**4. Proposed Encryption Algorithm Based on Numerical Method and Rotation–Translation Equation**

Here, we describe an encryption algorithm based on space contraction and numerical method. Any nonlinear function or polynomial *f* with one variable can be defined as part of a key.

We consider plaintext *P* with byte length of *L*. The initial values *x*<sup>0</sup> and *y*<sup>0</sup> from Equation (4), and an initial iteration number *M*0, are determined. The rotation–translation formula, Equation (4), is iterated for *M*<sup>0</sup> times.

The proposed algorithm based on numerical method and space contraction is given below:


**Remark 1.** *In Step 2 of Encryption algorithm, it is desirable that the function or polynomial f is such that each equation of the type f*(*z*) − *ci* = <sup>0</sup> *has at least one real root, and it is easy to determine the initial approximation, which guarantees the convergence of the iterative process.*
