TY - GEN

T1 - Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

AU - Ballentine, Sean

AU - Guillevic, Aurore

AU - Lorenzo-Garcia, Elisa

AU - Martindale, Chloe

AU - Massierer, Maike

AU - Smith, Benjamin

AU - Top, Jaap

PY - 2017/11/16

Y1 - 2017/11/16

N2 - Schoof’s classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof’s algorithm. While we are currently missing the tools we need to generalize Elkies’ methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this chapter, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.

AB - Schoof’s classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof’s algorithm. While we are currently missing the tools we need to generalize Elkies’ methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this chapter, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.

M3 - Conference Contribution (Conference Proceeding)

SP - 63

EP - 94

BT - Algebraic Geometry for Coding Theory and Cryptography

ER -