International Association for Cryptologic Research

International Association
for Cryptologic Research

IACR News item: 12 October 2015

Ping Ngai Chung, Craig Costello, Benjamin Smith
ePrint Report ePrint Report
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 L\\\'opez 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.

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