(1) School of Computer Science, Shaanxi Normal University, Xi’an, 710062, China (2) Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, China

Yao’s millionaires’ problem is a fundamental problem in secure multiparty computation, and its solutions have become building blocks of many secure multiparty computation solutions. Unfortunately, most protocols for millionaires’ problem are constructed based on public cryptography, and thus are inefficient. Furthermore, all protocols are designed to solve the basic millionaires’ problem, that is, to privately determine which of two natural numbers is greater. If the numbers are real, existing solutions do not directly work. These features limit the extensive application of the existing protocols. This study introduces and refines the first symmetric cryptographic protocol for the basic millionaires’ problem, and then extends the symmetric cryptographic protocol to privately determining which of two real numbers is greater, which are called the extended millionaires’ problem, and proposes corresponding protocols. We further prove, by a well accepted simulation paradigm, that these protocols are private. Constructed based on symmetric cryptography, these protocols are very efficient.