# Symmetric Key Encryption Based on Rotation-Translation Equation

^{*}

## Abstract

**:**

## 1. Introduction

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

- Lack of rules on how to choose the function f and suitable iterative method so that the convergence of the process is always guaranteed.
- Vulnerability to attack because in these types of algorithms the same letter is encoded with the same real number of each occurrence in the plaintext.

## 3. On Numerical Methods and Rotation–Translation Equation

#### 3.1. On Numerical Methods for Solving Nonlinear Equations

- the convergence speed of the iteration,
- an interval of convergence and the rules for choosing the initial approximations.

- need to have an interval $[a,b]$ containing a single root of f, and
- the derivatives ${f}^{\prime}$ and ${f}^{\u2033}$ must not have zeros in the interval $[a,b]$.

#### 3.2. Base of Rotation–Translation Equation

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

- Read the symbols from the plaintext data and get the ASCII values of the different symbols;
- Construct a system of L nonlinear equations by subtracting the ASCII values from the function f and equate with zero;
- Solve individually the nonlinear equations and put the results ${\alpha}_{i}$ into an array B;
- The loop of Equation (4) continues, and as an output, two real numbers ${x}_{i}$ and ${y}_{i}$ are generated. We take the sum of ${x}_{i}$ and ${y}_{i}$ to produce the real number ${d}_{i}={x}_{i}+{y}_{i}$, which is put into an array R.
- Return to Step 4 until a stream of real numbers R with length L is reached.
- We get the sum of the two arrays B and R to produce E, the output array of real numbers.

**Remark**

**1.**

#### 4.1. Approaches for Choosing a Nonlinear Function

#### 4.1.1. Nonlinear Function

#### 4.1.2. Polynomial Function

#### 4.2. An Example of Encryption

- $\left|f\right({z}_{k}\left)\right|\le \u03f5$, and
- $|{z}_{k}-{z}_{k+1}|\le \u03f5$,

#### 4.3. Brute-Force Attack Analysis

#### 4.4. Statistical Test Analysis of the Proposed Encryption

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Real Roots Counting of Polynomials

- $m\le p$;
- $p-m$ is an even integer.

#### Appendix A.2. Bounds of Real Roots of Polynomials

**Theorem**

**A1**(Cauchy)

**.**

**Theorem**

**A2**(Cauchy)

**.**

**Theorem**

**A3.**

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Letter | ASCII | NM | JM | Array B | Array R | Array E |
---|---|---|---|---|---|---|

(Char) | Code | Iterations | Iterations | Reached Root (${\mathit{\alpha}}_{\mathit{i}}$) | ${\mathit{d}}_{\mathit{i}}$ | ${\mathit{e}}_{\mathit{i}}={\mathit{\alpha}}_{\mathit{i}}+{\mathit{d}}_{\mathit{i}}$ |

S | 83 | 6 | 3 | 2.596938615169214 | 1.13761418319195 | 3.73455279836116 |

h | 104 | 6 | 3 | 2.707594514758099 | 3.83246813052273 | 6.54006264528082 |

u | 117 | 6 | 3 | 2.767550880788345 | 2.58986907946370 | 5.35741996025204 |

m | 109 | 6 | 3 | 2.731316748315844 | 4.91511042783787 | 7.64642717615371 |

e | 101 | 6 | 3 | 2.692927857503279 | 2.09200715087053 | 4.78493500837380 |

n | 110 | 6 | 3 | 2.735958159508397 | 7.52948634851868 | 10.2654445080271 |

94 | 6 | 3 | 2.657327240630354 | 0.10782288092075 | 2.76515012155110 | |

U | 85 | 6 | 3 | 2.608365856583876 | 5.70292778455835 | 8.31129364114222 |

n | 110 | 6 | 3 | 2.735958159508397 | 1.30796668243219 | 4.04392484194058 |

i | 105 | 6 | 3 | 2.712409561369016 | 7.38162948307289 | 10.0940390444419 |

v | 118 | 6 | 3 | 2.771941812496392 | 0.13478345799064 | 2.90672527048704 |

e | 101 | 6 | 3 | 2.692927857503279 | 5.23813805178474 | 7.93106590928801 |

r | 114 | 6 | 3 | 2.754198397484480 | 1.66157719938621 | 4.41577559687069 |

s | 115 | 6 | 3 | 2.758679632039476 | 7.40857522965636 | 10.1672548616958 |

i | 105 | 6 | 3 | 2.712409561369016 | 0.16954303210037 | 2.88195259346938 |

t | 116 | 6 | 3 | 2.763130305077092 | 6.32301454738480 | 9.08614485246189 |

y | 121 | 6 | 3 | 2.784941120909602 | 0.81874887373720 | 3.60368999464681 |

NIST Test | p-Value | Success Rate |
---|---|---|

Monobit | 0.556460 | 992/1000 |

Block frequency | 0.010093 | 981/1000 |

Cumulative sums forward | 0.399442 | 993/1000 |

Cumulative sums reverse | 0.299736 | 993/1000 |

Runs | 0.605916 | 986/1000 |

Longest run of ones | 0.605916 | 988/1000 |

Rank | 0.830808 | 988/1000 |

Fourier | 0.200115 | 980/1000 |

Non overlapping templates | 0.498222 | 990/1000 |

Overlapping templates | 0.859637 | 992/1000 |

Universal | 0.653773 | 988/1000 |

Approximate entropy | 0.693142 | 988/1000 |

Serial one | 0.894918 | 990/1000 |

Serial two | 0.282626 | 986/1000 |

Linear complexity | 0.051942 | 995/1000 |

State | p-Value | Success Rate |
---|---|---|

−4 | 0.696617 | 610/614 |

−3 | 0.746463 | 606/614 |

−2 | 0.211467 | 610/614 |

−1 | 0.501472 | 606/614 |

+1 | 0.933509 | 607/614 |

+2 | 0.584363 | 605/614 |

+3 | 0.873629 | 610/614 |

+4 | 0.672912 | 608/614 |

State | p-Value | Success Rate |
---|---|---|

−9 | 0.283657 | 608/614 |

−8 | 0.444875 | 607/614 |

−7 | 0.699986 | 609/614 |

−6 | 0.775401 | 607/614 |

−5 | 0.876173 | 610/614 |

−4 | 0.921867 | 607/614 |

−3 | 0.135745 | 607/614 |

−2 | 0.036332 | 610/614 |

−1 | 0.574229 | 612/614 |

+1 | 0.345203 | 609/614 |

+2 | 0.366645 | 607/614 |

+3 | 0.517714 | 610/614 |

+4 | 0.024235 | 612/614 |

+5 | 0.990938 | 612/614 |

+6 | 0.447934 | 610/614 |

+7 | 0.232430 | 609/614 |

+8 | 0.193732 | 611/614 |

+9 | 0.659297 | 611/614 |

ENT Test | Input of Bits | Input of Bytes |
---|---|---|

Entropy | 1.000000 | 7.999999 |

Optimum compression | Reduce size by 0% | Reduce size by 0% |

${\chi}^{2}$ square | 0.16, exceed 68.56 % | 242.28, exceed 70.66% |

Arithmetic mean value | 0.5000 | 127.5055 |

Monte Carlo for $\pi $ | 3.141226994 (error 0.01%) | 3.141226994 (error 0.01%) |

Serial correlation | −0.000002 | 0.000180 |

Test Name | Raw | Processed | Evaluation |
---|---|---|---|

BCFN(2,13):! | R = +0.0 | “pass” | normal |

BCFN(2+0,13−0) | R = −0.7 | p = 0.608 | normal |

BCFN(2 + 1,13 − 0) | R = +2.3 | p = 0.172 | normal |

BCFN(2 + 2,13 − 1) | R = −0.1 | p = 0.504 | normal |

BCFN(2 + 3,13 − 1) | R = −2.2 | p = 0.812 | normal |

BCFN(2 + 4,13 − 2) | R = −4.4 | p = 0.968 | normal |

BCFN(2 + 5,13 − 3) | R = −1.1 | p = 0.669 | normal |

BCFN(2 + 6,13 − 3) | R = −4.1 | p = 0.960 | normal |

BCFN(2 + 7,13 − 4) | R = +4.8 | p = 0.032 | normal |

BCFN(2 + 8,13 − 5) | R = +3.3 | p = 0.093 | normal |

BCFN(2 + 9,13 − 5) | R = −0.3 | p = 0.524 | normal |

BCFN(2 + 10,13 − 6) | R = −4.3 | p = 0.981 | normal |

BCFN(2 + 11,13 − 6) | R = −0.9 | p = 0.614 | normal |

BCFN(2 + 12,13 − 7) | R = +1.6 | p = 0.219 | normal |

BCFN(2 + 13,13 − 8) | R = −2.7 | p = 0.914 | normal |

DC6-9x1Bytes-1 | R = −1.0 | p = 0.795 | normal |

Gap-16:! | R = +0.0 | “pass” | normal |

Gap-16:A | R = +0.0 | p = 0.614 | normal |

Gap-16:B | R = −3.2 | p = 0.987 | normal |

(Low1/8)BCFN(2,13):! | R = +0.0 | “pass” | normal |

(Low1/8)BCFN(2+0,13 − 1) | R = −1.7 | p = 0.754 | normal |

(Low1/8)BCFN(2+1,13 − 2) | R = +1.0 | p = 0.336 | normal |

(Low1/8)BCFN(2+2,13 − 3) | R = +1.7 | p = 0.243 | normal |

(Low1/8)BCFN(2+3,13 − 3) | R = −0.7 | p = 0.605 | normal |

(Low1/8)BCFN(2+4,13 − 4) | R = +2.7 | p = 0.138 | normal |

(Low1/8)BCFN(2+5,13 − 5) | R = −0.3 | p = 0.528 | normal |

(Low1/8)BCFN(2+6,13 − 5) | R = −0.9 | p = 0.626 | normal |

(Low1/8)BCFN(2+7,13 − 6) | R = −2.3 | p = 0.838 | normal |

(Low1/8)BCFN(2+8,13 − 6) | R = −2.4 | p = 0.853 | normal |

(Low1/8)BCFN(2+9,13 − 7) | R = +1.6 | p = 0.223 | normal |

(Low1/8)BCFN(2+10,13 − 8) | R = +3.1 | p = 0.096 | normal |

(Low1/8)DC6-9x1Bytes-1 | R = −0.5 | p = 0.730 | normal |

(Low1/8)Gap-16:! | R = +0.0 | “pass” | normal |

(Low1/8)Gap-16:A | R = −0.1 | p = 0.675 | normal |

(Low1/8)Gap-16:B | R = −1.7 | p = 0.888 | normal |

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**MDPI and ACS Style**

Stoyanov, B.; Nedzhibov, G.
Symmetric Key Encryption Based on Rotation-Translation Equation. *Symmetry* **2020**, *12*, 73.
https://doi.org/10.3390/sym12010073

**AMA Style**

Stoyanov B, Nedzhibov G.
Symmetric Key Encryption Based on Rotation-Translation Equation. *Symmetry*. 2020; 12(1):73.
https://doi.org/10.3390/sym12010073

**Chicago/Turabian Style**

Stoyanov, Borislav, and Gyurhan Nedzhibov.
2020. "Symmetric Key Encryption Based on Rotation-Translation Equation" *Symmetry* 12, no. 1: 73.
https://doi.org/10.3390/sym12010073