In last years, low-dimensional and high-dimensional chaotic systems have been implemented in cryptography. The efficiency and performance of these nonlinear systems play an important role in limited hardware implementations. In this context, low-dimensional chaotic systems are more attractive than high-dimensional chaotic systems to produce the pseudorandom key stream used for encryption purposes. Although low-dimensional chaotic maps present some security disadvantages when they are used in cryptography, they are highly attractive due its simple structure, discrete nature, less arithmetic operations, high output processing, and relatively easy to implement in a digital system. In this paper, we proposed both a pseudorandomly enhanced logistic map (PELM) and its application in a novel pseudorandom number generator (PRNG) algorithm, which produces pseudorandom stream with excellent statistical properties. The proposed PELM is compared with logistic map by using histograms and Lyapunov exponents to show its higher benefits in pseudorandom number generator. In contrast to recent schemes in the literature, we present a comprehensive security analysis over the proposed pseudorandom number generator based on pseudorandomly enhanced logistic map (PRNG–PELM) from a cryptographic point of view to show its potential use in secure communications. In addition, the randomness of the PRNG–PELM is verified with the most complete random test suit of National Institute of Standards and Technology (NIST 800-22) and with TestU01. Based on security results, few arithmetic operations required, and high output rate, the proposed PRNG–PELM scheme can be implemented in secure encryption applications, even in embedded systems with limited hardware resources.

A novel pseudorandom number generator based on pseudorandomly enhanced logistic map

The present invention is related with a computer-implemented method for generating a pseudo-random bit generator including the steps: a. Defining a multi-modal map by the equation: fβ=β(dr+1−x)(x−dr), x ∈ ℑr; b. Set the value of k ∈ +, and obtaining the values of βj, for j=from at least 1, to the final value k by the following equations: β1=4k; βj=(j)(β1); for 2≦j≦k and taking the values of βj and split the space into 2j regions δ1j, to δ2jj which are determined by values k1j to k(2j)−1j.

Pseudo-random bit generator based on multi-modal maps

A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption.

Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption

In this paper we study the dynamical behav- ior of the one-dimensional discrete-time system, the so-called iterated map. Namely, a bimodal quadratic map is introduced which is obtained as an amplifica- tion of the difference between well-known logistic and tent maps. Thus, it is denoted as the so-called differ- ence map. The difference map exhibits a variety of be- haviors according to the selection of the bifurcation parameter. The corresponding bifurcations are studied by numerical simulations and experimentally. The sta- bility of the difference map is studied by means of Lyapunov exponent and is proved to be chaotic ac- cording to Devaney's definition of chaos. Later on, a design of the electronic implementation of the differ- ence map is presented. The difference map electronic circuit is built using operational amplifiers, resistors and an analog multiplier. It turns out that this elec- tronic circuit presents fixed points, periodicity, chaos and intermittency that match with high accuracy to the corresponding values predicted theoretically.

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Difference map and its electronic circuit realization

This work proposes a modified logistic map based on the system previously proposed by Han in 2019. The constructed map exhibits interesting chaos related phenomena like antimonotonicity, crisis, and coexisting attractors. In addition, the Lyapunov exponent of the map can achieve higher values, so the behavior of the proposed map is overall more complex compared to the original. The map is then successfully applied to the problem of random bit generation using techniques like the comparison between maps, X O R , and bit reversal. The proposed algorithm passes all the NIST tests, shows good correlation characteristics, and has a high key space.

https://www.mdpi.com/2073-8994/12/5/829/pdf

A Two-Parameter Modified Logistic Map and Its Application to Random Bit Generation

Pseudo-random bit sequence have a wide range of applications in the field of cryptography and communications. For the good chaotic dynamical properties of chaotic systems sequence such as randomness and initial sensitivity, chaotic systems have a strong advantage in generating the pseudo-random bit sequence. However, in practical use, the dynamical properties of chaotic systems will be degraded because of the limited calculation accuracy and it even could cause a variety of security issues. To improve the security, in full analyses of the pseudo-random bit generator proposed in our former paper, we point out some problems in our former design and redesign a better pseudo-random bit generator base on it. At the same time, we make some relevant theoretical and experimental analyses on it. The experiments show that the design proposed in this paper has good statistical properties and security features.

A design of pseudo-random bit generator based on single chaotic system

Investigating immobilization efficiency of Pb in solution and loess soil using bio-inspired carbonate precipitation.

Feasibility study of applying electrokinetic technology coupled with enzyme-induced carbonate precipitation treatment to Cu- and Pb-contaminated loess remediation

Artificial glacier melt reduction is gaining increasing attention because of rapid glacier retreats and the projected acceleration of future mass losses. However, quantifying the effect of artificial melt reduction on glaciers in China has not been currently reported. Therefore, the case of Urumqi Glacier No.1 (eastern Tien Shan, China) is used to conduct a scientific evaluation of glacier cover efficiency for melt reduction between 24 June and 28 August 2021. By combining two high-resolution digital elevation models derived from terrestrial laser scanning and unmanned aerial vehicles, albedo, and meteorological data, glacier ablation mitigation under three different cover materials was assessed. The results revealed that up to 32% of mass loss was preserved in the protected areas compared with that of the unprotected areas. In contrast to the unprotected glacier surface, the nanofiber material reduced the glacier melt by up to 56%, which was significantly higher than that achieved by geotextiles (29%). This outcome could be attributed to the albedo of the materials and local climate factors. The nanofiber material showed higher albedo than the two geotextiles, dirty snow, clean ice, and dirty ice. Although clean snow had a higher albedo than the other materials, its impact on slowing glacier melt was minor due to the lower snowfall and relatively high air temperature after snowfall in the study area. This indicates that the efficiencies of nanofiber material and geotextiles can be beneficial in high-mountain areas. In general, the results of our study demonstrate that the high potential of glacier cover can help mitigate issues related to regions of higher glacier melt or lacking water resources, as well as tourist attractions.

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Quantifying the Artificial Reduction of Glacial Ice Melt in a Mountain Glacier (Urumqi Glacier No. 1, Tien Shan, China)

Inappropriate irrigation could trigger migration of heavy metals into surrounding environments, causing their accumulation and a serious threat to human central nervous system. Traditional site remediation technologies are criticized because they are time-consuming and featured with high risk of secondary pollution. In the past few years, the microbial-induced carbonate precipitation (MICP) is considered as an alternative to traditional technologies due to its easy maneuverability. The enzyme-induced carbonate precipitate (EICP) has attracted attention because bacterial cultivation is not required prior to catalyzing urea hydrolysis. This study compared the performance of lead (Pb) and copper (Cu) remediation using MICP and EICP respectively. The effect of the degree of urea hydrolysis, mass and species of carbonate precipitation, and chemical and thermodynamic properties of carbonates on the remediation efficiency was investigated. Results indicated that ammonium ion (NH4 +) concentration reduced with the increase in lead ion (Pb2+) or copper ion (Cu2+) concentration, and for a given Pb2+ or Cu2+ concentration, it was much higher under MICP than EICP. Further, the remediation efficiency against Cu2+ is approximately zero, which is way below that against Pb2+ (approximately 100%). The Cu2+ toxicity denatured and even inactivated the urease, reducing the degree of urea hydrolysis and the remediation efficiency. Moreover, the reduction in the remediation efficiency against Pb2+ and Cu2+ appeared to be due to the precipitations of cotunnite and atacamite respectively. Their chemical and thermodynamic properties were not as good as calcite, cerussite, phosgenite, and malachite. The findings shed light on the underlying mechanism affecting the remediation efficiency against Pb2+ and Cu2+.

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Yi-Xin Xie

Papers

A design of pseudo-random bit generator based on single chaotic system

Investigating immobilization efficiency of Pb in solution and loess soil using bio-inspired carbonate precipitation.

Feasibility study of applying electrokinetic technology coupled with enzyme-induced carbonate precipitation treatment to Cu- and Pb-contaminated loess remediation

Quantifying the Artificial Reduction of Glacial Ice Melt in a Mountain Glacier (Urumqi Glacier No. 1, Tien Shan, China)

Immobilizing lead and copper in aqueous solution using microbial- and enzyme-induced carbonate precipitation