SIEA: Secure Image Encryption Algorithm Based on Chaotic Systems Optimization Algorithms and PUFs
Abstract
:1. Introduction
1.1. Related Works
1.2. Contribution of Proposed Method
2. Statement of Problem
- R1: Random numbers should not show any statistical weakness.
- R2: Knowing the subsets of random numbers should not allow the calculation or prediction of predecessor and consecutive random numbers.
- R3: It should not be possible to calculate previous random numbers if the internal state value of RSU is known, even if the internal state value is not known.
- R4: It should not be possible to calculate future random numbers if the internal state value of RSU is known, even if the internal state value is unknown.
3. Details of Proposed Architecture
3.1. Part I: Physical Unclonable Function Module
3.2. Part II: Determination of Initial Conditions and Control Parameters of Chaotic Systems Module
3.3. Part III: Random Number Generator Module
3.4. Part IV: Application Programming Interface
3.5. Part V: Encryption Module
4. Analysis of Proposed Architecture
4.1. Statistical Analysis
4.2. Provable Security Analysis
4.3. Analysis of Key Generation
Algorithm 1 Pseudo code for chaotic bit generator for logistic map |
Input: xValue, ControlParam Output: sequence 1. begin 2. sequence [1:1000000] 3. for i = 1 to 1000000 4. begin 5. xValue = ControlParam * xValue * (1 − xValue) 6. if (xValue < 0.5) 7. sequence[i] = 0 8. else 9. sequence[i] = 1 10. end if 11. end for 12. return sequence 13. end |
5. Conclusions
- The proposed key generator module successfully meets all the requirements (R1, R2, R3 and R4) needed for cryptographical applications.
- The developed key generator module has a high bit output rate (1:1).
- It was shown that the most suitable initial conditions and control parameters that meet the statistical randomness requirements for chaotic systems can be determined with the help of optimization algorithms.
- A user-friendly image encryption algorithm was designed.
- It was shown that the correlation problem in digital images can be overcome by using the space-filling curve transformation method.
- The cryptanalysis of the image encryption algorithm was proven using not only statistical measurements, but also a provable security approach. This approach addressed security concerns via proof with mathematical techniques.
- The proposed encryption architecture is based on a key generator fed from two different entropy sources and cryptographic primitives such as the SHA3 mod function and XOR operator, whose security has been proven as a result of long-term cryptanalysis studies. These design choices specifically address critical security threats such as known-plaintext [79] and chosen-ciphertext [13] attacks.
- Hardware was used as a PUF structure in the study. The dependency of this hardware can be a problem. This hardware dependency can be reduced by using alternative PUF resources in the future.
- The computing realization of chaotic systems is a critical issue, especially considering the problem of digital deterioration. In order to avoid this problem, the calculation sensitivity of the machine in the proposed study should be such that it does not cause this problem. This dependency can be considered as a disadvantage.
- Optimization algorithms are used to determine the initial conditions. It has been evaluated that the computational complexity of optimization algorithms can be interpreted as a disadvantage, although the process of determining the initial conditions with optimization algorithms can be operated offline.
- The cryptology science is a challenge between attacker and designer. Technological advances always keep the possibility of attack alive. Although the security of the proposed method has been proven from different angles, there is a need for a continuous cryptanalysis studies against vulnerabilities that may occur in the future.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Option Group | Options |
---|---|
Chaotic system type | Discrete-time systems (logistic, tent, sine, circle); continuous-time systems (lorenz, rossler, chua, chen); hyperchaotic (hyper_lorenz, hyper_rossler, hyper_chua, hyper_chen); fractional-order systems |
Optimization algorithm | Differential evolution (DE), particle swarm optimization (PSO), symbiosis organisms search (SOS) algorithm, gravitational search algorithm (GSA), harmony search algorithm (HS), golden sine algorithm II (GoldSA-II) |
Statistical test approach | NIST; AIS; chi-square |
Scenarios | Explanation |
---|---|
Option 1 | Let X be the number of state variables of the chaotic system. By applying mode X to the proposed random number generator system outputs, it is decided which state variable is selected by generating a value in the range [0, X]. |
Option 2 | Classical rnd () function can be used to decide which state variable is selected by generating a value in the range of [0, X]. |
Option 3 | It is decided which state variable is selected by generating value in the range of [0, X] using PUF outputs. |
Option 4 | More than one state variable can be selected at the same time. |
Option 5 | Direct selection of specific case variables in line with the best practice samples |
Scenarios | Explanation |
---|---|
Option 1 | The calculated state variable value of chaotic system is compared with a fixed value. If the state variable value is less than the specified fixed value, a value of 0 is generated; if the state variable is greater than or equal to the specified fixed value, a value of 1 is generated. In this way, state variable values are converted into bit values. |
Option 2 | The first three digits after the comma of the calculated state variable value of the chaotic system are selected (can be selected with different values). The selected three-digit values are converted to numerical values between 0 and 255 by applying mod 256. Using the obtained value, an 8 bit length random array of bit values is generated. |
Option | Value |
---|---|
Chaotic system | Logistic Map |
Initial value | 0.468326113906509 |
Control parameter | 4 |
Optimization algorithm | Golden Sine Algorithm II |
Statistical test | NIST |
State variable selection | Logistic map has only one state variable |
Transformation function | Threshold function (Table 3, option 1, threshold value = 0.5) |
Space-filling curve pattern | Figure 6a |
Image | (a) | (b) | (c) | (d) | (e) | (f) | (g) | (h) |
---|---|---|---|---|---|---|---|---|
NPCR | 0.9960 | 0.9957 | 0.9959 | 0.9961 | 0.9960 | 0.9959 | 0.9960 | 0.9959 |
UACI | 0.3349 | 0.3349 | 0.3341 | 0.3346 | 0.3347 | 0.3344 | 0.3351 | 0.3342 |
NIST Test Name | Sample Sequence 1 | Sample Sequence 2 | Sample Sequence 3 |
---|---|---|---|
Monobit test | P = 0.79641 | P = 0.94579 | P = 0.68034 |
Frequency within block test | P = 0.7009 | P = 0.63344 | P = 0.79594 |
Runs test | P = 0.74286 | P = 0.75504 | P = 0.94088 |
Longest run ones in a block test | P = 0.011128 | P = 0.19989 | P = 0.48956 |
Binary matrix rank test | P = 1 | P = 1 | P = 1 |
DFT test | P = 0.80431 | P = 0.17442 | P = 0.08449 |
Nonoverlapping template matching | P = 0.46496 | P = 0.18217 | P = 0.68789 |
Overlapping template matching | P = 0.5839 | P = 0.5839 | P = 0.5839 |
Maurer’s universal test | P = 0.56922 | P = 0.56656 | P = 0.57023 |
Linear complexity test | P = 1 | P = 1 | P = 1 |
Serial test | P = 0.91661 | P = 0.95029 | P = 0.91625 |
Approximate entropy test | P = 0.88844 | P = 0.6324 | P = 0.93674 |
Cumulative sums test | P = 1 | P = 1 | P = 1 |
Random excursion test | P = 0.83024 | P = 0.46499 | P = 0.97178 |
Random excursion variant test | P = 0.45337 | P = 0.90175 | P = 0.51685 |
NIST Test Name | OneRNG | ChaosKey | Linux CSPRNG | Used PUF (Infinite Noise TRNG) |
---|---|---|---|---|
Open hardware | Yes | Yes | N/A | Yes |
Open software | Yes | Yes | Yes | Yes |
Operating principle | RF & Avalanche noise | Reverse biased p-n junction | User-input and timing | Modular entropy multiplication |
Live health monitor | No | No | No | Yes |
Requires firmware | Yes | Yes | N/A | No |
Output rate | 350 kbit/s | 10 Mbit/s | Only very few bit/s | >300 kbit/s |
Pocket-friendly | Yes | Yes | No | Yes |
Price | 40 USD | 40 USD | free | 35 USD |
NIST Test Name | Random Sequence 1 | Random Sequence 2 | Random Sequence 3 |
---|---|---|---|
Monobit test | P = 0.1197 | P = 0.2644 | P = 0.0536 |
Frequency within block test | P = 0.5171 | P = 0.6282 | P = 0.1408 |
Runs test | P = 0.4205 | P = 0.7090 | P = 0.4828 |
Longest run ones in a block test | P = 0.5725 | P = 0.9077 | P = 0.5391 |
Binary matrix rank test | P = 0.7053 | P = 0.9603 | P = 0.1408 |
DFT test | P = 0.3985 | P = 0.0862 | P = 0.0986 |
Nonoverlapping template matching | P = 0.7900 | P = 0.1831 | P = 0.5539 |
Overlapping template matching | P = 0.2946 | P = 0.8186 | P = 0.2737 |
Maurer’s universal test | P = 0.8511 | P = 0.3253 | P = 0.0455 |
Linear complexity test | P = 0.8635 | P = 0.0251 | P = 0.4263 |
Serial test | P = 0.6410 | P = 0.0747 | P = 0.3272 |
Approximate entropy test | P = 0.4704 | P = 0.4867 | P = 0.6237 |
Cumulative sums test | P = 0.1537 | P = 0.3002 | P = 0.0535 |
Random excursion test | P = 0.6779 | P = 0.7999 | P = 0.6208 |
Random excursion variant test | P = 0.1245 | P = 0.3353 | P = 0.0462 |
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Muhammad, A.S.; Özkaynak, F. SIEA: Secure Image Encryption Algorithm Based on Chaotic Systems Optimization Algorithms and PUFs. Symmetry 2021, 13, 824. https://doi.org/10.3390/sym13050824
Muhammad AS, Özkaynak F. SIEA: Secure Image Encryption Algorithm Based on Chaotic Systems Optimization Algorithms and PUFs. Symmetry. 2021; 13(5):824. https://doi.org/10.3390/sym13050824
Chicago/Turabian StyleMuhammad, Aina’u Shehu, and Fatih Özkaynak. 2021. "SIEA: Secure Image Encryption Algorithm Based on Chaotic Systems Optimization Algorithms and PUFs" Symmetry 13, no. 5: 824. https://doi.org/10.3390/sym13050824
APA StyleMuhammad, A. S., & Özkaynak, F. (2021). SIEA: Secure Image Encryption Algorithm Based on Chaotic Systems Optimization Algorithms and PUFs. Symmetry, 13(5), 824. https://doi.org/10.3390/sym13050824