Abstract. This paper studies τ -adic expansions of scalars, which are important in the design of scalar multiplication algorithms for Koblitz Curves, but are also less understood than their binary counterparts. At Crypto '97 Solinas introduced the width-w τ -adic non-adjacent form for use with Koblitz curves. It is an expansion of integers z = P ℓ i=0 ziτ i , where τ is a quadratic integer depending on the curve, such that zi = 0 implies zw+i−1 = . . . = zi+1 = 0, like the sliding window binary recodings of integers. We show that the digit sets described by Solinas, formed by elements of minimal norm in their residue classes, are uniquely determined. However, unlike for binary representations, syntactic constraints do not necessarily imply minimality of weight. Digit sets that permit recoding of all inputs are characterized, thus extending the line of research begun by Muir and Stinson at SAC 2003 to the Koblitz Curve setting. Two new digit sets are introduced with useful properties; one set makes precomputations easier, the second set is suitable for low-memory applications, generalising an approach started by Avanzi, Ciet, and Sica at PKC 2004 and continued by several authors since, including Okeya, Takagi and Vuillaume. Results by Solinas, and by Blake, Murty, and Xu are generalized. Termination, optimality, and cryptographic applications are considered. The most important application is the ability to perform arbitrary windowed scalar multiplication on Koblitz curves without storing any precomputations first, thus reducing memory storage to just one point and the scalar itself.