**2. Symmetric Key Encryption Algorithms Based on Numerical Methods**

One of the first published works that consider *symmetric key encryption algorithm based on numerical methods* is by Ghosh in [24] (see also [25,26]), where it is shown that any nonlinear function with one variable *f*(*z*) can be defined as a key. The encryption process then is defined as finding the solution of the equation

$$f(z) - c\_i = 0,\tag{1}$$

where *ci* represents the numerical code of the *i*th symbol in the plaintext (e.g., the ASCII code). The function *f*(*z*) must be chosen in such a way that the corresponding formula (1) has at least one real root for any *i*. Then, the set of roots {*z*<sup>∗</sup> *<sup>i</sup>* } represents the ciphertext. On the receiver side, each entry *z*∗ *<sup>i</sup>* is decoded by substituting it into *f*(*z*) giving rise to the plaintext character *ci* = *f*(*z*<sup>∗</sup> *<sup>i</sup>* ) (the value *f*(*z*∗ *<sup>i</sup>* ) must be appropriately rounded to recover *ci*). In [24], as a key function *f*(*z*), the authors use a cubic polynomial and, for the numerical solution of equations *f*(*z*) − *ci* = 0, they use the Newton's iterative method. We have to mention that, in solving nonlinear Equation (1), we can use different iterative methods. Analogous to this algorithm, an example of a public key cryptosystem based on numerical methods is considered in [27].

It is important to say that the main weaknesses of such algorithms can be summarized in the following:


Our aim in the present work is to develop a new algorithm that solves the disadvantages mentioned above. In order to achieve this, the new scheme will be based on employing numerical iterative methods and rotation–translation formula.
