Welcome to the resource topic for
**2024/508**

**Title:**

Secure Multi-Party Linear Algebra with Perfect Correctness

**Authors:**
Jules Maire, Damien Vergnaud

**Abstract:**

We present new secure multi-party computation protocols for linear algebra over a finite field, which improve the state-of-the-art in terms of security. We look at the case of \emph{unconditional security with perfect correctness}, i.e., information-theoretic security without errors. We notably propose an expected constant-round protocol for solving systems of m linear equations in n variables over \mathbb{F}_q with expected complexity O(k(n^{2.5} + m^{2.5}+n^2m^{0.5})) where k > m(m+n)+1 (complexity is measured in terms of the number of secure multiplications required). The previous proposals were not error-free: known protocols can indeed fail and thus reveal information with probability \Omega(\textsf{poly}(m)/q).

Our protocols are simple and rely on existing computer-algebra techniques, notably the Preparata-Sarwate algorithm, a simple but poorly known ``baby-step giant-step’’ method for computing the characteristic polynomial of a matrix, and techniques due to Mulmuley for error-free linear algebra in positive characteristic.

**ePrint:**
https://eprint.iacr.org/2024/508

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