Polyadic Encryption

Abstract

A novel original procedure of encryption/decryption based on the polyadic algebraic structures and on signal processing methods is proposed. First, we use signals with integer amplitudes to send information. Then, we use polyadic techniques to transfer the plaintext into series of special integers. The receiver restores the plaintext using special rules and systems of equations.

1. Introduction

We propose a new approach to transfer hidden information in (continuous-time discrete-valued) signal processing (see, e.g., [1,2]) by considering the parameters of signals not as the ordinary integers [3,4], but as a special kind of integer numbers, polyadic integers, introduced in [5]. The polyadic integers form polyadic $\left(m,n\right)$-rings having (or closed with respect to) m additions and n multiplications [6]. In this way, preservation of the property to be in the same polyadic ring after signal processing will give various restrictions on the signal parameters. The main idea is to use these restrictions (as equations) to encrypt and decrypt a series of ordinary numbers using sets of signals prepared in special ways.

2. Polyadic Rings

We first remind that the polyadic ring or $\left(m,n\right)$-ring is a set with m-ary addition (being a m-ary group) and n-ary multiplication (being a n-ary semigroup), which are connected by the polyadic distributive law [6]. A polyadic ring is nonderived, if its full operations cannot be obtained as repetition of binary operations. As the simplest example of the binary ring or $\left(2,2\right)$-ring is the set ordinary integers $\mathbb{Z}$, the example of $\left(m,n\right)$-ring is the set of polyadic integers ${\mathbb{Z}}_{\left(m,n\right)}$ [5]. The concrete realization of polyadic integers is the set of representatives of the conguence class (residue class) of an integer a modulo b (with both a and b fixed)

$${\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}={\left[\left[a\right]\right]}_b=\left\{\left\{a+b·k\right\}\mid k\in \mathbb{Z},a\in {\mathbb{Z}}_{+},b\in \mathbb{N},0\le a\le b-1\right\}.$$

(1)

We denote a representative element by $x_k=x_k^{\left[a,b\right]}=a+b·k$, which are polyadic intergers $\left\{x_k^{\left[a,b\right]}\right\}\in {\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}$, because only m additions and n multiplications are possible in the nonderived case

$$\begin{array}{cc}\hfill \stackrel{m}{\overbrace{x_{k_1}+x_{k_2}+\dots +x_{k_m}}}& \in {\mathbb{G}}_{add\left(m\right)}^{\left[a,b\right]}\subset {\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \stackrel{n}{\overbrace{x_{k_1}{x}_{k_2}\dots x_{k_n}}}& \in {\mathbb{S}}_{mult\left(n\right)}^{\left[a,b\right]}\subset {\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]},k_i\in \mathbb{Z},\hfill \end{array}$$

(3)

where ${\mathbb{G}}_{add\left(m\right)}^{\left[a,b\right]}$ and ${\mathbb{S}}_{mult\left(n\right)}^{\left[a,b\right]}$ are the m-ary additive group and n-ary multiplicative semigroup of the polyadic ring ${\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}$.

It follows from (2) and (3) that more generally, the m-admissible sum consists of ${\ell }_m\left(m-1\right)+1$ summands and the n-admissible product contains ${\ell }_n\left(n-1\right)+1$ elements, where ${\ell }_m$ is a number of m-ary additions (m-polyadic power) and ${\ell }_n$ is a number of n-ary multiplications (n-polyadic power). Therefore, in general

$$\begin{array}{cc}\hfill \stackrel{{\ell }_m\left(m-1\right)+1}{\overbrace{x_{k_1}+x_{k_2}+\dots +x_{k_m}}}& \in {\mathbb{G}}_{add\left(m\right)}^{\left[a,b\right]}\subset {\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \stackrel{{\ell }_n\left(n-1\right)+1}{\overbrace{x_{k_1}{x}_{k_2}\dots x_{k_n}}}& \in {\mathbb{S}}_{mult\left(n\right)}^{\left[a,b\right]}\subset {\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]},k_i\in \mathbb{Z}.\hfill \end{array}$$

(5)

${\ell }_m\left(m-1\right)+1$

For instance, in the residue (congruence) class

$${\left[\left[3\right]\right]}_4=\left\{\dots -25,-21,-17,-13,-9,-5,-1,3,7,11,15,19,23,27,31,35,39\dots \right\}$$

(6)

we can add $4{\ell }_m+1$ representatives and multiply $2{\ell }_n+1$ representatives (${\ell }_m,{\ell }_n$ are polyadic powers) to retain in the same class ${\left[\left[3\right]\right]}_4$. If, for example, we take ${\ell }_m=2$, ${\ell }_n=3$, then we obtain closeness of polyadic operations

$$\begin{array}{cc}\hfill \left(7+11+15+19+23\right)-5-9-13-1& =47=3+4·11\in {\left[\left[3\right]\right]}_4,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \left(\left(7·3·11\right)·19·15\right)·31·27& =55103895=3+4·13775973\in {\left[\left[3\right]\right]}_4.\hfill \end{array}$$

(8)

Thus, we cannot add and multiply arbitrary quantities of representatives in ${\left[\left[3\right]\right]}_4$, only the admissible ones. This means that ${\left[\left[3\right]\right]}_4$ is really the polyadic $\left(5,3\right)$-ring ${\mathbb{Z}}_{\left(5,3\right)}^{\left[3,4\right]}$.

In general, a congruence class ${\left[\left[a\right]\right]}_b$ is a polyadic ring ${\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}$, if the following relations hold valid [5]

$$\left(m-1\right){\displaystyle \frac{a}{b}}=I^{\left(m\right)}\left(a,b\right)=I=integer,$$

(9)

$${\displaystyle \frac{a^n-a}{b}}=J^{\left(n\right)}\left(a,b\right)=J=integer,$$

(10)

where $I,J$ are called a (polyadic) shape invariants of the congruence class ${\left[\left[a\right]\right]}_b$, e.g., for the congruence class ${\left[\left[3\right]\right]}_4$, the shape invariants (9) and (10) are $I=3$ and $J=6$, correspondingly.

In

Table 1. The polyadic ring ${\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}$ of the fixed residue class ${\left[\left[a\right]\right]}_b$: the arity shape ${\mathrm{\Phi }}_{\left(m,n\right)}^{\left[a,b\right]}$.

$\mathit{a}\setminus \mathit{b}$	2	3	4	5	6	7	8	9	10
1	$\begin{array}{c}m=\mathbf{3}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{4}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{5}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{7}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{8}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{9}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{10}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$	$\begin{array}{c}m=\mathbf{11}\\ n=\mathbf{2}\\ I=1\\ J=0\end{array}$
2		$\begin{array}{c}m=\mathbf{4}\\ n=\mathbf{3}\\ I=2\\ J=2\end{array}$		$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{5}\\ I=2\\ J=6\end{array}$	$\begin{array}{c}m=\mathbf{4}\\ n=\mathbf{3}\\ I=1\\ J=1\end{array}$	$\begin{array}{c}m=\mathbf{8}\\ n=\mathbf{4}\\ I=2\\ J=2\end{array}$		$\begin{array}{c}m=\mathbf{10}\\ n=\mathbf{7}\\ I=2\\ J=14\end{array}$	$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{5}\\ I=1\\ J=3\end{array}$
3			$\begin{array}{c}m=\mathbf{5}\\ n=\mathbf{3}\\ I=3\\ J=6\end{array}$	$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{5}\\ I=3\\ J=48\end{array}$	$\begin{array}{c}m=\mathbf{3}\\ n=\mathbf{2}\\ I=1\\ J=1\end{array}$	$\begin{array}{c}m=\mathbf{8}\\ n=\mathbf{7}\\ I=3\\ J=312\end{array}$	$\begin{array}{c}m=\mathbf{9}\\ n=\mathbf{3}\\ I=3\\ J=3\end{array}$		$\begin{array}{c}m=\mathbf{11}\\ n=\mathbf{5}\\ I=3\\ J=24\end{array}$
4				$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{3}\\ I=4\\ J=12\end{array}$	$\begin{array}{c}m=\mathbf{4}\\ n=\mathbf{2}\\ I=2\\ J=2\end{array}$	$\begin{array}{c}m=\mathbf{8}\\ n=\mathbf{4}\\ I=4\\ J=36\end{array}$		$\begin{array}{c}m=\mathbf{10}\\ n=\mathbf{4}\\ I=4\\ J=28\end{array}$	$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{3}\\ I=2\\ J=6\end{array}$
5					$\begin{array}{c}m=\mathbf{7}\\ n=\mathbf{3}\\ I=5\\ J=20\end{array}$	$\begin{array}{c}m=\mathbf{8}\\ n=\mathbf{7}\\ I=5\\ J=11160\end{array}$	$\begin{array}{c}m=\mathbf{9}\\ n=\mathbf{3}\\ I=5\\ J=15\end{array}$	$\begin{array}{c}m=\mathbf{10}\\ n=\mathbf{7}\\ I=5\\ J=8680\end{array}$	$\begin{array}{c}m=\mathbf{3}\\ n=\mathbf{2}\\ I=1\\ J=2\end{array}$
6						$\begin{array}{c}m=\mathbf{8}\\ n=\mathbf{3}\\ I=6\\ J=30\end{array}$			$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{2}\\ I=3\\ J=3\end{array}$
7							$\begin{array}{c}m=\mathbf{9}\\ n=\mathbf{3}\\ I=7\\ J=42\end{array}$	$\begin{array}{c}m=\mathbf{10}\\ n=\mathbf{4}\\ I=7\\ J=266\end{array}$	$\begin{array}{c}m=\mathbf{11}\\ n=\mathbf{5}\\ I=7\\ J=1680\end{array}$
8								$\begin{array}{c}m=\mathbf{10}\\ n=\mathbf{3}\\ I=8\\ J=56\end{array}$	$\begin{array}{c}m=\mathbf{6}\\ n=\mathbf{5}\\ I=4\\ J=3276\end{array}$
9									$\begin{array}{c}m=\mathbf{11}\\ n=\mathbf{3}\\ I=9\\ J=72\end{array}$

, the mapping

$${\mathrm{\Phi }}_{\left(m,n\right)}^{\left[a,b\right]}:\left(a,b\right)⟶\left(m,n\right)$$

(11)

of the congruence class parameters to the polyadic ring arities (we call it the arity shape) and shape invariants is presented for their lowest values. The arity shape mapping (11) is injective and non-surjective (empty cells), and it cannot be expressed in closed form. Moreover, e.g., the congruence classes ${\left[\left[2\right]\right]}_4$, ${\left[\left[2\right]\right]}_8$, ${\left[\left[3\right]\right]}_9$, ${\left[\left[4\right]\right]}_8$, ${\left[\left[6\right]\right]}_8$ and ${\left[\left[6\right]\right]}_9$ do not correspond to any ring, while the same $\left(6,5\right)$-ring can be described by different congruence classes ${\left[\left[2\right]\right]}_5$, ${\left[\left[3\right]\right]}_5$, ${\left[\left[2\right]\right]}_{10}$, and ${\left[\left[8\right]\right]}_{10}$.

The polyadic arity shape ${\mathrm{\Phi }}_{\left(m,n\right)}^{\left[a,b\right]}$ (11) is the main tool in the encryption/decryption procedure, described below.

3. Polyadic Encryption/Decryption Procedure

Let us consider the initial plaintext as a series of ordinary integer numbers (any plaintext can be transformed to that by the corresponding encoding procedure)

$$\mathbf{T}=y_1,y_2,\dots y_r,y_j\in \mathbb{Z}.$$

(12)

We propose a general encryption/decryption procedure, when each of $y=y_j$ is connected with the various parameters of signal series, and the latter are transfered to the receiver, who then restores y using special rules and systems of equations known to him only.

The main idea is to examine such signals which have parameters as polyadics integers, that is, they are in the polyadic ring ${\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}$ (1). This can be treated as a polyadic generalization of the (binary) discretization technique (in which the parameters are ordinary integers $\mathbb{Z}$), and so we call it the polyadic discretization. Its crucial new feature is the possibility to transfer information (e.g., arities, minimal allowed number of additions and multiplications) using signal parameters, as it will be shown below.

Here, we apply this idea to signal amplitudes (such signals are called the continuous-time discrete-valued or quantized analog signal [7,8]) and their addition only. This means that we look on the additive part of the polyadic ring ${\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}$ which is a nonderived (allowed to add exactly m terms, no fewer) m-ary group ${\mathbb{G}}_{add\left(m\right)}^{\left[a,b\right]}$ (2). In this way, we denote the single ith signal shape as

$${\mathrm{\Psi }}_i^{\left({\ell }_f\right)}=A_i^{\left({\ell }_f\right)}·f^{\left({\ell }_f\right)}\left(t\right),$$

(13)

where $A_i^{\left({\ell }_f\right)}$ is the amplitude of the normalized (in some manner) ith signal $f^{\left({\ell }_f\right)}\left(t\right)$, t is time, and the natural ${\ell }_f\in \mathbb{N}$ corresponds to the special kind of signal (by consequent numerating sine/cosine, triangular, rectangular, etc.).

First, we assume that the amplitude $A_i^{\left({\ell }_f\right)}$ is in polyadic ring, i.e., it is a representative of the congruence class ${\left[\left[a\right]\right]}_b$, $b\in \mathbb{N},0\le a\le b-1$, and therefore, it has the form (1)

$$A_i^{\left({\ell }_f\right)}=a+b·k_i^{\left({\ell }_f\right)}\in {\mathbb{G}}_{add\left(m\right)}^{\left[a,b\right]}\subset {\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]},k_i^{\left({\ell }_f\right)}\in \mathbb{Z}.$$

(14)

Second, we identify the number of signal species ${\ell }_f$ with the m-polyadic power ${\ell }_m$ from (4)

$${\ell }_f={\ell }_m.$$

(15)

In this picture, for the signal species ${\ell }_f$, we prepare the sum of ${\ell }_f\left(m-1\right)+1$ signals as

$${\mathrm{\Psi }}_{tot}^{\left({\ell }_f\right)}=\sum _{i=1}^{{\ell }_f\left(m-1\right)+1}{\mathrm{\Psi }}_i^{\left({\ell }_f\right)}=A_{tot}^{\left({\ell }_f\right)}·f^{\left({\ell }_f\right)}\left(t\right)$$

(16)

where the total amplitude $A_{tot}^{\left({\ell }_f\right)}$ becomes different for distinct species ${\ell }_f$ and after usage of (14) has the general form

$$\begin{array}{cc}\hfill A_{tot}^{\left({\ell }_f\right)}& =\sum _{i=1}^{{\ell }_f\left(m-1\right)+1}{A}_i^{\left({\ell }_f\right)}=a·\left({\ell }_f\left(m-1\right)+1\right)+b·K\left(m,{\ell }_f\right)\in {\mathbb{G}}_{add\left(m\right)}^{\left[a,b\right]}\subset {\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill K\left(m,{\ell }_f\right)& =\sum _{i=1}^{{\ell }_f\left(m-1\right)+1}{k}_i^{\left({\ell }_f\right)}.\hfill \end{array}$$

(18)

Thus, we observe that the total amplitude (17) of the signal (13) contains all parameters of the m-ary group ${\mathbb{G}}_{add\left(m\right)}^{\left[a,b\right]}$ (the additive part of the polyadic ring ${\mathbb{Z}}_{\left(m,n\right)}^{\left[a,b\right]}$ (1)). This allows us to use the combination of signals (in the above particular case sums) to transfer securely the plaintext variables $y_j$ (12) from sender to recepient, if we encode each of them $y=y_j$ by the polyadic ring parameters $y⟶\left(a,b,m\right)$. The recepient obtains the set of the total amplitudes $A_{tot}^{\left({\ell }_f\right)}$ and treats (17) as the system of equations for parameters $\left(a,b,m\right)$, and then after the decoding $\left(a,b,m\right)⟶y⟶y=y_j$ for each j obtains the initial plaintext $\mathbf{T}$ (12). Schematically, we can present the proposed encryption/decryption procedure as

$$\begin{array}{cc}\hfill & \mathrm{snd}:y\stackrel{\mathrm{encoding}\mathrm{by}\mathrm{snd}}{⟶}\left(a,b,m\right)\stackrel{\mathrm{summing}}{⟶}{A}_{tot}^{\left({\ell }_f\right)}\stackrel{\mathrm{preparing}\mathrm{signals}}{⟶}{\mathrm{\Psi }}_{tot}^{\left({\ell }_f\right)}\stackrel{\mathrm{transferring}\mathrm{to}\mathrm{rcp}}{⟶}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & \mathrm{rcp}:\mathrm{system}\mathrm{of}\mathrm{equations}{A}_{tot}^{\left({\ell }_f\right)}(17)\stackrel{\mathrm{solving}\mathrm{by}\mathrm{rcp}}{⟶}\left(a,b,m\right)\stackrel{\mathrm{decoding}\mathrm{by}\mathrm{rcp}}{⟶}y\hfill \end{array}$$

(20)

The security of this procedure is goverened not by one key, as in the standard cases, but by the system (17) and Table 1, and by connection between a kind of signal ${\ell }_f$ and m-polyadic power (15), which are all unknown to the third party.

4. Example

Let us consider a concrete example of the proposed encryption/decryption procedure for the congruence class ${\left[\left[a\right]\right]}_b$ and one kind of signal ${\ell }_f={\ell }_m\equiv \ell$, where ℓ is m-polyadic power. Each of such class gives (by Table 1) the arity m as the plaintext entry to transfer $y=m$. Next, we should choose the shape of the function $K\left(m,\ell \right)$ (18), which is, in general, arbitrary. In the simplest case, we take the same linear function

$$k_i^{\left(\ell \right)}=i-1,$$

(21)

for all m-polyadic powers ℓ, but any functional dependence in (21) can be chosen, and it is different for different ℓ, which increases security of the procedure. The choice (21) gives

$$K\left(m,\ell \right)={\displaystyle \frac{1}{2}}\ell \left(m-1\right)\left(\ell \left(m-1\right)+1\right).$$

(22)

So, the total amplitudes for different polyadic powers become

$$A_{tot}^{\left(\ell \right)}\equiv B_{\ell }=a·\left(\ell \left(m-1\right)+1\right)+{\displaystyle \frac{b}{2}}\ell \left(m-1\right)\left(\ell \left(m-1\right)+1\right)·$$

(23)

The recepient obtains the set of signals with amplitudes (23) as polyadic integers

$${\mathrm{\Psi }}_{tot}^{\left(\ell \right)}=B_{\ell }·f^{\left(\ell \right)}\left(t\right).$$

(24)

To obtain the values of three variables $\left(a,b,m\right)$, one needs three equations, i.e., three total amplitudes with different arbitrary polyadic powers $\ell ={\ell }_1,{\ell }_2,{\ell }_3\in \mathbb{N}$. Because the general solution is too cumbersome, we choose the first three consequent polyadic powers $\ell =1,2,3$, while any three natural numbers are possible to increase the security. This gives the following system of quadratic equations

$$\begin{array}{cc}\hfill a·m+b·{\displaystyle \frac{\left(m-1\right)m}{2}}& =B_1,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill a·\left(2m-1\right)+b·\left(m-1\right)\left(2m-1\right)& =B_2,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill a·\left(3m-2\right)+b·{\displaystyle \frac{3\left(m-1\right)\left(3m-2\right)}{2}}& =B_3.\hfill \end{array}$$

(27)

The general solution of the system is

$$m={\displaystyle \frac{7B_1-4B_2+B_3\mp \sqrt{B_1^2-8B_2{B}_1-2B_3{B}_1+16B_2^2+B_3^2-8B_2{B}_3}}{4\left(3B_1-3B_2+B_3\right)}},$$

(28)

$$a=3B_1-3B_2+B_3,$$

(29)

$$\begin{array}{cc}\hfill b& ={\displaystyle \frac{1}{\left(3B_1-3B_2+B_3\right)\left(2B_2-B_1-B_3\right)}}\times \hfill \\ \hfill & \left[\left(11B_1^2-16B_2{B}_1+6B_3{B}_1-4B_2^2+4B_2{B}_3-B_3^2\right)\left(3B_1-3B_2+B_3\right)\right.\hfill \\ \hfill & -{\displaystyle \frac{105}{2}}{B}_1^3+{\displaystyle \frac{333}{2}}{B}_1^2{B}_2-162B_1{B}_2^2+48B_2^3-{\displaystyle \frac{113}{2}}{B}_1^2{B}_3\hfill \\ \hfill & +107B_1{B}_2{B}_3-46B_2^2{B}_3-{\displaystyle \frac{35}{2}}{B}_1{B}_3^2+{\displaystyle \frac{29}{2}}{B}_2{B}_3^2-{\displaystyle \frac{3}{2}}{B}_3^3\hfill \\ \hfill & \pm \left({\displaystyle \frac{15}{2}}{B}_1^2-{\displaystyle \frac{39}{2}}{B}_1{B}_2+12B_2^2+7B_1{B}_3-{\displaystyle \frac{17}{2}}{B}_2{B}_3+{\displaystyle \frac{3}{2}}{B}_3^2\right)\times \hfill \\ \hfill & \left.\sqrt{B_1^2-8B_2{B}_1-2B_3{B}_1+16B_2^2+B_3^2-8B_2{B}_3}\right].\hfill \end{array}$$

(30)

The sign in (28) and (30) should be chosen so that the solutions are ordinary integers.

In this particular case, for instance, the congruence class ${\left[\left[3\right]\right]}_4$ (6) has 5-ary addition, as follows from Table 1. So, if the sender wants to securely submit one element $y=m=5$ from his plainintext (12), he applies the proposed encryption procedure and prepares three sums of (quantized analog) signals (having integer amplitudes) corresponding to the polyadic powers $\ell =1,2,3$, as follows (using (16) and (23)):

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{tot}^{\left(1\right)}& =55·f^{\left(1\right)}\left(t\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{tot}^{\left(2\right)}& =171·f^{\left(2\right)}\left(t\right),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{tot}^{\left(3\right)}& =351·f^{\left(3\right)}\left(t\right),\hfill \end{array}$$

(33)

where $f^{\left(1,2,3\right)}\left(t\right)$ are different (or the same) normalized signals. The recepient receives three (quantized analog) signals (31)–(33), and because he knows the normalized signals, he immediately obtains the integer values $B_1=55$, $B_2=171$, $B_3=351$. Inserting them into the system of quadratic equations (known to him ahead) (25)–(27), he (directly or using (28)–(30)) derives the values $m=5$, $a=3$, $b=4$ and the desired element $y=m=5$ from the initial plaintext (12). The same procedure should be provided for each element $y_j=y$ of the plaintext (12), which completes its decryption.

5. Conclusions

Thus, we proposed a principally new encryption/decryption procedure based on exploiting the signal processing. The main idea is to consider the signal parameters as polyadic integers being representative of the fixed polyadic ring, which is treated as some kind of polyadic discretization depending on the ring integer characteristics. They allow us to transfer numerical information from sender to recepient by submitting special sets of signals with genuine properties agreed before. The recepient knows the rules and equations to solve, to decrypt the initial plaintext. Security is achieved by using a mathematical process where public information is exchanged openly, but this information is useless without a corresponding piece of private, secret information that is never transmitted.