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2.3 / RANDOM AND PSEUDORANDOM NUMBERS generate, create quick response code none on .net projects VB.NET approach is beyon QR Code ISO/IEC18004 for .NET d the reach of human analysts and their computers. However, a number of effective algorithms exist that test the primality of a number by using a sequence of randomly chosen integers as input to relatively simple computations.

If the sequence is sufficiently long (but far, far less than 210150), the primality of a number can be determined with near certainty. This type of approach, known as randomization, crops up frequently in the design of algorithms. In essence, if a problem is too hard or time-consuming to solve exactly, a simpler, shorter approach based on randomization is used to provide an answer with any desired level of confidence.

UNPREDICTABILITY In applications such as reciprocal authentication and session key generation, the requirement is not so much that the sequence of numbers be statistically random but that the successive members of the sequence are unpredictable. With true random sequences, each number is statistically independent of other numbers in the sequence and therefore unpredictable. However, as is discussed shortly, true random numbers are not always used; rather, sequences of numbers that appear to be random are generated by some algorithm.

In this latter case, care must be taken that an opponent not be able to predict future elements of the sequence on the basis of earlier elements.. TRNGs, PRNGs, and PRFs Cryptographic app qr codes for .NET lications typically make use of algorithmic techniques for random number generation. These algorithms are deterministic and therefore produce sequences of numbers that are not statistically random.

However, if the algorithm is good, the resulting sequences will pass many reasonable tests of randomness. Such numbers are referred to as pseudorandom numbers. You may be somewhat uneasy about the concept of using numbers generated by a deterministic algorithm as if they were random numbers.

Despite what might be called philosophical objections to such a practice, it generally works. As one expert on probability theory puts it [HAMM91], For practical purposes we are forced to accept the awkward concept of relatively random meaning that with regard to the proposed use we can see no reason why they will not perform as if they were random (as the theory usually requires). This is highly subjective and is not very palatable to purists, but it is what statisticians regularly appeal to when they take a random sample they hope that any results they use will have approximately the same properties as a complete counting of the whole sample space that occurs in their theory.

Figure 2.7 contrasts a true random number generator (TRNG) with two forms of pseudorandom number generators. A TRNG takes as input a source that is effectively random; the source is often referred to as an entropy source.

In essence, the entropy source is drawn from the physical environment of the computer and could include things such as keystroke timing patterns, disk electrical activity, mouse movements, and instantaneous values of the system clock. The source, or combination of sources, serves as input to an algorithm that produces. CHAPTER 2 / SYMMETRIC ENCRYPTION AND MESSAGE CONFIDENTIALITY Source of true randomness Contextspecific Seed values Seed Conversion to binary Deterministic algorithm Deterministic algorithm Random bit stream Pseudorandom bit stream Pseudorandom value (a) TRNG (b) PRNG (c) PRF TRNG = true rando m number generator PRNG = pseudorandom number generator PRF = pseudorandom function . Figure 2.7 Random and Pseudorandom Number Generators random binary out put. The TRNG may simply involve conversion of an analog source to a binary output. The TRNG may involve additional processing to overcome any bias in the source.

In contrast, a PRNG takes as input a fixed value, called the seed, and produces a sequence of output bits using a deterministic algorithm.Typically, as shown in Figure 2.7, there is some feedback path by which some of the results of the algorithm are fed back as input as additional output bits are produced.

The important thing to note is that the output bit stream is determined solely by the input value or values, so that an adversary who knows the algorithm and the seed can reproduce the entire bit stream. Figure 2.7 shows two different forms of PRNGs, based on application.

Pseudorandom number generator: An algorithm that is used to produce an open-ended sequence of bits is referred to as a PRNG. A common application for an open-ended sequence of bits is as input to a symmetric stream cipher, as discussed in the following section. Pseudorandom function (PRF): A PRF is used to produce a pseudorandom string of bits of some fixed length.

Examples are symmetric encryption keys and nonces. Typically, the PRF takes as input a seed plus some context specific values, such as a user ID or an application ID. A number of examples of PRFs will be seen throughout this book.

Other than the number of bits produced, there is no difference between a PRNG and a PRF. The same algorithms can be used in both applications. Both require a seed and both must exhibit randomness and unpredictability.

Furthermore, a PRNG application may also employ context-specific input..
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