Reducing $n$ in $(n,k)$ secret sharing while keeping it effective
This sounds related to Wyner's "wiretap channel": Wyner showed that an asymmetry between receiver and eavesdropper could be used to achieve information-theoretic security. There have been many, many follow-up works since then. 

Reference:
A. D. Wyner, The wire-tap channel. Bell Sys. Tech. J., vol. 54, pp. 1355-1387, 1975.