Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval

We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position, needs code length $m=exp\left(\Omega\left(\frac{n}{2^{b}{\sum_{i=0}^{b}}(^{l}_{i})}\right)\right)$ Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries, where the user only needs b bits from each of the two l-bit answers, unknown to the servers, satisfies $t=\Omega \left(\frac{n}{2^{b}\sum_{i=0}^{b}(^{l}_{i})}\right)$ This implies that several known PIR schemes are close to optimal. Our results generalize those of Goldreich et al. [8], who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf [12], our classical bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2-input function can be computed from a quantum superposition of both inputs.