The use of cloud technologies for processing confidential data requires a solution to the data security problem. One of the mechanisms to solve it is homomorphic encryption. However, homomorphic encryption only allows the arithmetic operations of addition and multiplication to be performed over encrypted numbers. Consequently, when implementing algorithms for matrix algebra, artificial neural networks, and deep learning, it becomes necessary to implement the comparison operation in a homomorphic cipher. In this paper, we study methods for comparison operations in a homomorphic cipher. One of the methods is to use the subtraction of numbers and determine the sign of a number. Two families of polynomials and their composition are used to approximate the sign function. We provide estimates of the accuracy of the sign function approximation obtained as a result of modeling and show that they are 1.98 times better than the state-of-the-art theoretical estimate for polynomials fn(x). Also, we show that the theoretical estimate of gn(x) is not applicable for n ≤ 4.