Solving p-adic polynomial systems via iterative eigenvector algorithms

Abstract

In this article, we describe an implementation of a polynomial system solver to compute the approximate solutions of a 0-dimensional polynomial system with finite precision p-adic arithmetic. We also describe an improvement to an algorithm of Caruso, Roe, and Vaccon for calculating the eigenvalues and eigenvectors of a p-adic matrix.

Super-linear convergence in the p-adic QR-algorithm

“…This is useful in practical cryptography to select curves with good properties for cryptosystems. Another application is in solving a 0-dimensional system of polynomial equations over Q p [Kul20,BL12].…”

“…We next show how to improve the iterative computation of the block Schur form introduced in [Kul20]. Our main theorem is: Theorem (5.1.1).…”

“…If the characteristic polynomial of M modulo p is square-free and factors completely then Algorithm 5.1 computes a Schur form T and a matrix U ∈ GL n (Z p ) such that M U = U T + O(p N ) in at most If the assumptions of the theorem above are not met, our algorithm will attempt to use the accelerated convergence strategy anyway. The timings in Section 6 demonstrate a significant improvement over both the classical method and the basic QR-iteration in [Kul20] in computing a weak block Schur form (see Definition 5.0.1), even when the hypothesis on the characteristic polynomial is not satisfied. Furthermore, the slowdown of convergence can be detected dynamically in Algorithm 4.3.…”

“…Furthermore, the slowdown of convergence can be detected dynamically in Algorithm 4.3. Should this occur we fallback to running the QR-iteration with linear convergence, as in [Kul20].…”

“…We state some basic definitions for our discourse. We follow [Kul20] for terminology, and direct the reader to [CRV15, Car17, Ked10] for more details. We can identify a subgroup of GL n (Q p ) where every matrix is well-conditioned, serving the analogous role to O n (R) in the real setting.…”