Solving norm equations over global function fields using compact representations
Date
2023-12-13
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Abstract
This thesis provides contributions to improving the efficiency of solving norm equations over global function fields F and providing theoretical and empirical evidence of the improvement. We present two new algorithms for solving norm equations over F; one is an exhaustive search algorithm inspired by the sole existing method in [21], and the other is an algorithm using principal ideal testing via index calculus. We employed compact representations in both of the novel algorithms to represent solutions in less space and also to improve their efficiency. In order to compare the asymptotic and practical efficiency of the novel algorithms to the existing one, we performed rigorous complexity analyses and empirical analyses of the algorithms. We provided the asymptotic complexities of the algorithm as the number of bit operations required. We also analyzed for the first time the complexity of the algorithm for computing compact representations in [15] as a part of the complexity analyses of the novel algorithms. The asymptotic complexities of both of our novel algorithms were reduced by double exponential factors in the size of F, when compared to the existing algorithm in [21]. The new algorithm with principal ideal testing requires even fewer bit operations than the new exhaustive search algorithm in terms of the size of F. Empirical testing results demonstrate our novel algorithms are significantly faster than the existing one in practice as expected from the complexity analyses.
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Keywords
global function fields, compact representations, Diophantine equations, norm equations, principal ideal tests
Citation
Leem, S. (2023). Solving norm equations over global function fields using compact representations (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.