## Cryptology ePrint Archive: Report 2015/983

Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

Ping Ngai Chung and Craig Costello and Benjamin Smith

Abstract: We give a general framework for uniform, constant-time one- and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus~2 curves that operate by projecting to the $x$-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper signed'' output back on the curve or Jacobian. This extends the work of López and Dahab, Okeya and Sakurai, and Brier and Joye to genus~2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre- and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus~2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic.

Category / Keywords: implementation / elliptic curve cryptography, hyperelliptic curve cryptography, scalar multiplication, signatures

Date: received 11 Oct 2015, last revised 19 Oct 2015

Contact author: smith at lix polytechnique fr

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2015/983

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