# Computing the invariant structure of integer matrices: fast algorithms into practice

@inproceedings{Pauderis2013ComputingTI, title={Computing the invariant structure of integer matrices: fast algorithms into practice}, author={Colton Pauderis and Arne Storjohann}, booktitle={ISSAC '13}, year={2013} }

We present a new heuristic algorithm for computing the determinant of a nonsingular n x n integer matrix. Extensive empirical results from a highly optimized implementation show the running time grows approximately as n3 log n, even for input matrices with a highly nontrivial Smith invariant structure. We extend the algorithm to compute the Hermite form of the input matrix. Both the determinant and Hermite form algorithm certify correctness of the computed results.

## 6 Citations

Fast heuristic algorithms for computing relations in the class group of a quadratic order, with applications to isogeny evaluation

- Mathematics, Computer Science
- 2016

Novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded are presented.

Deterministic Unimodularity Certification and Applications for Integer Matrices

- Computer Science, Mathematics
- 2013

A deterministic method — “double-plus-one” lifting — is presented to compute the highorder residue R as well as a succinct representation of B to give a heuristic, but certified, algorithm for computing the determinant and Hermite normal form of a square, nonsingular integer matrix.

Common Factors in Fraction-Free Matrix Decompositions

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It is shown that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors in theLUandQRmatrix decompositions using exact computations.

Matrix factoring by fraction-free reduction

- Computer ScienceArXiv
- 2016

It is shown that existing fraction-free QR (Gram-Schmidt) algorithms create a common factor in the last column of Q, which relates the existence of row factors in LU decomposition to factors appearing in the Smith normal form of the matrix.

Enhancing Goldreich, Goldwasser and Halevi’s scheme with intersecting lattices

- Computer Science, MathematicsJ. Math. Cryptol.
- 2019

This work aims to present a new technique to enhance the security of the Goldreich, Goldwasser and Halevi (GGH) scheme by modifying the public key which hides the structure of the corresponding private key.

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