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Generalized Inversive Congruential Pseudorandom Numbers
An approach to nonlinear congruential methods of generating uniform pseudorandom numbers in the interval primes p_1,\dots ,p_r \ge 5 will be present here. Let \mathbb_ = \ . For integers a,b \in \mathbb_ with gcd (a,m) = 1 a generalized inversive congruential sequence (y_)_ of elements of \mathbb_ is defined by : y_ = : y_\equiv a y_^ + b \pmod m \textn \geqslant 0 where \varphi(m)=(p_-1)\dots (p_-1) denotes the number of positive integers less than ''m'' which are relatively prime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ... to ''m''. Example Let take m = 15 = 3 \times 5\, a=2 , b=3 and y_0= 1. Hence \varphi (m)= 2 \times 4=8 \, and the sequence (y_)_=(1,5,13,2,4,7,1,\dots ) is not maximum. The result below shows that these sequences are closely re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Number Generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random number generation, random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value, called the PRNG's ''random seed, seed'' (which may include truly random values). Although sequences that are closer to truly random can be generated using hardware random number generators, ''pseudorandom number generators'' are important in practice for their speed in number generation and their reproducibility. PRNGs are central in applications such as simulations (e.g. for the Monte Carlo method), electronic games (e.g. for procedural generation), and cryptography. Cryptographic applications require the output not to be predictable from earlier outputs, and more cryptographically-secure pseudorandom number generator, elabora ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversive Congruential Generator
Inversive congruential generators are a type of nonlinear congruential pseudorandom number generator, which use the modular multiplicative inverse (if it exists) to generate the next number in a sequence. The standard formula for an inversive congruential generator, modulo some prime ''q'' is: : x_0 = \text, : x_ = \begin (ax_i^ + c) \bmod q & \text x_i \ne 0, \\ c & \text x_i = 0. \end Such a generator is denoted symbolically as and is said to be an ICG with parameters ''q'', ''a'', ''c'' and seed ''seed''. Period The sequence (x_n)_ must have x_i = x_j after finitely many steps, and since the next element depends only on its direct predecessor, also x_ = x_ etc. The maximum possible period for the modulus ''q'' is ''q'' itself, i.e. the sequence includes every value from 0 to ''q'' − 1 before repeating. A sufficient condition for the sequence to have the maximum possible period is to choose ''a'' and ''c'' such that the polynomial f(x) = x^2 - cx - a \in \mathbb F_q /math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Little Theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as a^p \equiv a \pmod p. For example, if and , then , and is an integer multiple of . If is not divisible by , that is, if is coprime to , then Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols: a^ \equiv 1 \pmod p. For example, if and , then , and is a multiple of . Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.. History Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Random Number Generators
Random number generators are important in many kinds of technical applications, including physics, engineering or mathematical computer studies (e.g., Monte Carlo simulations), cryptography and gambling (on game servers). This list includes many common types, regardless of quality or applicability to a given use case. Pseudorandom number generators (PRNGs) The following algorithms are pseudorandom number generators. Cryptographic algorithms Cipher algorithms and cryptographic hashes can be used as very high-quality pseudorandom number generators. However, generally they are considerably slower (typically by a factor 2–10) than fast, non-cryptographic random number generators. These include: * Stream ciphers. Popular choices are Salsa20 or ChaCha (often with the number of rounds reduced to 8 for speed), ISAAC, HC-128 and RC4. * Block ciphers in counter mode. Common choices are AES (which is very fast on systems supporting it in hardware), TwoFish, Serpent and Camel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
