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Rational Sieve
In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field sieve works. Method Suppose we are trying to factor the composite number ''n''. We choose a bound ''B'', and identify the ''factor base'' (which we will call ''P''), the set of all primes less than or equal to ''B''. Next, we search for positive integers ''z'' such that both ''z'' and ''z+n'' are ''B''-smooth — i.e. all of their prime factors are in ''P''. We can therefore write, for suitable exponents a_i, z=\prod_ p_i^ and likewise, for suitable b_i, we have z+n=\prod_ p_i^. But z and z+n are congruent modulo n, and so each such integer ''z'' that we find yields a multiplicative relation (mod ''n'') among the elements of ''P'', i.e. :\prod_ p_i^ \equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form :\exp\left( \left(\sqrt + o(1)\right)(\ln n)^(\ln \ln n)^\right) =L_n\left .html"_;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in_L-notation.html" ;"title="">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation">">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation), where is the natural logarithm. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots). The p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithmic Efficiency
In computer science, algorithmic efficiency is a property of an algorithm which relates to the amount of computational resources used by the algorithm. An algorithm must be analyzed to determine its resource usage, and the efficiency of an algorithm can be measured based on the usage of different resources. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or continuous process. For maximum efficiency it is desirable to minimize resource usage. However, different resources such as time and space complexity cannot be compared directly, so which of two algorithms is considered to be more efficient often depends on which measure of efficiency is considered most important. For example, bubble sort and timsort are both algorithms to sort a list of items from smallest to largest. Bubble sort sorts the list in time proportional to the number of elements squared (O(n^2), see Big O notation), but only requires a small amount of extra mem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor Base
In computational number theory, a factor base is a small set of prime numbers commonly used as a mathematical tool in algorithms involving extensive sieving for potential factors of a given integer. Usage in factoring algorithms A factor base is a relatively small set of distinct prime numbers ''P'', sometimes together with -1. Say we want to factorize an integer ''n''. We generate, in some way, a large number of integer pairs (''x'', ''y'') for which x \neq \pm y, x^2 \equiv y^2 \pmod, and x^2 \pmod \texty^2 \pmod can be completely factorized over the chosen factor base—that is, all their prime factors are in ''P''. In practice, several integers ''x'' are found such that x^2 \pmod has all of its prime factors in the pre-chosen factor base. We represent each x^2 \pmod expression as a vector of a matrix with integer entries being the exponents of factors in the factor base. Linear combinations of the rows corresponds to multiplication of these expressions. A linear dependence re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers. Definition A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, resp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Of Squares
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms. Derivation Given a positive integer ''n'', Fermat's factorization method relies on finding numbers ''x'' and ''y'' satisfying the equality :x^2 - y^2 = n We can then factor ''n'' = ''x''2 − ''y''2 = (''x'' + ''y'')(''x'' − ''y''). This algorithm is slow in practice because we need to search many such numbers, and only a few satisfy the equation. However, ''n'' may also be factored if we can satisfy the weaker congruence of squares condition: :x^2 \equiv y^2 \pmod :x \not\equiv \pm y \,\pmod From here we easily deduce :x^2 - y^2 \equiv 0 \pmod :(x + y)(x - y) \equiv 0 \pmod This means that ''n'' divides the product (''x'' + ''y'')(''x'' − ''y''). Thus (''x'' + ''y'') and (''x'' − ''y'') each contain factors of ''n'', but those factors can be trivial. In this case we need to find another ''x'' and ''y''. Computing the greatest common divisors of (''x'' +&t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the largest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
