  The development of Internet technology enables an analysis on the whole
population rather than a certain number of samples, and leads to increasing
requirement for privacy protection. Local differential privacy (LDP) is an
effective standard of privacy measurement; however, its large variance of mean
estimation causes challenges in application. To address this problem, this
paper presents a new LDP approach, an improved Christofides mechanism.
  It compared four statistical survey methods for conducting surveys on
sensitive topics -- modified Warner, Simmons, Christofides, and the improved
Christofides mechanism. Specifically, Warner, Simmons and Christofides
mechanisms have been modified to draw a sample from the population without
replacement, to decrease variance. Furthermore, by drawing cards without
replacement based on modified Christofides mechanism, we introduce a new
mechanism called the improved Christofides mechanism, which is found to have
the smallest variance under certain assumption when using LDP as a measurement
of privacy leakage. The assumption is do satisfied usually in the real world.
Actually, we decrease the variance to 28.7% of modified Christofides
mechanism's variance in our experiment based on the HCOVANY dataset -- a real
world dataset of IPUMS USA. This means our method gets a more accurate estimate
by using LDP as a measurement of privacy leakage. This is the first time the
improved Christofides mechanism is proposed for LDP framework based on
comparative analysis of four mechanisms using LDP as the same measurement of
privacy leakage.
