This research presents, and clarifies the application of two permutation algorithms, based on chaotic map systems, and applied to a file of speech signals. They are the Arnold cat map-based permutation algorithm, and the Baker’s chaotic map-based permutation algorithm. Both algorithms are implemented on the same speech signal sample. Then, both the premier and the encrypted file histograms are documented and plotted. The speech signal amplitude values with time signals of the original file are recorded and plotted against the encrypted and decrypted files. Furthermore, the original file is plotted against the encrypted file, using the spectrogram frequencies of speech signals with the signal duration. These permutation algorithms are used to shuffle the positions of the speech files signals’ values without any changes, to produce an encrypted speech file. A comparative analysis is introduced by using some of sundry statistical and experimental analyses for the procedures of encryption and decryption, e.g., the time of both procedures, the encrypted audio signals histogram, the correlation coefficient between specimens in the premier and encrypted signals, a test of the Spectral Distortion (SD), and the Log-Likelihood Ratio (LLR) measures. The outcomes of the different experimental and comparative studies demonstrate that the two permutation algorithms (Baker and Arnold) are sufficient for providing an efficient and reliable voice signal encryption solution. However, the Arnold’s algorithm gives better results in most cases as compared to the results of Baker’s algorithm.

We have been in the middle of a technological revolution in recent years, which involves stable and secure multimedia transmission [

Speech conversations are considered a vital part of our lifestyle. During the last few years, the fast and extensive growth of network technology has increased speech conversations through the public switch telephone network, cell phones, satellites, and the internet [

Leakage prevention is vital due to the regular flow of audio signals worldwide over transmission devices [

The criteria to meet multimedia protection needs have led to the invention of excellent encryption mechanisms (e.g., chaos-based). Chaos-based encryption mechanisms are considered perfect for practical use because they provide a secured combination of speed, high protection, complexity, rational overhead computing, and computational power [

The two permutation algorithms are based on Baker’s map [

Both chaotic maps are two-dimensional (2D) chaotic invertible maps. The Baker’s map is a chaotic bijection of a square I unit (

This paper is arranged as follows: Section 2 presents the chaotic maps (Baker’s chaotic map and Arnold’s cat map) used in our research. Section 3 describes the algorithms’ application on speech signals. Additionally, it discusses speech patterns, spectrograms, and histograms for permuted speech signals. Section 4 examines the measurements and analyses of performing encryption and decryption time, CC, SD, and LLR measures. Section 5 is the conclusion of the paper.

A succinct overview of the two chaotic mapping systems used in this work is clarified in this section. These are Baker’s map and Arnold’s map.

Chaotic map systems are sensible to the initial parameters. If specific parameters are used, the system operates in several orbits, which are complicated and strenuous to analyze and compute. The system output sequences have strong randomness, weak correlation, and unpredictability [

One of the chaotic map systems that rely on the encryption process’s permutations is Baker’s chaotic map. Chaotic structures use maps to rearrange the elements in the file as a whole or as a file block. Baker’s chaotic map is one of the 2D invertible chaotic maps that has been introduced. We follow Baker’s chaotic map to shuffle locations for speech signals in the audio file. Baker’s map,

The map acts on the unit square depicted in

In the generalized Baker map, the square is divided into vertical _{0} = 0, whereas _{i}_{i}

The discretized Baker map is an effective tool for randomizing objects within a square matrix. Let _{key}_{i}_{1} = 0:

The messy permutation is implemented as follows in phases:

A square matrix (

Items in every rectangle are reorganized into a row within the produced permutation matrix. Rectangles are possessed from right to left, starting with higher rectangles and then lower ones. For every column within each rectangle, the scan begins from the bottom left corner toward the top items.

As clarified in _{1} = 2, _{2} = 4, and _{3} = 2), whereas

Shang et al. [

There are positive numbers for _{m+1}, _{m+1}) is the new location of the premier item location (_{m}_{m}

The outcomes of implementing the two algorithms (based on Baker’s chaotic map system and Arnold’s cat map system) on speech signals are discussed in this section. The premier speech signal file of a time domain is a conversation between women and men. Its samples are demonstrated in

Baker’s algorithm is implemented on the premier speech file three times with different choices for the key (

The two algorithms were applied with different cases of keys and dimensions on the original speech signals. The results of the speech signal pattern histograms and spectrograms of the permuted speech signals are registered and plotted for each case for each algorithm compared to the plots of the premier speech file clarified in

A spectrogram represents an optical image of the signal frequency spectrum as it varies over time when applied to an audio signal. Additionally, it is usually depicted as an image with the density shown by altering the color or brightness.

However, the spectrograms of permuted speech for Arnold’s algorithm are entirely different from the premier speech spectrograms for all cases (i.e., implementing Arnold’s algorithm is better than implementing Baker’s algorithm).

The histogram is used for continuous data, whereas bins represent data domains. Additionally, histograms are a convergent portrayal of the distribution of categorical or numerical data. Both encryption procedures are used as permutation methods (scrambling of signal positions) for speech files. The histogram for the permuted speech signal file corresponds to the histogram for the premier speech signal file in

In contrast, the length of the original speech signal file is 60,416. This exceptional case is illustrated in

The empirical outcomes and comparative study are presented using various empirical and statistical tests for encryption and decryption methods. These tests include the encryption or decryption time, CC between samples of the premier file and signals of the encrypted file, SD test, and LLR test.

In this test, the time in seconds for the encryption (permutation) and decryption was estimated for applying both algorithms on the premier speech signal file in all situations of

Baker Algorithm | Arnold Algorithms | |||||
---|---|---|---|---|---|---|

Enc. | 0.0470 | 0.0470 | 0.0620 | 0.3120 | 0.0890 | 0.0470 |

Dec. | 0.1560 | 0.0520 | 0.0470 | 0.3580 | 0.0990 | 0.0460 |

As illustrated in

If premier and encrypted files are extremely dependent, the CC is 1 (i.e., an encryption procedure is unsuccessful in hiding the premier signal information). If the CC is 0, then the premier voice signals and their encryptions are entirely different. Consequently, encryption protocol success means lower CC values [

where

Algorithm | Correlation coefficient for encrypted speech signals | ||
---|---|---|---|

Baker | 0.0112 | 0.0134 | −0.0041 |

Arnold | −0.0086 | −0.0051 | 0.000383 |

The SD is a form of measurement implemented in the frequency domain on the frequency spectrum of the premier and encrypted voice signals. It is measured in decibels to show how far the spectrum of encrypted signals is from that of the premier speech signals. It is possible to calculate the SD using

where _{s}_{y}

Algorithm | Spectral distortion for encrypted speech signals | ||
---|---|---|---|

Baker | 13.9122 | 14.0185 | 14.0239 |

Arnold | 13.9700 | 14.0000 | 13.9586 |

The LLR metric for speech signals is based on the supposition that every portion can be interpreted using a linear all-pole predictive model as in

where _{m}

where

Algorithm | Log-likelihood ratio for encrypted speech signals | ||
---|---|---|---|

Baker | 0.1867 | 0.3910 | 0.6637 |

Arnold | 0.6776 | 0.5921 | 0.8303 |

The LLR outcomes for the decrypted speech signal file are 0 for both algorithms for all cases of

In this paper, two permutation algorithms based on chaotic map systems are discussed. The Baker and Arnold algorithms are implemented on a premier speech file to secure it using a permutation procedure for the signal locations. The permuted speech files generated by implementing both algorithms were examined and compared to each other using the following comparative analyses and empirical tests: the time for encryption and decryption, histogram, CC, SD, and LLR. The encryption/decryption times for both algorithms (Baker and Arnold) are very good and less than 0.36 s, whereas the encryption and decryption times for both are very close in all cases. Both algorithms achieve low (near 0) CC values, but the CC outcomes are better using Arnold’s algorithm than Baker’s algorithm in all situations. Both algorithms achieve very convergent perfect values for SD in all situations.

Additionally, both algorithms attain perfect results for LLR in all situations, but the LLR outcomes are better with Arnold’s algorithm than Baker’s algorithm in all situations. Moreover, permuted speech signal patterns and spectrograms were plotted and compared to the original speech signals, illustrating that Arnold’s algorithm is better and more stable than Baker’s algorithm. The decryption results for both algorithms produce a decrypted speech file that fully corresponds to the premier speech file. The final results reveal that the two permutation algorithms are robust algorithms to provide a productive and settled method to encrypt speech files. However, Arnold’s algorithm provides good results in most cases compared to the results of Baker’s algorithm.