# Dynamic Random Graph Protection Scheme Based on Chaos and Cryptographic Random Mapping

^{1}

^{2}

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## Abstract

**:**

## 1. Background

- (1)
- Non-interactivity: since network parameters are spontaneously and randomly changed, it is difficult to connect and communicate data with other nodes without exchanging one’s own network parameters with those of other nodes.
- (2)
- Randomness: Because chaotic algorithms and linear shift registers have randomness over small sets and may have mapping degeneracy problems over infinitely large sets, it may be hard to satisfy the requirement that random graph whose length is approximately infinite in theory also have randomness.
- (3)
- Uncorrelatedness: As the attacker is likely to have the ability to infer the previous or next network parameters by the change pattern of the current network parameters, and may also have the ability to infer the change rule of the network parameters of other nodes, the random graph has to satisfy the requirement that each column element and each row element are uncorrelated with each other when it is dynamically generated.
- (4)
- Distributivity: Since the network hopping is generally carried out in a multi-node network environment, the requirement of dynamically generating the same random map on multiple hosts in different locations needs to be satisfied.

- (1)
- We propose the idea of mapping chaos to cryptography and mathematically show that cryptographic ciphertexts have very good randomness and security.
- (2)
- We present a random graph scheme (CandCRM) based on chaos and cryptographic mapping. the random graph generated by CandCRM is effective against network attacks due to its excellent randomness and non-correlation.
- (3)
- The CandCRM scheme has good application. In a hopping network, it is suitable for deployment on multiple hopping hosts operating independently with little interaction between hosts.

## 2. Solution Design

**R**= (x

_{ij})

_{z}

_{×n}), we propose a random graph protection solution based on chaotic and cryptographic random mapping. The overall solution framework is shown in Figure 1 and consists of three parts: (1) system initialization; (2) random graph calculation; (3) update of network parameters. Firstly, the system initialization comprises the initialization of the total controller and nodes, including the assignment time period (T), the initial value (μ, X

_{0}) of logistic mapping, the initial value of the encryption algorithm (key sequence K = {k

_{0}, k

_{1}, k

_{i},…, k

_{θ}}), and the set of candidate network parameters (

**V**= {v

_{i}|0 ≤ I ≤ +

**Z**}), as shown in steps ① and ③ of Figure 1. Next, the listener program is always listening and once the start synchronization signal is received, the hopping period (T − 0.1~T + 0.1) is started and timed while the random graph is being calculated, as shown in step ⑤.

#### 2.1. Chaos and Cryptographic Random Mapping (CandCRM) Algorithm

_{k}(m) (m = X

_{i}, k = k

_{i}) for encryption mapping, the plaintext X

_{i}is a chaotic sequence, and the encryption secret key k

_{i}is also a chaotic sequence, so that it satisfies the requirements of randomness, irrelevance, non-interactivity, and distribution at the same time.

_{i}|1 ≤ i ≤ θ} and K = {k

_{i}|1 ≤ i ≤ θ}, and that the chaotic sequences satisfy the randomness and irrelevance requirements; and (2) assume that the encryption algorithm satisfies cryptographic security.

_{k}(m) encrypts the chaotic sequence item X

_{i}and the secret key (k

_{i}) to generate the cipher text (c

_{i}), then modulo V using c

_{i}(first collate c

_{i}as a fixed number of points) to obtain the modulus (a

_{i}), then match a

_{i}with the subscript of V elements to obtain the element corresponding to the subscript of V, and write it as r

_{i}= V[a

_{i}], and check whether the current matched element collides with the already matched element, if so, regenerate X, and encrypt, modulo and map again until no more collision occurs (see Algorithm 1).

_{i}generated by the encryption algorithm Enc

_{k}(m) is random and uncorrelated. In terms of the law of probability distribution, when because the probability distribution of the input chaotic sequences is uniform, random and uncorrelated, after encrypting them, then the probability distribution of the output cipher text is similarly uniform when random, as stated in Theorems 1 and 2.

**Theorem**

**1**

**.**Suppose that g(x) is an invertible function. Let X be an element on the domain of definition of g. X is randomly selected according to uniform distribution (that is, X is a random variable), then the corresponding output g(x) is random and uniformly distributed in the elements of the upper domain.

**Proof**

**of**

**Theorem**

**1.**

^{−1}of g, the element g(x) of the random above domain is equal to b as long as the element X of the random definition domain is equal to g

^{−1}(b). The probability of X = g

^{−1}(b) is 1 divided by the number of elements of the definition domain, so the probability of g(X) = b is also 1 divided by the number of elements of the definition domain. □

**Theorem**

**2**

**.**Define the function f

_{clear}=

_{m}(k) as f

_{clear}=

_{m}(k) = f(m,k) for each plaintext m, if the function f

_{m}is invertible for each plaintext m, then the cryptosystem using the encryption function f is perfectly confidential.

**Proof**

**of**

**Theorem**

**2.**

_{m}(k). By Theorem 1, the probability distribution of this random variable is uniform. □

- (1)
- Improved non-interactivity capability in a network environment. The traditional random mapping algorithm is more commonly used in the case of interaction, while the improved algorithm requires almost no interaction, and it satisfies the randomness of random mapping in both space and time.
- (2)
- It has improved the resistance to network attacks, such as resistance to known plaintext and cipher text attacks, key attacks, etc. Since the chaotic sequence generated by the chaotic algorithm itself has randomness, when the terms of the chaotic sequence are used as plaintext and cipher text, the probability distribution of the cipher text generated by encrypting the plaintext and the key multiple times is uniform and random, so it can effectively defend against network assault.

#### 2.1.1. CandCRM Algorithm Implementation

_{1},…,X

_{i},…,X

_{θ}}, Enc

_{k}(X

_{i}) encrypts X

_{i}, generates cipher text c

_{i}, and collates c

_{i}into fixed points, modulo cipher text c

_{i}with V, i.e., a

_{i}= Mod(c

_{i},V), chooses the corresponding element V[a

_{i}] from the set of network parameters V = {V

_{1},V

_{2},…, V

_{N}}, a

_{i}=1,…,N,

**R**+ = V[a

_{i}], and eliminates V[a

_{i}] from V. Finally,

**R**is output.

Algorithm 1 CandCRM algorithm |

Input: array A = {a_{1}, a_{2},…, a_{n}}, i, j, X, V, k_{i}; else then let i = i + 1.; |

Output: R; |

Initialization, j = 1, A = 0, c_{i} = 0; |

1: Compute the cipher text c_{i} = Enc_{k}(X[j]), a_{i} = Mod(c_{i},V); |

2: If j = = 1, then execute step 6. |

else then execute step 3. |

3: i = 1; |

4: If (a_{i} = A[i]) = = 1, then compute c_{i} = Enc_{k}(X[j]), a_{i} = Mod(c_{i},V) and return to perform step 3. |

5: if (i < j + 1) = = 1, execute step 6. |

else then execute step 4: |

6: j = a_{i}, R = V[A[j]], j = j + 1; |

7: if (j = N + 1) = = 1, then output R |

else return to execution step 1. |

#### 2.1.2. Security Analysis

- (1)
- Irreversibility

**Theorem**

**3.**

**R**, inferring the latest periodic element h

_{i,j−1}from the current element h

_{ij}is difficult because the computational complexity of inferring h

_{i,j−1}is 2O(2

^{l}) (where, l is the word length of secret key).

**Proof**

**of**

**Theorem**

**3.**

**R**, h

_{i,j−1}can only be deduced using exhaustive method, and the computational complexity of cryptographic random mapping for hi,j are 2O(2

^{l}). It is difficult to obtain h

_{i,j−1}in polynomial time, so it is difficult to introduce h

_{i,j−1}based on h

_{i,j}, and thus has irreversibility. □

- (2)
- Unpredictability

**Theorem**

**4.**

**R**, it is difficult to predict the elements h

_{i,j}

_{+}

_{1}in the next period based on the current elements h

_{i,j}, because the computational complexity of h

_{i,j}

_{+}

_{1}is estimated to be 2O(2

^{l}).

**Proof**

**of**

**Theorem**

**4.**

_{i,j}, h

_{i,j}

_{+}

_{1}can only be inferred by exhaustive method, and the computational complexity of cryptographic random mapping of h

_{i,j}is also 2O(2

^{l}). It is difficult to obtain h

_{i,j}

_{+}

_{1}in polynomial time, so it is difficult to predict h

_{i,j}

_{+}

_{1}and thus possesses unpredictability. □

- (3)
- Resistance to external and internal attacks. The hopping network has its own characteristics to actively resist various external attacks such as DoS, DDoS, and traffic analysis, and random graph R can resist both external and internal attacks. In addition, assuming that the algorithm operates in a TrustZone environment with secure hardware and the initial values of the algorithm are passed in a secure channel, the internal attack is mainly on the random graph itself, and according to Theorems 3 and 4 above, the attacker cannot infer the value of the previous period from the current value of the random graph, nor can he predict the value of the next period, so the random graph can defend against the internal attack in this case. The random graph is thus resistant to internal attacks in this case.

## 3. Experimental Analysis

#### 3.1. Data set and Environment Description

_{i}}, i = 1, …, 256 and v

_{i}is enumeration type values, the other is a set of pseudo random sequences, expressed as Y(t) = {Y

_{1}(t), Y

_{2}(t), …, Y

_{10}(t), t∈T}, Y

_{1}(t), Y

_{2}(t), …, Y

_{10}(t).

#### 3.2. Parameter Setting and Evaluation Index

- (1)
- Balance check: the balance reflects the uniform distribution of the random sequence, so it is necessary to check whether the balance of the random sequence is reasonable, that is, to check the difference between the “maximum” and “minimum” values in the sequence, the formula is:

_{0}and n

_{1}, respectively, n = n

_{0}+ n

_{1}. The calculated value is compared with the χ

^{2}value with 1 degree of freedom, and the significant effect is taken as a = 0.05 in the test. From the χ

^{2}distribution table, it can be found that the χ

^{2}value with a significant effect of 0.05 is 3.841. If the calculated value is less than 3.841, the random graph passes the balance check.

- (2)
- The variance checks, as an indicator of the equilibrium check of the random series, is given by:

- (3)
- The autocorrelation checks, which can be expressed as an autocorrelation function, is given by:

_{aa}(0) is the correlation between the sequence and its own sequence 0 displacement elements, R

_{aa}(1) is the correlation between the sequence and its own sequence 1 displacement elements, R

_{aa}(l) is the correlation between the sequence and its own series l displacement elements.

- (4)
- The cross-correlation check, which can be expressed as an autocorrelation function, is given by:

_{ab}(0) is the correlation of the sequence with another sequence of 0 displacement elements, R

_{ab}(1) is the correlation of the sequence with another sequence of 1 displacement elements, and R

_{ab}(l) is the correlation of the sequence with another sequence of l displacement elements.

^{5}ms). In the simulated experimental environment, one iteration of the algorithm corresponds to one hopping period. In the experiments, CandCRM was performed for 1 × 10

^{4}iterations with different initial values and different control parameters, and the v

_{i}frequency as well as the frequency was counted. In Figure 4, the results show that the frequencies and frequencies maintain a uniform distribution. On the basis of the above statistical results, we made the following additional check:

#### 3.2.1. Balance Check

_{1}(t), Y

_{2}(t), …, Y

_{10}(t) are much lower than the reference value of 3.841 for the initial values X

_{0}= 0.35 and X

_{0}= 0.36 and the control parameters μ = 3.7 and μ = 3.8, which indicates the balance effect of CandCRM well. In addition, we compare CandCRM with three other network parameter mapping methods, which are m-linear shift register (m-LSR) based on m-sequence [20,21,22], pseudo-random number generator (PRNG) [23], and function f(N

_{i},k) of the network parameter randomization method [24]. The experimental comparison results are shown in Figure 6. Compared with the remaining three methods, the method in this paper has obvious advantages in terms of balance.

#### 3.2.2. Variance Check

^{4}iterations of CandCRM were completed, we counted the frequency of each network parameter (v

_{i}) and the variance of the frequency, as shown in Figure 7 and Figure 8. The results show that the mean values of the frequencies of the network parameters are equal for different initial values and different control parameters. Also, variance tests were performed on the frequencies at different initial values and different control parameters, and the frequencies of the network parameters deviate from their means to a lesser extent and are generally smooth.

#### 3.2.3. Autocorrelation Check

^{4}iterations to generate 256 network parameters (r

_{i}) for Y

_{1}(t), Y

_{2}(t), …, Y

_{10}(t), respectively, and to check the level of autocorrelation between these elements, we carry out an autocorrelation correlation check. As shown in Figure 9, the autocorrelation levels of these random variables are always below 0.15 for different initial values and different control parameters, indicating that they pass the autocorrelation test.

#### 3.2.4. Cross-Correlation Check

## 4. Discussion

#### Equilibrium Check in Extreme Cases

_{0}= 0.38 and µ = 4.0 are set as the initial values of the LMAs and the initial values of the logistic mapping generator in the CandCRM scheme. Next, iterate through the LMAs and the logistic mapping generator 1 × 10

^{7}times, respectively. It should be noted that starting from the uneven results generated by the LMAs and the logistic mapping generator, the output values of the LMAs are modelled directly to

**V**, from which the corresponding values are obtained through the modelling results. The values generated by the iterations of the logistic mapping generator are used as parameters of the encryption algorithm in the CandCRM, and the cipher text is encrypted with the parameters and processed into fixed-point numbers, and the fixed-point numbers are used to modulo

**V**. The values are obtained from

**V**through the modulo results. The statistical values are shown in Figure 11 and the results show that CandCRM is better balanced than LMAs.

## 5. Conclusions and Future Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Frequency and frequency comparison under different parameters: (

**a**,

**c**) frequency comparison of network parameters, (

**b**,

**d**) comparison of frequency statistics.

**Figure 6.**(

**a**,

**b**) Comparison of balance check results between different methods. (

**a**,

**b**) Comparison of variance levels between different methods.

**Figure 10.**(

**a**,

**b**) Comparison of check results R

_{ab}(0); (

**c**,

**d**) Comparison of intercorrelation check results R

_{ab}(l).

**Figure 11.**(

**a**,

**b**) Comparison of balance levels and variance between CandCRM and LMAs. (

**a**) Comparison of balance level between CandCRM and LMAs, (

**b**) comparison of variance between CandCRM and LMAs.

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**MDPI and ACS Style**

Fang, Z.; Xu, Z.
Dynamic Random Graph Protection Scheme Based on Chaos and Cryptographic Random Mapping. *Information* **2022**, *13*, 537.
https://doi.org/10.3390/info13110537

**AMA Style**

Fang Z, Xu Z.
Dynamic Random Graph Protection Scheme Based on Chaos and Cryptographic Random Mapping. *Information*. 2022; 13(11):537.
https://doi.org/10.3390/info13110537

**Chicago/Turabian Style**

Fang, Zhu, and Zhengquan Xu.
2022. "Dynamic Random Graph Protection Scheme Based on Chaos and Cryptographic Random Mapping" *Information* 13, no. 11: 537.
https://doi.org/10.3390/info13110537