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Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems
Qianxue Wang, Simin Yu, Chengqing Li, Jinhu Lü, Xiaole Fang, Christophe Guyeux, Jacques M. Bahi
TL;DR
Digital quantization and finite precision degrade chaotic dynamics, motivating a systematic construction method for higher-dimensional digital chaotic systems. The paper proposes random-sequence-controlled HDDCS, proves Devaney chaos through state-network and dynamical analyses, and demonstrates FPGA-based image cryptography. It concludes that fixed-precision operations avoid quantization loss within the proposed system.
Problem
Finite-precision digital implementations degrade chaotic dynamics, while systematic theory for analyzing and overcoming these effects remains lacking.
Method
The paper constructs HDDCS in finite precision using a chaos-generation strategy controlled by random sequences and analyzes its state network and dynamical properties.
Results
HDDCS has a strongly connected state network, dense periodic points, topological transitivity, and satisfies Devaney’s definition of chaos.
Takeaways & Limitations
The proposed framework generates Devaney chaos in finite-precision digital devices and supports FPGA-based image cryptography without quantization loss from fixed-precision operations.
Abstract
from arXiv · showhide
Traditionally, chaotic systems are built on the domain of infinite precision in mathematics. However, the quantization is inevitable for any digital devices, which causes dynamical degradation. To cope with this problem, many methods were proposed, such as perturbing chaotic states and cascading multiple chaotic systems. This paper aims at developing a novel methodology to design the higher-dimensional digital chaotic systems (HDDCS) in the domain of finite precision. The proposed system is based on the chaos generation strategy controlled by random sequences. It is proven to satisfy the Devaney's definition of chaos. Also, we calculate the Lyapunov exponents for HDDCS. The application of HDDCS in image encryption is demonstrated via FPGA platform. As each operation of HDDCS is executed in the same fixed precision, no quantization loss occurs. Therefore, it provides a perfect solution to the dynamical degradation of digital chaos.
I. INTRODUCTION
The paper addresses finite-precision degradation in digital chaos by developing higher-dimensional digital chaotic systems controlled by random sequences. It establishes a general HDDCS framework for finite-precision devices and outlines theoretical analysis and FPGA application.
- Motivation: Finite-precision operations introduce rounding, quantization, and overflow errors that can create large differences between theoretical and actual chaotic orbits.These effects may degrade digital chaotic systems and cause security flaws in chaos-based encryption.
- Proposed approach: The paper develops a general framework for constructing higher-dimensional digital chaotic systems in finite-precision devices.The system uses a chaos-generation strategy controlled by random sequences.
- Analysis and application: The paper analyzes HDDCS through state-network construction, chaos proofs, Lyapunov-exponent calculation, and FPGA-based image-encryption implementation.The stated goal is to generate Devaney chaos directly in real digital devices.
- Finite-precision representation: The framework represents the system using m-dimensional states with N-bit fixed precision and finite integer and fractional parts.The input random numbers are constrained by the limited accuracy of N-bit representation.
- Proposed approach: HDDCS updates finite-precision state components through iterative functions whose selected bits are determined by independent random sequences.Bitwise AND, NOT, and OR operations are used in the iterative formulation.
B. Comparative study of RDCS, IDCS, CBDS, and HDDCS
The comparison distinguishes RDCS, IDCS, CBDS, and HDDCS by dimension, external-input requirements, update behavior, precision domain, and number representation. HDDCS extends random-bit-controlled digital chaos to higher dimensions while remaining suitable for digital devices.
- Precision domain: RDCS uses infinite precision, while IDCS, CBDS, and HDDCS use finite precision suitable for digital computers and devices.The implementation environment separates RDCS from the other three systems.
- Data representation: HDDCS extends the processed data from integer-domain representations to finite integer and fractional parts.The paper specifies N = P + Q for the finite-precision representation.
- Dimension: HDDCS extends IDCS and CBDS from one-dimensional systems to any finite dimension.The comparison identifies higher-dimensional operation as a principal distinction from these systems.
- External inputs: RDCS does not require external inputs, whereas IDCS, CBDS, and HDDCS do.This comparison concerns the systems’ input requirements.
- Update behavior: RDCS updates all bits at each iteration, IDCS updates one bit, and CBDS uses multiple random bitwise operations; HDDCS applies the CBDS-like approach in higher dimensions.The comparison describes update granularity and the role of random bitwise operations.
C. Network analysis of the state space of HDDCS
HDDCS state networks represent finite-precision state transitions as directed graphs. The constructed network is strongly connected, meaning every state can reach every other state, supporting the subsequent chaos proof.
- State-network representation: HDDCS state networks model possible state combinations as nodes and state mappings as directed edges.For HDDCS, nodes are combinations of the component states, with edges defined by the transition function.
- Connectivity result: The HDDCS state network is strongly connected, so every node is reachable from every other node.This connectivity property is established for the associated state network of the system.
- Illustrative transition system: For the illustrated 2-D system, the transition diagram is derived from the concrete state transitions of system (8).The example uses N = 2 with P = 2 and Q = 0; the uncontrolled system uses bitwise XOR.
- Random-sequence-controlled transitions: The iterative HDDCS updates use random sequences generated by a true random number generator, and their concrete transitions produce the strongly connected network.The transition relation is obtained from the iterative form associated with Eqs. (6)–(8).
III. CHAOTIC PERFORMANCE OF HDDCS
The paper establishes chaotic performance for HDDCS in finite precision by proving that its map satisfies Devaney’s definition of chaos. This requires dense periodic points and transitivity in the definitional domain.
- Devaney chaos: The HDDCS map is proven to satisfy Devaney’s definition of chaos in finite precision.The proof addresses the two stated properties required in this section.
- Periodic points: Periodic points of HDDCS are required to be dense in its definitional domain.The section states this property as one component of the Devaney-chaos result.
- Transitivity: The HDDCS map is transitive on its definitional domain.Transitivity is stated alongside dense periodic points as part of the chaos definition used here.
A. Dense periodic points of HDDCS
The paper proves that HDDCS has dense periodic points in the metric space (E, d), constructing periodic sequences that return to prescribed states after finite iterations.
- Theorem 1 establishes that HDDCS periodic points are dense in the metric space (E, d).
- The proof constructs sequences whose first k0 entries approximate a target point, with m × 2^-Nk0 < ε.Matching sufficiently many leading sequence entries makes the metric distance arbitrarily small.
- After k0 iterations, the constructed trajectory returns to the target state (x̂1, x̂2, …, x̂m), yielding a periodic point.
- Strong connectivity supplies a path between intermediate and target states, allowing the sequence construction to close into a periodic orbit.
- The resulting periodic points are summarized by the diagram in Fig. 4.
B. Transitive property of HDDCS
The paper proves that the HDDCS map GF is topologically transitive: every pair of nonempty open sets can be connected by some iterate of GF.
- Theorem 2 establishes that GF is topologically transitive in the metric space (E, d).
- Topological transitivity requires an iterate Gn0 that maps one nonempty open set U′ into an intersection with another, U′′.
- Strong connectivity provides a finite path from the intermediate state to the target state, producing an iterate whose image intersects U′′.
- The paper concludes that GF is transitive in (E, d), and combines this result with dense periodic points to establish Devaney chaos.
IV. LYAPUNOV EXPONENTS OF A CLASS OF HDDCS
The paper estimates the Lyapunov exponents of HDDCS for the case N = P, equivalently Q = 0.
- Lyapunov exponents of HDDCS are estimated under the condition N = P (Q = 0).
A. The general expression of equivalent decimal for GF
The section constructs equivalent decimal representations for finite-precision multidimensional digital states from binary sequences, random sequences, bit updates, and sequence shifts.
- An m-dimensional array of N-bit integers is represented in binary form before conversion to equivalent decimal values.
- Random number sequences are expressed as corresponding decimal fractions using weighted binary terms.
- The multidimensional iterative system is expressed through component functions F1(·), F2(·), …, Fm(·) over the state vector X.
- The decimal representation is obtained by combining the contributions from the relevant sequence expressions and shifted values.
- The resulting digital state updates multiple random bits and shifts one value in every one-sided infinite sequence.
B. The mathematical expression for ∂gi(y1,y2,...,ym)
The section derives the partial derivatives of the component functions and uses them to obtain the Jacobian matrix for the digital system.
- Partial derivatives are evaluated from finite differences of each component function with respect to individual state variables.
- The resulting Jacobian matrix is the diagonal matrix diag(2^N, 2^N, · · ·, 2^N).
C. Estimating the Lyapunov exponents
The section estimates the Lyapunov exponents of HDDCS from the iterated Jacobian matrices.
- The Lyapunov exponent of a specific HDDCS is estimated from the logarithmic growth of the iterated matrix product Φ_n = J_n.
V. FPGA-BASED REAL-TIME APPLICATION OF 3D-DCS
The paper implements 3D-DCS on FPGA and applies it to RGB image encryption, using random-sequence control, synchronized transmitter–receiver states, and fixed-precision digital operations. Experiments with a 640 × 480 image show correct decryption only when the two systems’ states exactly match, supporting the reported feasibility of the implementation.
- Image-encryption method: The RGB image is split into R, G, and B components, whose pixel values are scrambled separately using 3D-DCS.
- 3D-DCS design: The 3D-DCS example uses N = 32 and three random sequences s, u, and v; ISAAC is adopted for demonstration.
- 3D-DCS design: The three output channels pass the NIST randomness test suite, while their mutual correlations and autocorrelation strengths are reported as very low.
- FPGA implementation: Three oscillator-ring TRNG blocks provide control inputs, while the 3D-DCS processing block updates states in clock-driven feedback.
- Hardware system: The transmitter encrypts RGB data before public-channel transmission, and the receiver uses synchronized 3D-DCS states to recover pixels through separate XOR operations.
- Experimental result: 640 × 480 image experiments show that correct decryption occurs only when transmitter and receiver 3D-DCS states exactly match.
- Experimental result: FPGA-based image cryptography implementation confirms the reported feasibility and applicability of the HDDCS approach.