We recall the main developments in non-uniform random variate generation. Complexity of nonuniform random number. Complexity Questions in Non-Uniform.

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class -- an object receives a probability essentially proportional to an exponential of its size.

As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice. We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs.

We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms.

Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large.

As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to 1 � O(�log n)/k). We describe efficient constructions of small probability spaces that approximate the independent distribution for general random variables. Previous work on efficient constructions concentrate on approximations of the independent distribution for the special case of uniform boolean-valued random variables.

Our results yield efficient constructions of small sets with low discrepancy in high dimensional space and have applications to derandomizing randomized algorithms. 1 Introduction The problem of constructing small sample spaces that 'approximate' the independent distribution on n random variables has received considerable attention recently (cf.

[6, Chor Goldreich] [8, Karp Wigderson], [11, Luby], [1, Alon Babai Itai], [13, Naor Naor], [2, Alon Goldreich Hastad Peralta], [3, Azar Motwani Naor]). The primary motivation for this line of research is that random variables that are 'approximately' independent suffices for the analysis of many interesting randomized algorithm and hence c. We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1=2 + ffl, where ffl is an arbitrary positive constant. We also consider the problem of testing if one of these access structures is a sub-structure of an arbitrary access structure and we show that this problem is NP-complete. We provide several general lower bounds on information rate and average information rate of graphs.

In particular, we show that any graph with n vertices admits a secret sharing scheme with information rate Omega Gammate/3 n)=n). 1 Introduction A secret sharing scheme is a method to distribute a secret s among a set of participants P in such a way that only qualified subsets of P can reconstruct the value of s whereas any other subset of P; non-qualified to know s; cannot. We define the universal type class of a sequence x n, in analogy to the notion used in the classical method of types. Two sequences of the same length are said to be of the same universal (LZ) type if and only if they yield the same set of phrases in the incremental parsing of Ziv and Lempel (1978). We show that the empirical probability distributions of any finite order of two sequences of the same universal type converge, in the variational sense, as the sequence length increases. Consequently, the normalized logarithms of the probabilities assigned by any kth order probability assignment to two sequences of the same universal type, as well as the kth order empirical entropies of the sequences, converge for all k.

We study the size of a universal type class, and show that its asymptotic behavior parallels that of the conventional counterpart, with the LZ78 code length playing the role of the empirical entropy. We also estimate the number of universal types for sequences of length n, and show that it is of the form exp((1+o(1))γ n/log n) for a well characterized constant γ. We describe algorithms for enumerating the sequences in a universal type class, and for drawing a sequence from the class with uniform probability. As an application, we consider the problem of universal simulation of individual sequences. A sequence drawn with uniform probability from the universal type class of x n is an optimal simulation of x n in a well defined mathematical sense. Daws at cs.ru.nl Abstract.

We present a language-theoretic approach to symbolic model checking of PCTL over discrete-time Markov chains. The probability with which a path formula is satisfied is represented by a regular expression. A recursive evaluation of the regular expression yields an exact rational value when transition probabilities are rational, and rational functions when some probabilities are left unspecified as parameters of the system. This allows for parametric model checking by evaluating the regular expression for different parameter values, for instance, to study the influence of a lossy channel in the overall reliability of a randomized protocol. We describe efficient constructions of small probability spaces that approximate the joint distribution of general random variables.

Previous work on efficient constructions concentrate on approximations of the joint distribution for the special case of identical, uniformly distributed random variables. Preliminary version has appeared in the Proceedings of the 24th ACM Symp.

On Theory of Computing (STOC), pages 10--16, 1992. Of Electrical Engineering--Systems, Tel--Aviv University, Ramat--Aviv, Tel--Aviv 69978, Israel. Email: guy@eng.tau.ac.il. Z Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Email: oded@wisdom.weizmann.ac.il. Research partially supported by grant No.

89-00312 from the United StatesIsrael Binational Science Foundation (BSF), Jerusalem, Israel. X International Computer Science Institute, Berkeley, CA 94704, USA. Email: luby@icsi.berkeley.edu. Research supported in part by National Science Founda.

We consider secret sharing schemes in which the dealer is able (after a preprocessing stage) to activate a particular access structure out of a given set and/or to allow the participants to reconstruct different secrets (in different time instants) by sending them the same broadcast message. In this paper we establish a formal setting to study secret sharing schemes of this kind.

The security of the schemes presented is unconditional, since they are not based on any computational assumption. We give bounds on the size of the shares held by participants, on the size of the broadcast message, and on the randomness needed in such schemes.

1 Introduction A secret sharing scheme is a method of dividing a secret s among a set P of participants in such a way that: if the participants in A ` P are qualified to know the secret then by pooling together their information they can reconstruct the secret s; but any set A of participants not qualified to know s has absolutely no information on the.

When a cubical die is rolled, a random number between 1 and 6 is obtained. Random number generation is the generation of a sequence of or symbols that cannot be reasonably predicted better than by a chance, usually through a hardware random-number generator (RNG). Various have led to the development of several different methods for generating data, of which some have existed since ancient times, among whose ranks are well-known 'classic' examples, including the rolling of,, the of, the use of stalks (for ) in the, as well as countless other techniques. Because of the mechanical nature of these techniques, generating large numbers of sufficiently random numbers (important in statistics) required a lot of work and/or time.

Thus, results would sometimes be collected and distributed as. Nowadays, after the advent of computational random-number generators, a growing number [ ] of government-run and lottery games have started [ ] using RNGs instead of more traditional drawing methods. RNGs are also used to determine the outcomes of modern. Several computational methods for random-number generation exist. Many fall short of the goal of true randomness, although they may meet, with varying success, some of the intended to measure how unpredictable their results are (that is, to what degree their patterns are discernible). However, carefully designed secure computationally based methods of generating random numbers also exist, such as those based on the, the, and others. Unity3d Obfuscator Keygen Crack Patch.

Main article: Random number generators have applications in,,,,, and other areas where producing an unpredictable result is desirable. Generally, in applications having unpredictability as the paramount, such as in security applications, are generally preferred over pseudo-random algorithms, where feasible. Random number generators are very useful in developing simulations, as is facilitated by the ability to run the same sequence of random numbers again by starting from the same. They are also used in – so long as the seed is secret. Sender and receiver can generate the same set of numbers automatically to use as keys. The generation of is an important and common task in computer programming. While cryptography and certain numerical algorithms require a very high degree of apparent randomness, many other operations only need a modest amount of unpredictability.

Some simple examples might be presenting a user with a 'Random Quote of the Day', or determining which way a computer-controlled adversary might move in a computer game. Weaker forms of randomness are used in and in creating and. Some applications which appear at first sight to be suitable for randomization are in fact not quite so simple. For instance, a system that 'randomly' selects music tracks for a background music system must only appear random, and may even have ways to control the selection of music: a true random system would have no restriction on the same item appearing two or three times in succession. Pseudo-random numbers [ ].

See also: There are two principal methods used to generate random numbers. The first method measures some physical phenomenon that is expected to be random and then compensates for possible biases in the measurement process. Example sources include measuring, thermal noise, and other external electromagnetic and quantum phenomena. For example, cosmic background radiation or radioactive decay as measured over short timescales represent sources of natural. The speed at which entropy can be harvested from natural sources is dependent on the underlying physical phenomena being measured. Thus, sources of naturally occurring 'true' entropy are said to be – they are rate-limited until enough entropy is harvested to meet the demand. On some Unix-like systems, including most, the pseudo device file will block until sufficient entropy is harvested from the environment.

Due to this blocking behavior, large bulk reads from, such as filling a with random bits, can often be slow on systems that use this type of entropy source. The second method uses computational that can produce long sequences of apparently random results, which are in fact completely determined by a shorter initial value, known as a seed value.

As a result, the entire seemingly random sequence can be reproduced if the seed value is known. This type of random number generator is often called a. This type of generator typically does not rely on sources of naturally occurring entropy, though it may be periodically seeded by natural sources. This generator type is non-blocking, so they are not rate-limited by an external event, making large bulk reads a possibility. Some systems take a hybrid approach, providing randomness harvested from natural sources when available, and falling back to periodically re-seeded software-based (CSPRNGs).

The fallback occurs when the desired read rate of randomness exceeds the ability of the natural harvesting approach to keep up with the demand. This approach avoids the rate-limited blocking behavior of random number generators based on slower and purely environmental methods. While a pseudorandom number generator based solely on deterministic logic can never be regarded as a 'true' random number source in the purest sense of the word, in practice they are generally sufficient even for demanding security-critical applications.

Indeed, carefully designed and implemented pseudo-random number generators can be certified for security-critical cryptographic purposes, as is the case with the and. The former is the basis of the /dev/random source of entropy on,,, and others. Also uses a pseudo-random number algorithm based on known as. Generation methods [ ] Physical methods [ ].

See also: and Even given a source of plausible random numbers (perhaps from a quantum mechanically based hardware generator), obtaining numbers which are completely unbiased takes care. In addition, behavior of these generators often changes with temperature, power supply voltage, the age of the device, or other outside interference. And a software bug in a pseudo-random number routine, or a hardware bug in the hardware it runs on, may be similarly difficult to detect. Generated random numbers are sometimes subjected to statistical tests before use to ensure that the underlying source is still working, and then post-processed to improve their statistical properties.

An example would be the TRNG9803 hardware random number generator, which uses an entropy measurement as a hardware test, and then post-processes the random sequence with a shift register stream cipher. It is generally hard to use statistical tests to validate the generated random numbers. Wang and Nicol proposed a distance-based statistical testing technique that is used to identify the weaknesses of several random generators.Li and Wang proposed a method of testing random numbers based on laser chaotic entropy sources using Brownian motion properties. Other considerations [ ] Random numbers uniformly distributed between 0 and 1 can be used to generate random numbers of any desired distribution by passing them through the inverse (CDF) of the desired distribution (see ).

Inverse CDFs are also called. To generate a pair of random numbers ( x, y), one may first generate the ( r, θ), where r~ and θ~ (see ).

Some 0 to 1 RNGs include 0 but exclude 1, while others include or exclude both. The outputs of multiple independent RNGs can be combined (for example, using a bit-wise operation) to provide a combined RNG at least as good as the best RNG used. This is referred to as. Computational and hardware random number generators are sometimes combined to reflect the benefits of both kinds.

Computational random number generators can typically generate pseudo-random numbers much faster than physical generators, while physical generators can generate 'true randomness.' Low-discrepancy sequences as an alternative [ ] Some computations making use of a random number generator can be summarized as the computation of a total or average value, such as the computation of integrals by the. For such problems, it may be possible to find a more accurate solution by the use of so-called, also called numbers.

Such sequences have a definite pattern that fills in gaps evenly, qualitatively speaking; a truly random sequence may, and usually does, leave larger gaps. Activities and demonstrations [ ] The following sites make available Random Number samples: • The resource pages contain a number of of random number generation using Java applets. • The Quantum Optics Group at the generates random numbers sourced from quantum vacuum. You can download a sample of random numbers by visiting their research page.

• makes available random numbers that are sourced from the randomness of atmospheric noise. • The at the harvests randomness from the quantum process of photonic emission in semiconductors. They supply a variety of ways of fetching the data, including libraries for several programming languages. • The Group at the Taiyuan University of technology generates random numbers sourced from chaotic laser.

You can obtain a sample of random number by visiting their. Backdoors [ ].

Main article: Since much cryptography depends on a cryptographically secure random number generator for key and generation, if a random number generator can be made predictable, it can be used as by an attacker to break the encryption. The NSA is reported to have inserted a backdoor into the certified. If for example an SSL connection is created using this random number generator, then according to it would allow NSA to determine the state of the random number generator, and thereby eventually be able to read all data sent over the SSL connection. Even though it was apparent that Dual_EC_DRBG was a very poor and possibly backdoored pseudorandom number generator long before the NSA backdoor was confirmed in 2013, it had seen significant usage in practice until 2013, for example by the prominent security company. There have subsequently been accusations that RSA Security knowingly inserted a NSA backdoor into its products, possibly as part of the program. RSA has denied knowingly inserting a backdoor into its products.

It has also been theorized that hardware RNGs could be secretly modified to have less entropy than stated, which would make encryption using the hardware RNG susceptible to attack. One such method which has been published works by modifying the dopant mask of the chip, which would be undetectable to optical reverse-engineering. For example, for random number generation in Linux, it is seen as unacceptable to use Intel's hardware RNG without mixing in the RdRand output with other sources of entropy to counteract any backdoors in the hardware RNG, especially after the revelation of the NSA Bullrun program. In 2010, by the information security director of the (MUSL), who surreptitiously installed backdoor on the MUSL's secure RNG computer during routine maintenance. During the hacks the man won a total amount of $16,500,000 by predicting the numbers correct a few times in year.

ASLR or Address Space Layout Randomization, a mitigation against rowhammer and related attacks on the physical hardware of memory chips has been found to be inadequate as of early 2017 by VUSec. The random number algorithm if based on a shift register implemented in hardware is predictable at sufficiently large values of p and can be reverse engineered with enough processing power. This also indirectly means that malware using this method can run on both GPUs and CPUs if coded to do so, even using GPU to break ASLR on the CPU itself. In popular culture [ ] The process of random number generation in games, especially in, is often referred to as being controlled by a 'Random Number God' or 'RN-Jesus'. The term was originally coined by players of the games and, and also references the belief that certain actions can either appease or anger the 'God', leading to number generation seemingly skewed for or against the player.

See also [ ].