A TwoParameter Modified Logistic Map and Its Application to Random Bit Generation
Abstract
:1. Introduction
2. The Proposed Map
3. Application to Random Bit Generation
 Step 1.
 First, two modified logistic maps ${x}_{0},{y}_{0}$, one classic logistic map ${z}_{0}$, as well as two bit sequences ${b}_{0},{d}_{0}$ are initialized, and the maps’ parameters are chosen.
 Step 2.
 In every iteration, the decimal part of ${x}_{i}+{y}_{i}$ is compared to the decimal part of ${10}^{6}({x}_{i}\xb7{y}_{i})$ and depending on the result a 0 or 1 is produced and saved in ${b}_{i}$. Similarly, the decimal part of ${10}^{6}({x}_{i}+{y}_{i})$ is compared to the decimal part of ${x}_{i}\xb7{y}_{i}$ and depending on the result a 0 or 1 is produced and saved in ${d}_{i}$.
 Step 3.
 For every 10 iterations, the value of the logistic map $z\left(\frac{i}{10}\right)$ is compared to the decimal part of ${x}_{i}+{y}_{i}$. Depending on the result, a bit reversal is performed on the last ten digits of b or d.
 Step 4.
 Once the desired bitstream length is reached, the obtained sequence is computed using $XOR(b,d)$.
Algorithm 1 The Proposed Random Bit Generator. 
Data: Initialize initial conditions: ${x}_{0},{y}_{0},{z}_{0}$, parameter values: ${r}_{x},{r}_{y},{r}_{z},{\beta}_{x},{\beta}_{y}$, Bit subsequences ${b}_{0},{d}_{0}$ and bitstream length: ℓ. 

4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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If $\mathit{P}\ge \mathit{\alpha}$, the Test is Successful  

No.  Statistical Test  pValue  Proportion  Result 
1  Frequency  0.289667  49/50  success 
2  Block Frequency  0.383827  48/50  success 
3  Cumulative Sums  0.419021  49/50  success 
4  Runs  0.122325  50/50  success 
5  Longest Run  0.383827  50/50  success 
6  Rank  0.616305  49/50  success 
7  FFT  0.191687  49/50  success 
8  NonOverlapping Template  0.991468  49/50  success 
9  Overlapping Template  0.739918  50/50  success 
10  Universal  0.699313  50/50  success 
11  Approximate Entropy  0.534146  50/50  success 
12  Random Excursions  0.407091  32/32  success 
13  Random Excursions Variant  0.066882  32/32  success 
14  Serial  0.171867  50/50  success 
15  Linear Complexity  0.911413  48/50  success 
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Moysis, L.; Tutueva, A.; Volos, C.; Butusov, D.; MunozPacheco, J.M.; Nistazakis, H. A TwoParameter Modified Logistic Map and Its Application to Random Bit Generation. Symmetry 2020, 12, 829. https://doi.org/10.3390/sym12050829
Moysis L, Tutueva A, Volos C, Butusov D, MunozPacheco JM, Nistazakis H. A TwoParameter Modified Logistic Map and Its Application to Random Bit Generation. Symmetry. 2020; 12(5):829. https://doi.org/10.3390/sym12050829
Chicago/Turabian StyleMoysis, Lazaros, Aleksandra Tutueva, Christos Volos, Denis Butusov, Jesus M. MunozPacheco, and Hector Nistazakis. 2020. "A TwoParameter Modified Logistic Map and Its Application to Random Bit Generation" Symmetry 12, no. 5: 829. https://doi.org/10.3390/sym12050829