# E-ART: A New Encryption Algorithm Based on the Reflection of Binary Search Tree

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation and Research Goals

#### 1.2. Contribution

- (1)
- A balanced tree data structure along with American Standard Code for Information Interchange (ASCII) values of text characters to encode data. This makes searching for a particular character value more efficient, as there is no need to visit every node when searching for a specific value. Thus, higher computational efficiency is achieved.
- (2)
- Dynamic keys based on a pseudo-random generator. Each character in the text document is encrypted with a different cryptographic key. The character’s position is used as a seed in the random number generation function to produce the pseudo-random number. This ensures a high level of security against classical and modern powerful attacks, which is traditionally ensured by increasing the key size, without scarifying performance.

## 2. Background and Related Work

## 3. Proposed Schema

#### 3.1. A. Key Derivation

#### 3.1.1. Initial Key

- Variable offset. It is calculated mathematically using the proposed tree properties. The left and right nodes are shown in Figure 1. It uses the N value derived from the initial key to calculate ${N}_{L}$ and its reflection node ${N}_{R}$ and then generate the value of $Offse{t}_{var}$. This value is added to the initial reflection value according to Equations (3) and (4) to add more complexity and prevent cryptanalysis attacks that take advantage of one-to-one mapping. It is computed as follows:

- Dynamic offset. It is produced automatically using a pseudo-random number and the second part of the initial key (Variance). The pseudo-random generator uses each character’s position in the text as a seed to generate a pseudo-random number of 64 or 128 bits. The pseudo-random number is then adjusted using the Variance value. This offset is added in the last step to produce the final encrypted characters and is changed for each character. This results in a high degree of robustness and resistance to known powerful attacks. The dynamic offset is calculated as follows:

#### 3.1.2. E-ART Structure

_{𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑟}

_{ef}can be computed from the original value 𝑉𝑎𝑙

_{𝑜𝑟𝑔}as follows:

- (1)
- The presence of special characters, such as space, carriage return, and other text formatting characters, which range from 0 to 32 in the ASCII table, is not addressed.
- (2)

- Initially, the input textual data are stored in an array of characters (plaintext list).
- Each character in the list is converted into its corresponding ASCII values and stored in $Va{l}_{org.}$
- The variable offset $Offse{t}_{var}$ is generated using N, the first value of the initial key, and the properties of the tree—R, ${N}_{L}$, and ${N}_{R}$—as shown in Equation (1).
- For each character in the list, the initial reflected value $Va{l}_{Initialref}$ for each character is calculated using Equation (3).
- For each character in the list, the dynamic offset is generated by a pseudo-random generator using Variance on the second value of the initial key and characters’ positions, as shown in Equation (2).
- Then, value X is generated by adding the variable offset $Offse{t}_{var}$ and constant offsets $Offse{t}_{const}$ to the initial reflected value using Equation (4).
- Value X changes based on the maximum length ($Le{n}_{max}$) and non-printable character range. If the value of X is greater than $Le{n}_{max}$, then apply the mod operation and then add $Offse{t}_{const}$, as shown in Equation (4).
- Then, the dynamic offset is added to the value of X using Equation (5) to generate the final reflection value $Va{l}_{Ref}.$
- $Va{l}_{Ref}$ is converted to the equivalent ASCII character to produce the encrypted character.
- Append the character to encrypted list.
- Once all characters in the plaintext are encrypted, the encrypted text file is generated.

Algorithm 1: E-ART Algorithm |

Input: R,$Offse{t}_{const}$, $Le{n}_{max}$, input_text, N, VarianceOutput: Encrypted text1: Initialization2: input_list = Read all words from input file 3: Get the $Va{l}_{org}$ for each character 4: Get the $Offse{t}_{var}$ from Equation (1) 5: while all words in input_list are not iterated, do6: word = pop word from input_list 7: for each character in word do8: Get $Va{l}_{Initialref}$ character from Equation (3) 9: Get $\mathrm{Dynamic}\text{}\mathrm{offset}$ from Equation (2) with 10: Let X = $Va{l}_{Initialref}$ + $Offse{t}_{var}$ + $Offse{t}_{const}$ 11: if X is greater than $Le{n}_{max}$12: $Va{l}_{Ref}$ [(X mod $Le{n}_{max}$) + $Offse{t}_{const}$ + $\mathrm{Dynamic}\text{}\mathrm{offset}]$ 13: else14: $Va{l}_{Ref}$ [X + $\mathrm{Dynamic}\text{}\mathrm{offset}]$ 15: end if16: end for17: append $Characte{r}_{value}$ of $Va{l}_{Ref}$ to EncryptedWord 18: append word or EncryptedWord to EncryptedList 19: end while20: Write all values from EncryptedList to output document |

- Initially, the input data are stored in an array of characters (ciphertext list).
- Each character in the list is converted into its corresponding ASCII value and stored in $Va{l}_{org}$.
- The variable offset is generated using N, the first value of the initial key, and the properties of the tree—R, ${N}_{L}$, and ${N}_{R}$—as shown in Equation (1).
- For each character in the list, the dynamic offset is regenerated by a pseudo-random generator using the same parameters, Variance, with the second value of the initial key and the characters’ positions as shown in Equation (2).
- Value X is generated by subtracting the dynamic offset from $Va{l}_{org}$.
- Then, we check: if subtraction of variable offset $Offse{t}_{var}$ and constant offset $Offse{t}_{const}$ from X is less than 0, then set Quotient to be equal to 1; otherwise, set Quotient value to be equal to 0.
- Generate the $Va{l}_{Ref}$ by multiplying $Le{n}_{max}$ and Quotient and then subtract X, variable offset $Offse{t}_{var}$ and constant offset $Offse{t}_{const}$.
- Generate decrypted value by subtracting $Va{l}_{Ref}$ from $Le{n}_{max}$ plus 1.
- Decrypted value is converted to the equivalent ASCII character to produce the decrypted character.
- Append the character to the decrypted list.
- Once all characters in the ciphertext are decrypted, the decrypted text file is generated.

Algorithm 2: Data Decryption Algorithm |

Input: R,$Offse{t}_{const}$, $Le{n}_{max}$, Encrypted Text, N, VarianceOutput: decrypted text1: Initialization2: input_list = Read all word from input file 3: Get the $Va{l}_{org}$ or each character 4: Get the $Offse{t}_{var}$ from Equation (1) 5: while all words in input_list are not iterated, do6: word = pop word from input_list 7: for each character in word do8: Get $\mathrm{Dynamic}\text{}\mathrm{offset}$rom Equation (2) 9: Let X = $Va{l}_{org}-$$\mathrm{Dynamic}\text{}\mathrm{offset}$ 10: if (X − $Offse{t}_{var}-$$Offse{t}_{const}$ < 011: Quotient = 1 12: else13: Quotient = 0 14: $Va{l}_{Ref}$ $Le{n}_{max}$[(xuotient + X) − $Offse{t}_{var}$$Offse{t}_{const}]$ 15: decrypted_value = ($Le{n}_{max-}$$Va{l}_{Ref}$) + 1 16: end if17: end for18: append $Characte{r}_{value}$ of decrypted_value to decryptedWord 19: append word or decryptedWord to decryptedList 20: end while21: Write all values from decryptedList to output document |

## 4. Experimental Evaluation

#### 4.1. A. Performance Analysis

#### 4.2. Security Analysis

#### 4.2.1. Avalanche Effect

#### 4.2.2. Bit Independence Criterion

#### 4.2.3. Frequency Analysis

#### 4.2.4. Randomness Verification

- Frequency test
- Block frequency test
- Runs test
- Cumulative sums forward test
- Cumulative sums backward test

#### 4.3. Security against Attacks

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Symbol | Definition |
---|---|

$Le{n}_{max}$ | Maximum ASCII character range (default is 127) |

$Va{l}_{org}$ | Original ASCII character value |

$Va{l}_{initialref}$ | Reflected value of the original ASCII character value |

$Offse{t}_{const}$ | Constant offset value used to avoid non-printable characters (32 by default) |

$Va{l}_{ref}$ | Reflected value of the original ASCII character value after adding the offset |

$Offse{t}_{var}$ | Variable offset computed based on the properties of the tree |

R | The root node of the tree |

${N}_{L}$ | Adjusted value of N so that it is within the range of maximum value |

${N}_{R}$ | Reflection value of ${N}_{L}$ |

Pseudo | Pseudo-random number generated based on the character’s position in the text |

$Characte{r}_{value}$ | Equivalent ASCII character for a given value |

File Size (KB) | Encryption | Decryption | ||
---|---|---|---|---|

Processing Time (ms) | Memory (MB) | Processing Time (ms) | Memory (MB) | |

200 | 1433 | 17 | 1378 | 17 |

400 | 1956 | 21 | 1363 | 18 |

600 | 2118 | 27 | 1637 | 25 |

800 | 2335 | 30 | 1645 | 31 |

1000 | 2528 | 33 | 1995 | 35 |

2000 | 3616 | 55 | 1754 | 56 |

File Size (KB) | Encryption | Decryption | ||
---|---|---|---|---|

Processing Time (ms) | Memory (MB) | Processing Time (ms) | Memory (MB) | |

200 | 1838 | 18 | 1992 | 19 |

400 | 2067 | 22 | 2444 | 25 |

600 | 2190 | 27 | 2750 | 29 |

800 | 2575 | 31 | 3183 | 37 |

1000 | 3034 | 34 | 3658 | 41 |

2000 | 4537 | 55 | 5500 | 49 |

File Size (KB) | Encryption | Decryption | ||
---|---|---|---|---|

Processing Time (ms) | Memory (MB) | Processing Time (ms) | Memory (MB) | |

200 | 123 | 14 | 162 | 7 |

400 | 189 | 23 | 250 | 15 |

600 | 253 | 23 | 320 | 20 |

800 | 413 | 29 | 385 | 29 |

1000 | 479 | 32 | 460 | 36 |

2000 | 1854 | 65 | 775 | 68 |

Technique | Hamming Distance | Avalanche Effect |
---|---|---|

E-ART | 57,852 | 50.1% |

AES | 39,345 | 49.2% |

DES | 39,425 | 49.3% |

Test | p-Value | Remarks |
---|---|---|

Frequency test | 0.7399 | Random |

Block frequency test | 0.7399 | Random |

Runs test | 0.0668 | Random |

Cumulative sums forward test | 0.1223 | Random |

Cumulative sums backward test | 0.5341 | Random |

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**MDPI and ACS Style**

Alabdullah, B.; Beloff, N.; White, M.
E-ART: A New Encryption Algorithm Based on the Reflection of Binary Search Tree. *Cryptography* **2021**, *5*, 4.
https://doi.org/10.3390/cryptography5010004

**AMA Style**

Alabdullah B, Beloff N, White M.
E-ART: A New Encryption Algorithm Based on the Reflection of Binary Search Tree. *Cryptography*. 2021; 5(1):4.
https://doi.org/10.3390/cryptography5010004

**Chicago/Turabian Style**

Alabdullah, Bayan, Natalia Beloff, and Martin White.
2021. "E-ART: A New Encryption Algorithm Based on the Reflection of Binary Search Tree" *Cryptography* 5, no. 1: 4.
https://doi.org/10.3390/cryptography5010004