An improved negative binomial approximation with high accuracy to the negative hypergeometric probability for order-preserving encryption

Encryption is a well established technology to sensitive data. However, once encrypted, data can no longer be easily range queried or indexed efficiently. Order-preserving symmetric encryption (OPE) helps with this problem whose encryption function preserves numerical ordering of the plaintexts. It supports efficient logarithmic-complexity search over encrypted content without sacrificing security. In Eurocrypt 2009, the connection between OPE and the negative hypergeometric (NHG) distribution was given by Boldyreva et al. and the authors pointed out that an efficient sampling algorithm for the NHG distribution was required when designing a practical OPE scheme. However, the existence of such an algorithm is open for it needs a high accuracy approximation to the NHG distribution. By using the Stirling's formula, we develop a new negative binomial (NB) approximation to the NHG distribution and introduce a series of correction factors to this approximation to improve its accuracy. In addition, a result of the NB approximation to the NHG distribution is given in terms of the variation distance. It is the first time to quantify the quality of the NB approximation for fixed parameters. Some numerical examples are presented to illustrate the results obtained.