An efficient countermeasure against fault attacks for a right-to-left binary exponentiation algorithm was proposed by Boscher, Naciri and Prouff (WISTP, 2007). This countermeasure was later generalized by Baek (Int. J. Inf. Sec., 2010) to the 2w-ary right-to-left algorithms for any $w \geqslant 1$ (the case w = 1 corresponding to the method of Boscher, Naciri and Prouff). In this paper, we modify theses algorithms, devise new coherence relations for error detection, and reduce the memory requirements without sacrificing the performance or the security. In particular, a full register (in working memory) can be gained compared to previous implementations. As a consequence, the implementations described in this paper are particularly well suited to applications for which memory is a premium. This includes smart-card implementations of exponentiation-based cryptosystems.