Implementation of a Las Vegas Integer Matrix Determinant Algorithm

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Implementation of a Las Vegas Integer Matrix
Determinant Algorithm Arne Storjohann
Pascal Giorgi Zach
Olesh University of Waterloo University of
Waterloo University of Waterloo
Given an input of
output either the determinant of , or false if
the Hermite form of is nontrivial.
Investigate the practicality of high-order
lifting for integer matrix invariant computations.

• Step 1 System Solving
• Step 2 Determinant Reduction
• Step 3 Unimodularity Certification

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HNF
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HNF

• Let be the last row of
• Solve
• denominator of
• Cost

• p 2k where k
• l
• Use high-order lifting to compute
• Only compute where is a power of 2
• Cost

• Replace the last column of
• Use System Solver and the Extended GCD algorithm
• Cost

All tests were run on an Intel Itanium2 at 1.3GHz
with 64GB RAM using GMP 4.1.3 and ATLAS 3.6.0
approximated by timing one prime and
multiplying by the number of primes.