Coded Caching is an efficient technique to reduce peak hour network traffic. One limitation of known coded caching schemes is that the demands of all users are revealed to their peers in the delivery phase. Schemes that assure privacy for user demands are studied in recent past. Assuming that the users are equipped with caches of small memory sizes, the achievable rate under demand privacy constraints is investigated in this work. We present an MDS code based demand private coded caching scheme with $K$ users and $N$ files that achieves a memory rate pair $left(frac{1}{K(N-1)+1},Nleft(1-frac{1}{K(N-1)+1}right)right)$. The presented memory-rate pair meets the lower bound under demand-privacy requirements, proposed by Yan textit{et al.} in the recent work cite{c13}. By memory sharing this characterizes the exact rate-memory trade-off for the demand private coded caching scheme for cache memory $Min left[0,frac{1}{K(N-1)+1}right]$.
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