Simultaneous Diagonalization of Incomplete Matrices and Applications

TL;DR

This paper introduces algorithms for recovering diagonal matrices from incomplete samples, with applications in cryptanalysis, notably improving methods for solving the approximate common divisor problem and breaking cryptographic multilinear maps.

Contribution

The paper presents practical algorithms for simultaneous diagonalization of incomplete matrices, tailored to cases with low-rank unknown matrices, advancing cryptanalytic techniques.

Findings

Enhanced algorithms for approximate common divisor problem

Improved methods for breaking cryptographic multilinear maps

Practical solutions depending on low-rank structures

Abstract

We consider the problem of recovering the entries of diagonal matrices $\{U_a\}_a$ for $a = 1,\ldots,t$ from multiple "incomplete" samples $\{W_a\}_a$ of the form $W_a=PU_aQ$, where $P$ and $Q$ are unknown matrices of low rank. We devise practical algorithms for this problem depending on the ranks of $P$ and $Q$. This problem finds its motivation in cryptanalysis: we show how to significantly improve previous algorithms for solving the approximate common divisor problem and breaking CLT13 cryptographic multilinear maps.