How to Use Koblitz Curves on Small Devices?

Koblitz curves allow very efficient scalar multiplications because point doublings can be traded for cheap Frobenius endomorphisms by representing the scalar as a τ -adic expansion. Typically elliptic curve cryptosystems, such as ECDSA, also require the scalar as an integer. This results in a need for conversions between integers and the τ -adic domain, which are costly and prevent from using Koblitz curves on very constrained devices, such as RFID tags or wireless sensors. In this paper, we provide a solution to this problem by showing how complete cryptographic processes, such as ECDSA signing, can be completed in the τ -adic domain with very few resources, consequently outsourcing the expensive conversions to a more powerful party. We also provide small circuitries that require about 76 gate equivalents on 0.13μm CMOS and that are applicable for all Koblitz curves.