Abstract
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 \(\mathbb{F}_{q}\) of large characteristic from \({\widetilde{O}}(\log^8 q)\) to \({\widetilde{O}}(\log^5 q)\). Using our algorithm we compute a 256-bit prime-order Jacobian, suitable for cryptographic applications, and also the order of a 1024-bit Jacobian.
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Gaudry, P., Kohel, D., Smith, B. (2011). Counting Points on Genus 2 Curves with Real Multiplication. In: Lee, D.H., Wang, X. (eds) Advances in Cryptology – ASIACRYPT 2011. ASIACRYPT 2011. Lecture Notes in Computer Science, vol 7073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25385-0_27
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