In the recent past, the storage of images and data in the cloud has shown rapid growth due to the tremendous usage of multimedia applications. In this paper, a modulated version of the Ikeda map and key generation algorithm are proposed, which can be used as a chaotic key for securely storing images in the cloud. The distinctive feature of the proposed map is that it is hyperchaotic, highly sensitive to initial conditions, and depicts chaos over a wide range of control parameter variations. These properties prevent the attacker from detecting and extracting the keys easily. The key generation algorithm generates a set of sequences using a designed chaos map and uses the harmonic mean of the generated sequences as the seed key. Furthermore, the control parameters are modified after each iteration. This change in the control parameters after each iteration makes it difficult for an attacker to predict the key. The designed map was tested mathematically and through simulations. The performance evaluation of the map shows that it outperforms other chaotic maps in terms of its parameter space, Lyapunov exponent, bifurcation entropy. Comparing the designed chaotic map with existing chaotic maps in terms of average cycle length, maximum Lyapunov exponent, approximate entropy, and a number of iterations, it is found to be very effective. The existence of chaos is also proved mathematically using Schwartz’s derivative theorem. The proposed key generation algorithm was tested using the National Institute of Standards and Technology (NIST) randomness test with excellent results.

Recently, cryptographic systems have gained importance because of the broad usage of multimedia in various fields. Hence, it is essential to strengthen the security of multimedia content against illegitimate users. Recently, chaotic map-based security algorithms have gained much attention and are being widely used to conceal intelligible information that is present in images. Hence, it becomes essential to construct a chaotic map that offers excellent security. One of the primary reasons for emphasizing security in chaotic maps is the keyspace. The keyspace can be expanded by increasing the space of control parameter, initial value, or iteration time. The work presented here focuses on expanding the chaotic range of chaotic control parameters using a novel chaotic map and the development of a key generation algorithm to conceal information stored in the cloud.

In general, most prevailing image encryption techniques depend on chaotic maps [

However, many encryption techniques are crypt analyzed due to inadequate key space and weak keys. For instance, [

In this article, we propose a Sine Ikeda modulated map and a key generation algorithm to improve the security of images stored in the cloud. The designed map provides sufficient keyspace, optimum sensitivity to initial conditions and hyperchaotic behavior throughout the chaotic range. The proposed algorithm behaves like a one-time pad and offers considerable security. Moreover, the randomness of the key generated has been tested using the standard NIST test, and the results are found to be excellent. The designed chaotic key can be used to secure stored medical images, military-related images, intelligent transportation images [

The subsequent sections are framed as follows. Section 2 describes the basic and Sine Ikeda modulated maps. Section 3 describes the designed key generation algorithm. In Section 4, performance analysis for the chaotic maps and the key generation algorithm has been performed and Section 5 concludes the article.

An Ikeda map [

This map acts chaotic for values

The proposed chaotic map (Sine Ikeda modulated map) has been constructed by adding sine and Ikeda maps. The chaotic region of the map is stretched over a wide range. Thus, the keyspace of the designed map is expanded. Moreover, the total number of chaotic parameters and the sensitivity of the map to initial conditions increase compared to the Ikeda map. The proposed chaotic map can be defined mathematically as follows:

The control parameters

The chaotic nature of the proposed map can be proved mathematically by using Schwarzian derivative, considering the

The evidence of the concept is in [

The most important property of the chaotic map is its sensitive dependence on the initial conditions. _{1}) is plotted with the initial condition _{2}) is plotted with the initial condition _{1} and x_{2}.

The algorithm for key generation using the designed chaotic map is stated below in this section.

Step 1: Generate

Step 2: Calculate the harmonic means of the generated sequences using the formula

Step 3: Take the harmonic means

Step 4: Use the generated seed key and new values of random control parameters

Step 5: The generated set of sequences of

The key generated can be made more secure by randomly changing the control parameter values after each iteration. Thus, the adversary cannot detect the keys, even using phase plots. The key provided to each image to be encrypted can also be changed by providing the initial seed key as the harmonic mean derived from the image. This acts as a one-time pad, and the adversary cannot hack the key from the image.

In this segment, the bifurcation diagram of the Sine Ikeda modulated map, the Lyapunov exponent and the bifurcation entropy, which are the most important characteristics of the chaotic systems, are analyzed. The parameters used for analysis are

The bifurcation diagram illustrates the change in qualitative behavior of the map as the chaotic control parameters are varied.

The characteristics of discrete chaotic maps are determined using a tool termed the Lyapunov exponent. A positive Lyapunov exponent signifies the presence of chaos. The Lyapunov exponents are calculated using the Jacobian matrix-based method that is described in [

Similarly, the Jacobian matrix of the Ikeda map is computed using

The bifurcation entropy specifies the measure of uncertainty present in the bifurcation structure.

The randomness of the keys generated using the proposed sine-modulated Ikeda is verified using the NIST statistical test suite SP 800-22 [

Statistical analysis | Proportions | Result | |
---|---|---|---|

Block frequency | 0.702187 | 0.99 | Success |

Frequency (M = 128) | 0.446271 | 0.99 | Success |

Cumulative sums | 0.719874 | 0.99 | Success |

Runs | 0.947386 | 0.99 | Success |

Longest run of ones in block | 0.543892 | 0.99 | Success |

Non overlapping template | 0.778451 | 0.99 | Success |

Overlapping template | 0.218973 | 0.99 | Success |

Rank | 0.297543 | 0.99 | Success |

FFT | 0.227232 | 0.99 | Success |

Universal | 0.667239 | 0.99 | Success |

Approximate entropy | 0.799189 | 1.00 | Success |

Serial (m = 16) | 0.948756 | 0.99 | Success |

Linear complexity (M = 500) | 0.487254 | 0.99 | Success |

Random excursions | 0.297654 | 1.00 | Success |

Random excursions variant | 0.643001 | 1.00 | Success |

In this section, the designed chaotic map has been compared with the existing techniques in terms of sensitive dependence on initial conditions, average cycle length, maximum Lyapunov exponent, and approximate entropy.

Parameters | 2D LSCM [ |
2D SLMM [ |
Ikeda [ |
Proposed |
---|---|---|---|---|

No. of iterations to exhibit sensitive dependence on the initial condition | 20 | 15 | 35 | 2 |

The average cycle length (Precision |
5431 | 4858 | 237 | 7543 |

Maximum Lyapunov exponent | 1.403 | 0.540 | 1.232 | 7.032 |

Approximate entropy | 7.905 | 7.904 | 7.901 | 7.992 |

The first parameter from the table shows the number of iterations required for the chaotic maps to change their paths with a one-digit change in the initial condition. The initial conditions for all maps are assumed to be the same. The proposed chaotic map requires only two iterations compared to the existing map. This outcome shows that the sensitive dependence on the initial conditions is high compared to that of the existing 2D maps, which is the desirable characteristic of a chaotic key.

The second parameter is the cycle length. The cycle length of the chaotic map is important to decide on the dynamical degradation of the chaotic map. Also,

The last parameter listed in the table is the approximate entropy of the key generated. The entropy is used to measure the randomness of the key. The maximum entropy for an eight-bit key is 8. It can be inferred from the above table that the designed map has an entropy close to 8 compared to the existing technique. As a result, an attacker will find it difficult to decode and interpret the keys.

In this article, a novel chaotic map has been designed that exhibits chaos over a wide range of the chaotic parameters so that the keyspace is increased. The proposed map has excellent sensitivity to the initial conditions. The absence of the blank window in the bifurcation plot of the designed map reduces the possibility of an exhaustive search attack. The increased number of chaotic parameters further enhances the security of cryptographic systems. Furthermore, the uncertainty and the Lyapunov exponent in the chaotic region are maximal and exhibit hyper-chaotic properties compared to the existing maps. Also, NIST test results show that the designed chaotic map provides excellent randomness; this outcome enhances its applications in cybersecurity.