Wanted: Mentor for Project with Hyperelliptic Curves

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Simon Schwarz

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Apr 7, 2021, 5:30:49 PM4/7/21

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Hello all,

my name is Simon and I am currently a postgraduate student in Computer Science at the University of Cambridge.

As a GSoC 2021 project, I would be interested in extending SageMath's implementation of Jacobians of hyperelliptic curves. In particular, I'd like to add point counting methods over finite fields (relevant paper) and implement the Weil pairing (and possibly additional pairings) on hyperelliptic curves (relevant paper).

Note that SageMath already supports those over elliptic curves, but currently lacks support for hyperelliptic curves. In contrast, Magma already includes those methods (see Magma documentation here), so I think it would be a great addition if this functionality is also supported by SageMath.

If anybody can imagine being a mentor for this project please get in touch with me so we can discuss this project a bit further. Likewise, if you have any other questions do not hesitate to contact me.

Best regards,

Simon

David Roe

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Mar 29, 2022, 5:10:25 PM3/29/22

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I'm probably not able to mentor this summer, but I will note that Sage does include point counting on hyperelliptic curves using p-adic methods (https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/hyperelliptic_curves/hyperelliptic_finite_field.html). Certainly there's always more that can be done in terms of optimization.

There is an old ticket related to optimized arithmetic in Jacobians of hyperelliptic curves (https://trac.sagemath.org/ticket/23154), one on exposing the matrix of Frobenius from Pari (https://trac.sagemath.org/ticket/20309) and some related bugs (https://trac.sagemath.org/ticket/32024) that may be interesting to look at.