On Single Server Private Information Retrieval With Private Coded Side Information

Motivated by an open problem and a conjecture, this work studies the of single server private information retrieval with coded side (PIR-PCSI) that was recently introduced Heidarzadeh et al. The goal PIR-PCSI is to allow user efficiently retrieve desired message ${W}_{{\theta }}$ , which one notation="LaTeX">$K$ independent messages are stored at server, while utilizing linear combination uniformly chosen size- notation="LaTeX">$M$ subset ( notation="LaTeX">${\mathcal {S}}\subset [K]$ ) messages. settings PIR-PCSI-I PIR-PCSI-II correspond constraints notation="LaTeX">${\theta }$ generated from notation="LaTeX">$[K]\setminus {\mathcal {S}}$ notation="LaTeX">$ respectively. In each case, notation="LaTeX">$({\theta }, {S}})$ must be kept server. capacity defined as supremum over field sizes, achievable rates (number bits retrieved per bit download) characterized for in general, notation="LaTeX">$M>(K+1)/2$ notation="LaTeX">$(K-M+1)^{-1}$ . For notation="LaTeX">$2\leq M\leq (K+1)/2$ remains open, it conjectured even case We show equal notation="LaTeX">$2/K$ notation="LaTeX">$2 \leq M \frac {K+1}{2}$ strictly larger than value, does not depend on within parameter regime. Remarkably, half side-information found redundant. also characterize infimum (infimum fields instead supremum), coefficients. results generalized notation="LaTeX">$\theta \in [K]\setminus \mathcal {S}$ settings.