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Explicit 2-power torsion of genus 2 curves over finite fields

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  • We give an efficient explicit algorithm to find the structure and generators of the maximal 2-subgroup of the Jacobian of a genus 2 curve over a finite field of odd characteristic. We use the 2-torsion points as seeds to successively perform a chain of halvings to find divisors of increasing 2-power order. The halving loop requires a solution to certain degree 16 polynomials over the base field, and the termination of the algorithm is based on the description of the graph structure of the maximal 2-subgroup. The structure of our algorithm is the natural extension of the even characteristic case.
    Mathematics Subject Classification: Primary: 11G20, 14H40; Secondary: 14H55.


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