This paper discusses representations for computation on non-supersingular elliptic curves over binary fields, where computations are performed on the $x$-coordinates only. We discuss existing methods and present a new one, giving rise to a faster addition routine than previous Montgomery-representations. As a result a double exponentiation routine is described that requires 8.5 field multiplications per exponent bit, but that does not allow easy $y$-coordinate recovery. For comparison, we also give a brief update of the survey by Hankerson et al. and conclude that, for non-constrained devices, using a Montgomery-representation is slower for both single and double exponentiation than projective methods with $y$-coordinate.