Near Optimal Rate Homomorphic Encryption for Branching Programs

We initiate the study of good rate homomorphic encryption schemes. Based on previous work on securely evaluating (binary I/O) branching programs, we propose a leveled homomorphic encryption scheme for large-output polynomial-size branching programs (which we call LBP) that possesses near optimal-rate. The rate analysis of the new scheme is intricate: the best rate is achieved if a certain parameter s is set equal to the only positive root of a degree-m polynomial, where m is the length of the branching program. We employ the Newton-Puiseux algorithm to find a Puiseux series for this parameter, and based on this, propose a Θ(logm)-time algorithm to find an integer approximation to s. We also describe a rate-optimal 1-out-of-n CPIR based on rate-optimal homomorphic encryption. In concrete terms, when applied to say, a movie database with n = 2 elements of ` = 3.8 · 10bits, the client can privately download a movie with a communication rate of almost 0.99, hence sacrificing only about 1% of bandwidth for privacy. We also analyze the optimality of the rate efficiency of our scheme in a novel model that may be of independent interest. Our 1-out-of-n CPIR has rate 1 − 1.72 √ k/` · log2 n + O`(`), while we show that no black-box construction surpasses 1− √ k/`(logn/ log logn)+O`(` −1) in terms of rate, where ` is the length of the database elements and k the security parameter.