Random generation is what comes to mind in naturalistic settings.

Being unpredictable is useful for creativity, exploration, and good decision-making. Decades of research asking people to generate random sequences of numbers have concluded that people systematically deviate from randomness, but individuals do so in very idiosyncratic ways. However, there is no consensus on the cause of these rich individual differences: most theories postulate that people achieve this by spending cognitive resources monitoring their own output and changing the way they say items accordingly, whereas a Local Sampling account postulates that people tap into a general-purpose ability to produce samples, which they would use to make judgments and choices, and which are inherently somewhat unpredictable. Here we distinguish between these possibilities by asking people to generate sequences both at random and as they come to mind. We employ several measures tapping into different ways sequences could deviate from randomness. We find that, consistent with the Local Sampling explanation, people deviate from randomness in virtually identical ways in both sequences, with high individual differences pointing to a common cognitive process. We follow up these findings by computationally modeling human performance using several Local Sampling models. We find many participants had the same model as best-fitting in both sequences; with estimated parameters correlating strongly across tasks showing individual differences in how dependent on previous items sampling is. Overall, we conclude that random generation is better understood as employing a general-purpose faculty, Local Sampling, which is stable across time and tasks, with differences in the sampler’s features resulting in differences in random generation performance.

A random number sequence is a sequence of numbers that exhibit statistical randomness. In other words, all numbers in the sequence have an equal chance and it’s hard to predict the next number in the sequence. They are most commonly used in cryptography to strengthen the security. Same key can be used by sender and receiver to generate the same set of random numbers. The pseudo-random number generation is an important task in computer science with various security features. There are 2 types of random number generators; they are True Random Number Generators (TRNG) and Pseudo Random Number Generators (PRNG). In recent years, PRNGs are implemented in hardware or software using chaotic maps. Some of the chaotic maps used in PRNG are Logistic map, Lorenz map, Henon map, Skew Tent map, Saw tooth map, etc. The Tinker Bell map is a discrete-time chaotic dynamical system with four system parameters. It is a most commonly studied examples of dynamical systems that exhibit chaotic behaviour. Chaotic maps possess several drawbacks such as the discontinuity of ranges, non-uniform distribution, periodicity in chaotic range, and small key space. In this PRNG, the numbers are generating in an unpredictable manner. To improve the uniform distribution and periodicity in random number generation, combination of Logistic and Tinker bell map is proposed. The Logistic map is a polynomial mapping. The Tinker bell map is known to have shorter periods. It is notable for having chaotic solutions for certain parameter values and initial conditions. In the proposed work, the Tinker bell map perturbs Logistic map to increase the chaotic behaviour of Logistic map. Output of the Logistic map is taken as Pseudo random bit sequence or Pseudo random number. NIST statistical test suite is used to verify the randomness of the generated random sequence.

Random number sequences and RNGs play an important role in trusted computing environments and cryptographic applications. For example, we use random numbers in the generation of keys in TPM. In some web protocols, random numbers are applied to resist replay attacks. It is necessary to guarantee the quality of RNGs and their random sequences because deterministic factors are likely to be involved in the generation process. If a random number generator is not designed carefully, then the output number sequences may become predictable and bring high security risks. Thus, the design of random number generators that produce high-quality random number sequences has been a hot research topic in these decades. Recently, with the development of resource constrained environments, the demand of lightweight random number generators dramatically increases. People prefer to use the random number generators with extreme high efficiency in the on-the-fly applications. This will affect the security performance of the generators. In this paper, we design a random number generator which meets the current lightweight requirements in the resource-limited environments. Our design is originally based on a lightweight block cipher, and applies the property of random looking output of block cipher to the random number generators. We combine a traditional encryption mode with a novel structure for the random number generator, so that the trade-off between security and efficiency can be made perfectly. We also take a comprehensive security evaluation for our random number generator.

A "Random Number Generator" is a program that produces a number each time it is invoked. This number is, for a broad range of uses, "random." More exactly, it is unpredictable and has the same chance of being selected as any other number among those the generator can produce. A random generator is useful in the same way as flipping a coin can be useful: randomness is independent of human motives and desires, it is impartial and fair. Its applications include generating test data, Monte Carlo procedures, testing grammars, and making choices in programs for creating music, poetry, and other art. The definition of randomness is a philosophical and mathematical question, beyond the scope of this discussion. It would not be out of place, however, to explain that there is no such thing as a "random number." While a die may I and five-spot uppermost we would not say that therefore the number 5 is random. Put in a relatively precise way, we would say that a sequence of numbers is random if there is no possible program that could be written to predict what the next member of the sequence was going to be given all previous members of the sequence. We would also require that the sequence had certain statistical properties, for example, that each possible output was equally likely. A random generator is a device (such as an honest roulette wheel) that produces such a sequence. It follows that a computer program cannot generate a random sequence (if it could, then a faster computer with the same program could 'predict' the outputted numbers). This is generally true. The usual "random number generator" or "random function" provided at most installations is more properly a pseudorandom generator. It is, in fact, a program that produces a string of numbers that for most practical purposes appears random. It has good statistical properties and while it will eventually repeat itself and is certainly predictable, it is unlikely that you (or your program) would ever detect these "faults." There is a large and still growing literature on methods of programming pseudorandom functions, and one can probably be written in relatively short order (see for example, the Algorithms department of the Communications of the Association for Computing Machinery ). The random generator at your installation may have any of a number of forms. Usually it is a function or procedure whose name looks like RND, RAN, RANDOM, RANF, IRANF, etc. In APL it is simply a

Random number generators are built into most home computers now, so many more people are getting a chance to use them then ever before. Usually, a 'random number generator' manufactures, on demand, numbers t between 0 and 1, produced in such a manner as to be equally likely to reside anywhere in that interval. More precisely, if [a, b] is a subinterval of (0,1), then the probability that the next random number ( lies in [a, b] is b a. For an ideal random number generator that statement would be true for every [a, b] c (0,1). Given such a supply of random numbers, it is easy to produce other kinds of distributions by various manipulations. To choose a number X uniformly at random in the interval (7,13), for instance, we would write X = 7 + 6t, where ( is a random number. Again, to choose an integer L uniformly at random between 0 and n, just set L = [ (n + 1)J, in which ( is a random number and '1 ',is the greatest integer function. Sometimes we want numbers that aren't equally likely to turn up everywhere, but instead are more likely to lie in some places than in others. These can be obtained by twisting random numbers out of shape, and some fearsomely ingenious constructions have been suggested, over the years, in order to achieve various kinds of twists. See Chapter 3 of [3] for accounts of some of these. Among nonuniform distributions one of the most important is the exponential distribution, which describes positive real numbers X that are chosen in such a way that for every t > 0, the probability that X < t is 1 e. It is therefore more likely that X lies between l and 2 (the chance is e-l e-2 = .232), say, than between 8 and 9 (probability e-8 e-9 = .00021), etc. The exponential distribution occurs naturally in many situations. Two categories of these are waiting time problems and particle diffusion problems. In waiting line ('queueing') simulations it is frequently the case that it is more likely that we wait a shorter time and less likely a longer time. Hence in a computer simulation it is often desirable to select the delay times from the exponential distribution. Likewise, when a particle bounces around in a diffusing medium, it is more probable that it will travel a shorter distance D between successive collisions with molecules of the medium than a longer distance. Under reasonable assumptions it isn't hard to show that D obeys the exponential distribution. A question of mathematical interest and beauty, as well as of practical application (a happy combination) is then the following: given a random number generator, how shall we select numbers X that have the exponential distribution? Here's the quick answer (then we'll look at two other ways). First choose a random number (, then calculate X = log (. That's all. We claim that the numbers X that are so produced have the exponential distribution. To see that we need perform only the following calculation ('Prob' means 'the probability of'):

CHAPTER 12 - SIMULATIONS WITH RANDOM NUMBERS

Subject of research: crypto-resistant methods and tools of generating random sequences and hardware support of cryptographic transformations in IoT devices. The aim of the article is to study crypto-resistant methods and tools for generating and testing random sequences suitable for use in IoT devices with limited resources; determination of circuit implementations of random sequences hardware generators; formation of conclusions on the use of random number generators (RNG) in cryptographic protection systems of the IoT network. The article solves the following tasks: analysis of methods and hardware for generating random sequences to protect IoT solutions with limited resources; identification of safe and effective technologies for the implementation of RNG; classification of RNG attacks; analysis of the shortcomings of the practical use of statistical test packages to assess the quality of random sequences of RNG; evaluation of the speed of cryptoaccelerators of hardware support for cryptographic transformations; providing practical guidance on RNG for use in resource-constrained IoT devices. Research methods: method of structural and functional analysis of RNG and IoT devices, cryptographic methods of information protection, methods of random sequence generation, method of stability analysis of systems, methods of construction of autonomous Boolean networks and Boolean chaos analysis, methods of quality assessment of random sequences. Results of work: the analysis of technologies and circuit decisions of hardware RNG on characteristics: quality of numbers’ randomness and unpredictability of sequences, speed, power consumption, miniaturization, possibility of integral execution; providing practical recommendations for the use of RNG in cryptographic protection systems of the IoT network. The novelty of the study is the analysis of methods and hardware to support technologies for generating random sequences in the system of cryptographic protection of IoT solutions; classification of attacks on RNG and features of protection against them; identification of effective RNG technologies and circuit solutions for use in low-power IoT devices with limited computing resources; providing practical recommendations for the use of RNG in cryptographic protection systems of the IoT network. The analysis of technologies and circuit solutions allowed to draw the following conclusions: protection of IoT solutions includes: security of IoT network nodes and their connection to the cloud using secure protocols, ensuring confidentiality, authenticity and integrity of IoT data by cryptographic methods, attack analysis and network cryptographic stability; the initial basis for the protection of IoT solutions is the true randomness of the formed RNG sequences and used in algorithms for cryptographic transformation of information to protect it; feature of IoT devices is their heterogeneity and geographical distribution, limited computing resources and power supply, small size; The most effective (reduce power consumption and increase the generation rate) for use in IoT devices are RNG exclusively on a digital basis, which implements a three-stage process: the initial digital circuit, normalizer and random number flow generator; Autonomous Boolean networks (ABN) allow to create RNG with unique characteristics: the received numbers are really random, high speed – the number can be received in one measure, the minimum power consumption, miniature, high (up to 3 GHz) throughput of Boolean chaos; a promising area of ABN development is the use of optical logic valves for the construction of optical ABN with a bandwidth of up to 14 GHz; the classification of known classes of RNG attacks includes: direct cryptanalytic attacks, attacks based on input data, attacks based on the disclosure of the internal state of RNG, correlation attacks and special attacks; statistical test packages to evaluate RNG sequences have some limitations or shortcomings and do not replace cryptanalysis; Comparison of cryptoaccelerators with cryptographic transformation software shows their significant advantages: for AES block encryption algorithm, speeds increase by 10-20 times in 8/16-bit cryptoaccelerators and 150 times in 32-bit, growth hashing of SHA-256 in 32-bit cryptoaccelerators more than 100 times, and for the NMAS algorithm - up to 500 times.

In many tasks, human behavior is far noisier than is optimal. Yet when asked to behave randomly, people are typically too predictable. We argue that these apparently contrasting observations have the same origin: the operation of a general-purpose local sampling algorithm for probabilistic inference. This account makes distinctive predictions regarding random sequence generation, not predicted by previous accounts-which suggests that randomness is produced by inhibition of habitual behavior, striving for unpredictability. We verify these predictions in two experiments: people show the same deviations from randomness when randomly generating from non-uniform or recently-learned distributions. In addition, our data show a novel signature behavior, that people's sequences have too few changes of trajectory, which argues against the specific local sampling algorithms that have been proposed in past work with other tasks. Using computational modeling, we show that local sampling where direction is maintained across trials best explains our data, which suggests it may be used in other tasks too. While local sampling has previously explained why people are unpredictable in standard cognitive tasks, here it also explains why human random sequences are not unpredictable enough.

In many tasks, human behavior is far noisier than is optimal. Yet when asked to behave randomly, people are typically too predictable. We argue that these apparently contrasting observations have the same origin: the operation of a general-purpose local sampling algorithm for probabilistic inference. This account makes distinctive predictions regarding random sequence generation, not predicted by previous accounts—which suggests that randomness is produced by inhibition of habitual behavior, striving for unpredictability. We verify these predictions in two experiments: people show the same deviations from randomness when randomly generating from non-uniform or recently-learned distributions. In addition, our data show a novel signature behavior, that people’s sequences have too few changes of trajectory, which argues against the specific local sampling algorithms that have been proposed in past work with other tasks. Using computational modeling, we show that local sampling where direction is maintained across trials best explains our data, which suggests it may be used in other tasks too. While local sampling has previously explained why people are unpredictable in standard cognitive tasks, here it also explains why human random sequences are not unpredictable enough.

To evaluate the performance of adult young subjects in a Random Number Generation (RNG) task by controlling the response speed (RS). Sixty-nine university students of both sexes took part in the experiment (25.05 +/- 6.71 year-old). Participants were alloted into 3 groups which differed in RS rates to generate numbers: 1, 2 and 4 seconds to generate each number. A digital metronomer was used to control RS. Participants were asked to generate 100 numbers. The responses were measured through Evans's RNG Index. There were statistically significant differences among the groups [F (3, 68) = 7.120; p <.05]. Differences were localized between 1 and 2 seconds (p = 0.004) and between 1 and 4 seconds (p = 0.006). No differences were observed between 2 and 4 seconds (p = 0.985). The present results suggest that the response speed in production of random numbers influences the performance of the Random Numbers Generation task.

Random numbers have great application value in the fields of secure communications, which are commonly used as secret keys to encrypt the information. To guarantee that the information is absolutely secure in the current high-speed communication, the applied random keys should possess a generation speed not less than the encrypted data rate, according to one-time pad theory found by Shannon (Shannon C E 1949 Bell.Syst.Tech.J. 28 656) Pseudo-random numbers generated by algorithm may easily reach a fast speed, but a certain periodicity makes them difficult to meet the aforementioned demand of information security. Utilizing physical stochastic phenomena can provide reliable random numbers, called physical random number generators (RNGs). However, limited by the bandwidth of the conventional physical sources such as electronic noise, frequency jitter of oscillator and quantum randomness, the traditional physical RNG has a generation speed at a level of Mb/s typically. Therefore, real-time and ultrafast physical random number generation is urgently required from the view of absolute security for high-speed communication today. With the advent of wideband photonic entropy sources, in recent years lots of schemes for high-speed random number generation are proposed. Among them, chaotic laser has received great attention due to its ultra-wide bandwidth and large random fluctuation of intensity. The real-time speed of physical RNG based on chaotic laser is now limited under 5 Gb/s, although the reported RNG claims that an ultrafast speed of Tb/s is possible in theory. The main issues that restrict the real-time speed of RNG based on chaotic laser are from two aspects. The first aspect is electrical jitter bottleneck confronted by the electrical analog-to-digital converter (ADC). Specifically, most of the methods of extracting random numbers are first to convert the chaotic laser into an electrical signal by a photo-detector, then use an electrical ADC driven by radio frequency (RF) clock to sample and quantify the chaotic signal in electronic domain. Unfortunately, the response rate of ADC is below Gb/s restricted by the aperture jitter (several picoseconds) of RF clock in the sample and hold circuit. The second aspect comes from the complex post-processes, which are fundamental in current RNG techniques to realize a good randomness. The strict synchronization among post-processing components (e.g., XOR gates, memory buffers, high-order difference) is controlled by an RF clock. Similarly, it is also an insurmountable obstacle to achieve an accurate synchronization due to the electronic jitter of the RF clock. In this paper, we propose a method of ultrafast multi-bit physical RNG based on chaotic laser without any post-process. In this method, a train of optical pulses generated by a GHz mode-locked laser with low temporal jitter at a level of fs is used as an optical sampling clock. The chaotic laser is sampled in the optical domain through a low switching energy and high-linearity terahertz optical asymmetric demultiplexer (TOAD) sampler, which is a fiber loop with an asymmetrical nonlinear semiconductor optical amplifier. Then, the peak amplitude of each sampled chaotic pulse is digitized by a multi-bit comparator (i.e., a multi-bit ADC without sample and hold circuit) and converted into random numbers directly. Specifically, a proof-of-principle experiment is executed to demonstrate the aforementioned proposed method. In this experiment, an optical feedback chaotic laser is used, which has a bandwidth of 6 GHz. Through setting a sampling rate to be 5 GSa/s and selecting 4 LSBs outputs of the 8-bit comparator, 20 Gb/s (=5 GSa/s4 LSBs) physical random number sequences are obtained. Considering the ultrafast response rate of TOAD sampler, the speed of random numbers generated by this method has the potential to reach several hundreds of Gb/s as long as the used chaotic laser has a sufficient bandwidth.

There is great interest about the individual differences that influence the ability of dealing with risky decisions. In this light, an intriguing question is whether decision-making during risk is related to other cognitive abilities, especially executive functions. To investigate, in healthy subjects, the existence of a possible correlation between risk-taking and cognitive abilities, the balloon analogue risk task (BART) has been exploited to assess risk-taking propensity and the random number generation (RNG), to investigate cognitive functions. The risk-taking propensity is significantly correlated with the Cycling factor, a feature of RNG performance specifically related to the ability of updating and monitoring information. In particular, an excessive activity of monitoring (expressed by lower values of Cycling factor) is related to a more risk-averse behavior. An overlapping between the circuits involved in both RNG and BART, centered on the dorsolateral prefrontal cortex, could be the possible neurophysiological substrate for this correlation. This study suggests a relevant contribution of executive functions in risk-taking behavior. This could have relevant implications in neuroeconomics and neuropsychiatry of addiction and pathological gambling.

The problems of the Internet of Things, in particular the need to protect confidential data, lead to the use of encryption and decryption methods, which, in turn, rely on the use of random numbers. The use of HVDC can help ensure the security of this process, but it is important to use high-quality HVDC and thoroughly test them to avoid security vulnerabilities. Statistical analysis is used to assess the randomness of the generated sequences by testing them using statistical tests of randomness. These tests evaluate various statistical properties, such as the frequency distribution of the generated numbers, the distribution of consecutive numbers, and the presence of patterns. By performing a statistical analysis, you can identify any flaws or biases in the operation of the random number generator and improve its quality. The article checks the sequence for randomness using two- and three-dimensional statistics. A mathematical model is built, a theorem is given. A scheme of statistical analysis of sequences was built, based on which software engineering methods were used. Examples of using binary statistics for bit sequence analysis are given. The use of statistical analysis of the randomness of sequences generated by random (or pseudo-random) number generators is a popular trend, as it is critical to the reliability and security of various computing systems and applications, including the Internet of Things (IoT), cryptography, simulations, and scientific calculation. With the growing importance of cybersecurity and data privacy, the demand for reliable and secure random number generators is increasing, and the use of statistical analysis to assess the randomness of sequences generated by random (or pseudo-random) number generators is becoming increasingly critical.

Extending OPNET Modeler with External Pseudo Random Number Generators and Statistical Evaluation by the Limited Relative Error Algorithm

Data encryption is a way to increase security of data by using cryptographic keys generated from random numbers. Random Number generator (RNG), which produces such numbers, is classified as True Random Number Generator (TRNG) or Pseudorandom Number generator (PRNG). The first one uses physical phenomena to generate random numbers whereas the second one uses deterministic systems. A commonly used deterministic system in PRNG applications is the logistic map, a nonlinear system that under certain operating ranges becomes chaotic. However, the logistic map alone does not generate satisfactory random sequence because the sequence generated is not uniformly distributed, and it does not have good results in statistical tests, such as the NIST test suite. To solve this problem, this paper proposes an implementation in Field Programmable Gate Array (FPGA) of a PRNG, which uses the k – logistic map concept, discarding the most significant k decimal digits of an underlying orbit generated from the traditional logistic map equation. The results from synthesis and simulations show that the designed circuit is a low-cost and a high efficiency RNG solution.