Paper 2015/983 Original: eprint.iacr.org/2015/983 ↗

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

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.