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

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 xline 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 L´opez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to twodimensional 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 logarithmbased 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.