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Significance Statement
A discrete-time chaotic map, formed by the iteration of the output value in a transformation function, known as map, can be used for the generation of random numbers. Simple piecewise affine input-output (I/O) characteristics have been extensively used for the generation of random bits, e.g., the Bernoulli map and the tent map. The entropy source of a chaotic map is the inherent noise of the system, which is amplified in the positive gain feedback loop by the iteration of the output signal in the map function. The output of the system will be unpredictable after a few first output samples. Pipelining multiple stages of the chaotic map circuit can increase the overall open loop gain of the system. In conclusion, High Speed, capability of integration, and the high quality of the generated bits make the discrete-time chaotic maps excellent candidates for high speed embeddable random number generators.
Presented is a novel discrete-time chaotic mapping named zigzag map that demonstrates excellent chaotic behaviors with applications in Truly Random Number Generators (TRNGs). A TRNG is a device that generates an unbiased and independent stream of bits, which is an essential component in applications such as public key cryptography and digital signature schemes. With secure data processing such as banking and emails increasingly being used on small mobile devices, there is a growing demand for high-speed embedded truly random number generators to ensure the security of the data transmission at the hardware level. Our comprehensive investigation of the zigzag map reveals its critical chaotic characteristics and parameters.
We further present a circuit implementation for the zigzag map based on the current-mode technique. In practice, implementation variations can deteriorate the quality of the output sequence as a result of variation of the chaotic map parameters, such as slope and breaking points. In order to quantify the impact of variations on the map performance, we modeled the variations using a combination of our theoretical analysis of the entropy-rate (to appear in IEEE Transactions on Circuits and Systems II: Express Briefs) and Monte-Carlo simulations on the circuits in our follow-up work published and featured in Electronics Letters, vol. 48, no. 24, pp. 1537-1538, November 2012. Our results demonstrated that the presented circuit implementation is resilient to process variations. Henceforth, a truly random sequence is obtained that passes the NIST 800 − 22 statistical randomness tests, after very simple post-processing of the output bit sequence.
Hamid Nejati, Ahmad Beirami, Warsame H. Ali
Analog Integrated Circuits and Signal Processing, October 2012, Volume 73, Issue 1, pp 363-374.
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, USA 48109
School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, USA 30332
Electrical and Computer Engineering Department, Prairie View A&M University, Prairie view, USA 77446
Abstract
In this paper, we introduce a novel discrete chaotic map named zigzag map that demonstrates excellent chaotic behaviors and can be utilized in truly random number generators (TRNGs). We comprehensively investigate the map and explore its critical chaotic characteristics and parameters. We further present two circuit implementations for the zigzag map based on the switched current technique as well as the current-mode affine interpolation of the breakpoints. In practice, implementation variations can deteriorate the quality of the output sequence as a result of variation of the chaotic map parameters. In order to quantify the impact of variations on the map performance, we model the variations using a combination of theoretical analysis and Monte-Carlo simulations on the circuits. We demonstrate that even in the presence of the map variations, a truly random number generators based on the zigzag map passes all of the NIST 800-22 statistical randomness tests using simple post processing of the output data.
