B_j$ from block coefficients

Complexity

• PSMM: Encoding involves $N\cdot\mathcal{O}((k + t)m^2/k)$ operations; local multiply scales as $N\cdot\mathcal{O}(m^3/k^2)$ per agent; decoding uses $\mathcal{O}((m/k)^2 N^2)$ (naive) or near-linear (fast) methods.
• LA-PSMM: Encoding/decoding overheads are unchanged. Local multiplication reduces to $N\cdot\mathcal{O}(T_l m^2 / k)$, provided $T_l \ll (m/k)^3$.

Speedup is observed when $T_l \ll m/k$, with empirical results reaching up to $80\%$ local speedup for $m=4096$, and speedup scaling approximately linearly with $m$ (e.g., a $5 \times$ reduction for $m=8192$). This suggests LA-PSMM achieves substantial gains for large matrix dimensions and moderate partition factors (He et al., 14 Jan 2026).

6. Security Analysis and Theoretical Guarantees

The privacy of LA-PSMM is founded on:

• The masking lemma, ensuring that any $t-1$ polynomial evaluations are statistically independent of $A,B$, thus any subset of $t-1$ colluding agents observes fully random, independent shares.
• Operator-invariance, as learned bilinear expansions in LA-PSMM are constructed to be exactly equivalent to the standard multiplication for all field inputs, ensuring that the polynomial masking structure and critical “signal” exponents (those from which $A_i^\top B_j$ are recovered) remain unchanged.
• The recovery threshold remains $N \geq N^\star(k, t)$, with the Vandermonde interpolation problem remaining full-rank due to the polynomial structure and random block selection.

The LA-PSMM protocol thus inherits the perfect privacy, optimal recovery, and correctness guarantees of the original PSMM, while introducing no new vulnerabilities (He et al., 14 Jan 2026).

7. Empirical Evaluation and Implications

Experimental benchmarks, conducted over $\F_p$ for large prime $p$ and square matrices of size $512 \le m \le 8192$, demonstrate:

• For $k = 8$, $t = 4$, and learned rank $T_l \approx 0.1(m/k)$ (using AlphaTensor-style reinforcement learning), LA-PSMM local computation times are reduced to $\sim20\%$ of conventional PSMM for $m = 4096$.
• Speedup, defined as PSMM time divided by LA-PSMM time, increases almost linearly with $m$, reaching approximately $5\times$ for $m=8192$.
• Wall-clock times (excluding network latency) confirm the scalability of LA-PSMM’s local computation cost advantage for increasing matrix dimensions.

A plausible implication is that the practical cost of perfect secrecy can now be substantially reduced in large-scale collaborative or distributed settings, provided suitable low-rank bilinear decompositions are available and exact (He et al., 14 Jan 2026).

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Learning-augmented PSMM synthesizes block-masked, information-theoretically secure MPC protocols with advances in learning-based tensor decompositions. This union enables scalable, perfectly secure collaborative matrix multiplication under strong adversarial models, with computational efficiency improvements that scale with problem size, offering a highly practical primitive for coded computing with robust privacy-preserving guarantees (He et al., 14 Jan 2026).