Discrete-time chaotic-map truly random number generators: design, implementation, and variability analysis of the zigzag map

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 TRNG based on the zigzag map passes all of the NIST 800 − 22 statistical randomness tests using simple post processing of the output data.

A smooth chaotic map with parameterized shape and symmetry

“…We compare the performance of the tanh map with that of the modified modified Bernoulli map. An interesting future direction is to extend this comparison to modified versions of the tent map, such as the maps defined in [40][41][42]. Another direction for future research is the calculation of the invariant measure of the tanh map.…”

A <I>&beta;</I>-ary to binary conversion for random number generation using a <I>&beta;</I> encoder

“…When raw random numbers from a physical RNG have bias, i.e., imbalance of the rate of occurence of 0s and 1s, or have correlations among the successive bits, post-processing is applied to improve the randomness of the random numbers [3,4,[6][7][8]. The problem of converting from a sequence of numbers generated from a random process with known or unknown probability distribution to another sequence of random numbers with a prescribed target distribution is known as a random number generation problem in information theory [44].…”

“…On the other hand, there are demands, especially for security purposes, for a physical random number generator that uses measurements of a certain physical phenomenon. Examples of physical random number generation include a random number generator using a semiconductor laser [1] or one that uses an electronic circuit based on a chaotic map [2][3][4][5][6][7][8][9].…”

“…Callegari et al proposed an RNG using a 1.5 bit pipelined A/D converter [4], where taking EXOR of consecutive four output bits is recommended so that the resulting random numbers are robust to the non-ideality of circuit element. In [7], the issue of implementation variations of a PL map called zigzag map for chaos-based RNG is discussed. It was proved in [8] that there exists a sequence of post-processors that asymptotically generate independent and identically distributed (i.i.d.)…”

On Nonlinear Complexity and Shannon’s Entropy of Finite Length Random Sequences

“…The chaotic system is highly sensitive to its initial condition and parameters, together with its random-like and unpredictability, et al, is very useful in improving its security, and is better than the Linear Feedback Shift Register in the cryptographic properties. Therefore, in recent years, chaotic system is regarded as an important pseudorandom source in the design of random bit generators [4][5][6][7][8][9].…”