Abstract
We discuss chaos and its quality as measured through the 0-1 test for chaos. When the 0-1 test indicates deteriorating quality of chaos, because of the finite precision representations of real numbers in digital implementations, then the process may eventually lead to a periodic sequence. A simple method for improving the quality of a chaotic signal is to mix the signal with another signal by using the XOR operation. In this paper, such mixing of weak chaotic signals is considered, yielding new signals with improved quality (with K values from the 0-1 test close to 1). In some sense, such a mixing of signals could be considered as a two-layer prevention strategy to maintain chaos. That fact may be important in those applications when the hardware resources are limited. The 0-1 test is used to show the improved chaotic behavior in the case when a continuous signal (for example, from the Chua, Rössler or Lorenz system) intermingles with a discrete signal (for example, from the logistic, Tinkerbell or Henon map). The analysis is presented for chaotic bit sequences. Our approach can further lead to hardware applications, and possibly, to improvements in the design of chaotic bit generators. Several illustrative examples are included.
Highlights
Most well-known chaotic-based bit generators use a single source, either a continuous or a discrete one [1,2]
Due to a possible synchronization [3,4,5,6] or prediction of a bit sequence because of a finite length representation [7,8,9], such sequences of bits could be compromised. Those problems are based on the fact that, in the finite precision arithmetic, the output of a single input discrete generator becomes periodic, and nonchaotic, even if the length of the output sequence is of order 106
The methods of designing chaotic bit generators lack protection against problems involving finite lengths of bits in number representations, which may result in low entropy levels for the generated chaotic signals
Summary
Most well-known chaotic-based bit generators use a single source (input), either a continuous or a discrete one [1,2]. Due to a possible synchronization [3,4,5,6] or prediction of a bit sequence because of a finite length representation [7,8,9], such sequences of bits could be compromised Those problems are based on the fact that, in the finite precision arithmetic, the output of a single input discrete generator becomes periodic, and nonchaotic, even if the length of the output sequence is of order 106. In the quest to provide strong sequences of chaotic bits, we examine how two signals of mixed nature (continuous and discrete) behave when XORed. We claim that the quality of chaos is improved when various combinations of such signals are analyzed, as it is much more difficult to predict the chaotic binary output of the resulting signal. A deterioration of the quality of chaotic signals, for example, due to a failure in the chaotic system’s components or a Trojan insertion into the system, can be avoided by applying the method presented in this paper
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