Secure multi-party computation (SMC) is an important research direction of cryptography. In this paper

we study the secure computation of intervals. Using the Paillier encryption scheme

we design the protocols of relationship between an interval and a point (or an interval). Firstly

the outputs of protocols are ciphertexts. If we extend it to rational intervals

the protocols are safer and more efficient than existing protocols. And then

we study the multi-dimensional problems

that is

the threshold problems of multiple points (or intervals) and intervals

which are new problems in SMC. Since the outputs of the basic protocols are ciphertexts

the multi-dimensional problem protocols are more secure. We strictly prove the security of the protocols using the simulation paradigm method

analyze and demonstrate the efficiency of the protocols through experiments

and compare with the related work to illustrate that the protocols are efficient.