A universal algorithm for transforming chaotic sequences into uniform pseudo-random sequences

Abstract

We present a universal algorithm for transforming chaotic sequences of either chaotic map systems or chaotic differential dynamic systems into uniform pseudo-random sequences. Theoretically，the algorithm is based on bit-operations represented by floating-point algorithm，not aiming at any definite physical chaotic systems. It has been proved that，any real random variable generally has a type of natural tendency of homogenization which exponentially increases bitwise with random variable. As a result，any real chaotic sequence can be completely transformed into the pseudo-random sequence having uniform identical independent distribution. Adopting logistic map，Hénon map and Lorenz system as examples to test the universal validity of the algorithm，respectively，the experiments demonstrate that the algorithm is correct. We can reasonably expect that the universally valid algorithm should become the technological basis of standardized modular design of chaotic pseudo-random sequence generator in hardware implementation.

A universal algorithm to generate pseudo-random numbers based on uniform mapping as homeomorphism

“…From Tables 1 and 2, we conclude that the algorithm is suitable for any chaotic map or series and the results are very satisfactory while the algorithms shown in Refs. [1], [8]- [14] have no universality.…”

“…(i) Complex theories are involved when PRNGs are applied, which is nonbeneficial to computer hardware implementation [1,8,9] and the uniformity of the obtained random sequences cannot be guaranteed. For example, TD-ERCS chaotic system was constructed to generate the PRNG in Ref.…”

“…[8], but calculation speed was decreased and extreme difficulty came into being when FPGA was used because arcsine and arccosine functions were applied to guarantee a uniform distribution. [1,8] A universal algorithm was devised to obtain uniform random binary sequences and overcome the difficulty in implementing computer hardwares. [1] Unfortunately, only relatively uniform time series {x n } was obtained because bit-operations were applied too many times and chaotic properties were damaged.…”

“…[1,8] A universal algorithm was devised to obtain uniform random binary sequences and overcome the difficulty in implementing computer hardwares. [1] Unfortunately, only relatively uniform time series {x n } was obtained because bit-operations were applied too many times and chaotic properties were damaged. It is hard to obtain uniform pseudo random sequences of high quality with the time series {x n }.…”

“…(ii) PRNGs are undesirable because two most important properties of chaotic systems were not utilized sufficiently. [1][2][3]8,9] The two most important properties are sensitively dependent on initial conditions and the ergodicity (in other words, invariant density) of chaos orbits.…”