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 containing a single root of f, and
- the derivatives and must not have zeros in the interval .
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 into an array B;
- The loop of Equation (4) continues, and as an output, two real numbers and are generated. We take the sum of and to produce the real number , 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.
4.1. Approaches for Choosing a Nonlinear Function
4.1.1. Nonlinear Function
4.1.2. Polynomial Function
4.2. An Example of Encryption
- , and
- ,
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
- ;
- is an even integer.
Appendix A.2. Bounds of Real Roots of Polynomials
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Letter | ASCII | NM | JM | Array B | Array R | Array E |
---|---|---|---|---|---|---|
(Char) | Code | Iterations | Iterations | Reached Root () | ||
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% |
square | 0.16, exceed 68.56 % | 242.28, exceed 70.66% |
Arithmetic mean value | 0.5000 | 127.5055 |
Monte Carlo for | 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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Stoyanov, B.; Nedzhibov, G. Symmetric Key Encryption Based on Rotation-Translation Equation. Symmetry 2020, 12, 73. https://doi.org/10.3390/sym12010073
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 StyleStoyanov, Borislav, and Gyurhan Nedzhibov. 2020. "Symmetric Key Encryption Based on Rotation-Translation Equation" Symmetry 12, no. 1: 73. https://doi.org/10.3390/sym12010073