Counting Points on Genus 2 Curves with Real Multiplication
edited by: DongHoon Lee, Xiaoyun Wang
We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field $\mathbbF_q$ of large characteristic from $\widetildeO(\log^8 q)$ to $\widetildeO(\log^5 q)$ . Using our algorithm we compute a 256-bit prime-order Jacobian, suitable for cryptographic applications, and also the order of a 1024-bit Jacobian.