The Hamming weight of the non-adjacent-form under various input statistics

The Hamming weight of the non-adjacent form is studied in relation to the Hamming weight of the standard binary expansion. In particular, we investigate the expected Hamming weight of the NAF of a n-digit binary expansion with k ones where k is either fixed or proportional to n. The expected Hamming weight of NAFs of binary expansions with large (≥ n/2) Hamming weight is studied. Finally, the covariance of the Hamming weights of the binary expansion and the NAF is computed. Asymptotically, these Hamming weights become independent and normally distributed.