Base extension in residue number systems with low space
By a result of Chiu, Davida, and Litow [1], improved by Hesse, Allender, and Barrington [2], Chinese remainder representation base extension is computable in logarithmic space, and in fact in DLOGTIME-uniform $\mathrm{TC}^0$, when all the $p_i$ are “small” (i.e., given in unary). Since the effective size of the input is here $p^*+\sum_ip_i\le p^*+m\max_ip_i\le p^*+(\max_ip_i)^2$ (using the fact that the $p_i$ are distinct), this means it is computable in space
$$O\bigl(\max\{\log p^*,\log p_i:i=1,\dots,m\}\bigr).$$

References:

[1] A. Chiu, G. Davida, B. Litow, Division in logspace-uniform $\mathit{NC}^1$, RAIRO – Theoretical Informatics and Applications 35 (2001), no. 3, pp. 259–275.

[2] W. Hesse, E. Allender, D. Mix Barrington, Uniform constant-depth threshold circuits for division and iterated multiplication, Journal of Computer and System Sciences 65 (2002), no. 4, pp. 695–716.