Scalar multiplication on Koblitz curves can be very efficient due to the elimination of point doublings. Modular reduction of scalars is commonly performed to reduce the length of expansions, and τ-adic Non-Adjacent Form (NAF) can be used to reduce the density. However, such modular reduction can be costly. An alternative to this approach is to use a random τ-adic NAF, but some cryptosystems (e.g. ECDSA) require both the integer and the scalar multiple. This paper presents an efficient method for computing integer equivalents of random τ-adic expansions. The hardware implications are explored, and an efficient hardware implementation is presented. The results suggest significant computational efficiency gains over previously documented methods.