A new Pseudo random number generator based on generalized Newton complex map with dynamic key

Chaotic systems are widely used in various aspects such as information security, signal processing, and synchronous control. The structural complexity and the chaotic behavior of chaotic systems are two significant factors affecting their practical applications. In this paper, we propose a universal two-dimensional (2D) absolute-cosine chaotic model (ACCM). The 2D-ACCM is composed of a nonlinear bounded cosine function and an absolute value function. It can construct new chaotic maps with simple structures and complex chaotic behaviors on the basis of existing chaotic systems. To verify the effectiveness of the proposed system, we first choose two existing one-dimensional (1D) chaotic maps and one existing 2D chaotic map as the seed maps of the 2D-ACCM to generate two new maps, respectively. The results of chaotic behavior analysis show that these two new maps have more complex chaotic behavior and wider chaotic ranges than seed maps and some advanced chaotic maps. Then a hardware experiment platform based on a field-programmable gate array (FPGA) is used for the hardware implementation of the new maps. Finally, a simple chaos-based pseudo-random number generator (PRNG) is introduced to show the practical application. The experimental results show that the new maps can be easily implemented on the FPGA and the chaos-PRNGs can generate pseudo-random numbers with excellent randomness.

In recent years, a chaotic system is considered as an important pseudo-random source to pseudo-random number generators (PRNGs). This paper proposes a PRNG based on a modified logistic chaotic system. This chaotic system with fixed system parameters is convergent and its chaotic behavior is analyzed and proved. In order to improve the complexity and randomness of modified PRNGs, the chaotic system parameter denoted by floating point numbers generated by the chaotic system is confused and rearranged to increase its key space and reduce the possibility of an exhaustive attack. It is hard to speculate on the pseudo-random number by chaotic behavior because there is no statistical characteristics and infer the pseudo-random number generated by chaotic behavior. The system parameters of the next chaotic system are related to the chaotic values generated by the previous ones, which makes the PRNG generate enough results. By confusing and rearranging the output sequence, the system parameters of the previous time cannot be gotten from the next time which ensures the security. The analysis shows that the pseudo-random sequence generated by this method has perfect randomness, cryptographic properties and can pass the statistical tests.

Chaos has been widely combined with cryptography in the field of information security, especially, a considerable amount of studies of generating pseudo-random numbers based on chaotic systems have been proposed in recent decades. However, many of them are easy to be attacked via utilizing the nonlinear prediction method based on phase space reconstruction and other analysis. Furthermore, under the finite precision environment of computer simulation, there does not exist a random sequence which is truly non-periodic. Unfortunately, few researches had made a related analysis on the above two discussions. This paper is devoted to designing a pseudo-random number generator based on coupled map lattice with time-varying delay, analyzing the random properties of the generated pseudo-random numbers and discussing the dynamical degradation of the system under finite precision of computer simulation. The proposed scheme merely depends on the determining equation; thus, the algorithm itself is not complex, which does not impose high demand on computer hardware and its efficiency is excellent. In order to meet the requirements of using the proposed pseudo-random number generator in cryptography and other practical engineering applications, the proposed pseudo-random number generator is subjected to statistical tests utilizing the well-known test suites, such as NIST SP800-22 and TestU01. Moreover, other related properties, such as permutation entropy, invariant distribution, degradation of dynamical characteristics and parameter test, are also investigated. All results illustrate that the new pseudo-random number generator can generate a high percentage of available pseudo-random numbers for scientific computer simulation and practical applications in the field of information security.

Most commonly-used pseudo-random number generators (PNGs) in computer systems are based on linear recurrence. These deterministic PNGs have fast and compact implementations, andean ensure very long periods. However, the points generated by linear PNGs in fact have a regular lattice structure and are thus not suit able for applications that rely on the assumption of uniformly distributed pseudo-random numbers (PNs). In this paper we propose and evaluate several fast and compact linear, non-linear, and hybrid PNGs for a field- programmable gate array (FPGA). The PNGs have excellent equidistribution properties and very small autocorrelations, and have very long repetition periods. The distribution and long-range correlation properties of the new generators are efficiently, and much more rapidly, estimated at hardware speeds using designed modules within the FPGA. The results of these statistical tests confirm that the combination of several linear PNGs or the combination of even one small non-linear PNG with a linear PNG significantly improves the statistical properties of the generated PNs.

This article explores randomness and how general purpose computers generate it. It also discusses the security requirements that a pseudo-random number generator must satisfy for use in cryptographic applications. And also special attention is paid to the known and new methods of testing random bit sequences. Analyzing the effectiveness of pseudo-random sequence generators is a pressing problem for smart cards when using more advanced methods of encryption and information protection. The available methods show low flexibility and versatility in the means of finding hidden patterns in data. To solve this problem, it is proposed to use algorithms based on multivariate statistics. These algorithms combine all the advantages of statistical methods and are the only alternative for analyzing short and medium-length sequences. The paper considers the scheme of operation of pseudo-random number generators in limited devices. The main requirements for modern smart cards are highlighted. A criterion for checking the randomness of bit sequences of small length (up to 100 bits) is proposed. This approach is appropriate for testing a lightweight pseudo-random number generator in devices with certain resource constraints. The paper presents the compatible distributions of the number of 2-strings and the number of 3-strings of a fixed form of a random bit sequence, which make it possible to carry out a statistical analysis of local sections of this sequence. A possible application of the obtained formulas can be to test the hypothesis of the randomness of the arrangement of zeros and ones in a (0, 1) -sequence of finite length. Research has shown that even with limited resources and a limited entropy environment like a smart card, good quality pseudo-random sequences can be created that can satisfy all the requirements for pseudo-random number generators, even those used for general purpose computers. In the work, the set of statistical tests was expanded to include other tests that are not included in the statistical set of NIST tests, and to analyze the work of the proposed algorithms. The paper presents algorithms for testing a pseudo-random sequence using multivariate statistics to illustrate their possible application in a smart card environment.

Pseudorandom number generators (PRNGs) are fundamental components in cryptographic algorithms. The new concept of multi-PRNG introduced in this article consists of a unique generator capable of producing multiple streams of pseudorandom numbers. Multiscroll chaotic systems are known for generating multiple scrolls within a single attractor. With the aforementioned this article introduces a novel field-programmable gate array (FPGA) implementation of a multi-PRNG based on a multiscroll chaotic memristive Hopfield neural network (MHNN). The main contribution of this work is the generation of multiple spatially dependent PRNG streams from a chaotic multiscroll system by dividing the phase space of the attractor into sub-phase spaces. Each scroll in the multiscroll attractor functions as an independent PRNG. This innovative approach to generating multiple PRNGs from multiscroll chaotic systems is unprecedented in the existing literature. The 5-D MHNN chaotic model used in this work employs hyperbolic tangent and sine functions, which were implemented through a hardware-efficient CORDIC approach. Besides, The FPGA implementation to produce the chaotic time series leverages the Euler method with 32-bit fixed-point arithmetic, selected for its simplicity and low resource utilization. Finally, The randomness of the binary sequences produced by the multi-PRNG is rigorously validated using the NIST SP 800-22a and TestU01 suites, confirming their potential for cryptographic applications.

Many fields need random and pseudorandom numbers, especially in cryptographic applications. d-bit segment binary pseudorandom numbers can be more easily used for block encryption. This paper studies on the statistical test for binary d string pseudorandom number generator (PRNG). Three postulates on the randomness for ideal pseudorandom d-bit segment sequences have been proposed. Based on the FIPS140-2 tests, a statistical test suite for d-bit segment sequences generated by PRNG has been proposed. Using the test suite tests the 100 key streams generated by RC4 PRNG and Matlab PRNG, respectively. The test results show that in the two 100 key streams generated via the two PRNGs with different seeds, about 97% 8-bit strings have passed the test suite, respectively. Using the key streams generated via the two PRNGs encrypts an RGB image. The results have shown that the encrypted RGB images have significant stream encryption avalanche effect.KeywordsInformation TheoryPseudorandom Number StringRandomness PostulateFIPS 140-2 TestNew Test Standard

In this paper, after a short survey describing several known constructions recommended for generating sequences of pseudo-random numbers based on elliptic curves over finite fields of prime order, we propose a method of generating such sequences of points with algorithms driven by a chaotic map. Our construction improves randomness of the sequence generated since it combines good statistical properties of an ECPRNG (Elliptic Curve Pseudo-Random Number Generator) and a CPRNG (Chaotic Pseudo-Random Number Generator). Theoretical analysis shows that periods of the proposed constructions are longer than in the case of the ECPRNG without modulation by a chaotic map. In the second part of the paper we present numerical analysis of the proposed construction to obtain optimal parameters of the generator. We also use some tests from the NIST’s SP 800-22 Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications to analyze statistical properties of the proposed constructions for different values of parameters.

Sequences of pseudorandom numbers are needed for applications of stochastic simulation. Usually they are generated in the computer by a deterministic algorithm which produces a sequence of standard pseudorandom numbers, i.e. a sequence of numbers which “behave” as a realization of a sequence of independent identically distributed random variables having a uniform distribution on the unit interval. In a second step these standard pseudorandom numbers are often transformed in order to fit a given (non-uniform) distribution. For example, Devroy’s book [4] is devoted to this problem. This paper deals only with algorithms producing sequences of standard pseudorandom numbers. Such algorithms are called pseudorandom number generators. Many users of simulation techniques do not think about the pseudorandom number generators implemented in their computers and use the standard software at hand. This habitude is dangerous since the “stochastic quality” of the pseudorandom number sequences is fundamental for the results of stochastic simulation and many pseudorandom number generators in use have serious defects (see e.g. [2], [6], p. 10, and [26], p. 18). The present paper gives a survey on methods for generating deterministic sequences of real numbers which can be used as standard pseudorandom number sequences. Special emphasis is given to generation methods which have been studied by the author’s research group at Darmstadt Technical University.

The generalized feedback shift register pseudorandom number algorithm has several advantages over all other pseudorandom number generators. These advantages are: (1) it produces multidimensional pseudorandom numbers; (2) it has an arbitrarily long period independent of the word size of the computer on which it is implemented; (3) it is faster than other pseudorandom number generators; (4) the “same” floating-point pseudorandom number sequence is obtained on any machine, that is, the high order mantissa bits of each pseudorandom number agree on all machines— examples are given for IBM 360, Sperry-Rand-Univac 1108, Control Data 6000, and Hewlett-Packard 2100 series computers; (5) it can be coded in compiler languages (it is portable); (6) the algorithm is easily implemented in microcode and has been programmed for an Interdata computer.

Transmission of the information in any form requires security. Security protocols used for communication rely on the use of random numbers. Pseudo-random numbers are required with good statistical properties and efficiency. The use of a single chaotic map may not produce enough randomness. The turbulence is padded into the existing map to improve its chaotic behaviour and increase the periodicity. A Pseudo-random number generator (PRNG) with this architecture is devised to generate random bit sequences from secret keys. The statistical properties of newly constructed PRNG are tested with NIST SP 800–22 statistical test suite and were shown to have good randomness. To ensure its usability in cryptographic applications, we analysed the size of its key space, key sensitivity, and performance speed. The test results show that the newly designed PRNG has a 3.6% increase in key space and a 5% increase in its performance speed compared to existing chaotic PRNGs. The novel PRNG with faster performance is found suitable for lightweight cryptographic applications.

The article is dedicated to a detailed review of pseudorandom number generators (PRNGs), their operating principles, features, advantages, limitations, and various implementation methods. Pseudorandom number generators have become an integral part of modern information security, leveraging advanced cryptographic algorithms to ensure reliability and resistance to attacks. Pseudorandom numbers are widely used in cryptography, simulation, computer games, and many other fields. The main goal of the article is to analyze various methods of generating pseudorandom numbers, their features, advantages, and disadvantages, as well as to study their impact on cryptographic strength. The article begins with the basic principles of PRNG operation, focusing on linear congruential generators (LCGs) as the simplest example. Although LCGs are easy to implement, they have significant limitations in the context of cryptographic security, such as short periods and relative predictability. For higher security requirements, cryptographically secure pseudorandom number generators (CSPRNGs) based on block ciphers, hash functions, or other complex mathematical algorithms are recommended. Additional attention is given to generators such as ChaCha20 and NORX, which demonstrate high speed and resistance to cryptanalytic attacks. Theoretical cryptographic strength is also analyzed for generators based on modified algorithms, such as BBS and BM, supported by the complexity of computational problems. In conclusion, the article emphasizes the critical importance of understanding the operation of various PRNGs for information security professionals, as the choice of generator can significantly impact the overall security level of a system. A comparison of the efficiency and cryptographic strength of different pseudorandom number generation methods is presented in a table, supplemented by program algorithms, their characteristics, and examples of their use in cryptography.

Bit cyclic shift method to reinforce digital chaotic maps and its application in pseudorandom number generator

A robust and secure chaotic standard map based pseudorandom permutation-substitution scheme for image encryption

Recently, due to the initial sensitivity, ergodicity, boundedness, and unpredictability of chaos, it has been widely used in the fields of secure communication, signal processing, chaotic synchronization, etc. Moreover, due to the low complexity of hardware implementation, widespread attention to one-dimensional (1D) chaotic maps has been paid from researchers. However, some existing 1D maps have simple chaotic behaviors and discrete chaotic intervals, which have significantly affected the practical application. To address these shortcomings, this paper proposes a 1D chaotic model based on the absolute value function and the cosine function called the cosine-coupled chaotic model (CCCM). It can use existing 1D chaotic maps as seed maps to construct a series of 1D chaotic maps with continuous chaotic intervals and complex chaotic behaviors. To verify the effectiveness of the CCCM, we first generate three new chaotic maps using the CCCM, and then their chaotic properties are analyzed by using various measures including Lyapunov exponent (LE), sample entropy (SE), correlation dimension (CD), 0-1 test and sensitivity. The experimental results confirm that the new chaotic maps have wider chaotic intervals and more complex chaotic behaviors than the corresponding seed maps and some other advanced chaotic maps. Then we use a field programmable gate array (FPGA) as the hardware platform for simple hardware implementations of the new chaotic maps. And a new pseudo-random number generator (PRNG) is introduced to verify practical applications of the new maps. Finally, experimental results show that the new chaotic maps can be implemented on FPGA and the proposed PRNG has strong random properties.