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Pocklington Primality Test
In mathematics, the Pocklington–Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer. The test uses a partial factorization of N - 1 to prove that an integer N is prime. It produces a primality certificate to be found with less effort than the Lucas primality test, which requires the full factorization of N - 1. Pocklington criterion The basic version of the test relies on the Pocklington theorem (or Pocklington criterion) which is formulated as follows: Let N > 1 be an integer, and suppose there exist natural numbers and such that Then is prime. Here i \equiv j \pmod means that after finding the remainder of division by ''k'', ''i'' and ''j'' are equal; i \vert j means that ''i'' is a divisor for ''j''; and gcd is the greatest common divisor. Note: Equation () is simply a Fermat primality test. If we find ''any'' value of , not divisible by , such that equation () is false, we may immediately conclude that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Primality Test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is '' trial division'': given an input number, n, check whether it is divisible by any prime number between 2 and \sqrt n (i.e., whether the division leaves no remainder). If so, then n is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For any divisor p \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Henry Cabourn Pocklington
Henry Cabourn Pocklington FRS (28 January 1870, Exeter – 15 May 1952, Leeds) was an English physicist and mathematician. His primary profession was as a schoolmaster, but he made important contributions to number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ... with the discovery of Pocklington's primality test in 1914 and the invention of Pocklington's algorithm. He also derived the first equation for the current in a wire antenna, Pocklington's integral equation. References * English physicists 19th-century English mathematicians 20th-century English mathematicians Fellows of the Royal Society 1870 births 1952 deaths People from Exeter {{UK-physicist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Derrick Henry Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes. His peripatetic career as a number theorist, with him and his wife taking numerous types of work in the United States and abroad to support themselves during the Great Depression, fortuitously brought him into the center of research into early electronic computing. Early life Lehmer was born in Berkeley, California, to Derrick Norman Lehmer, a professor of mathematics at the University of California, Berkeley, and Clara Eunice Mitchell. He studied physics and earned a bachelor's degree from UC Berkeley, and continued with graduate studies at the University of Chicago. He and his father worked together on Lehmer sieves. Marriage During his studies at Berkeley, Lehmer met Emma Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Primality Certificate
In mathematics and computer science, a primality certificate or primality proof is a succinct, formal proof that a number is prime. Primality certificates allow the primality of a number to be rapidly checked without having to run an expensive or unreliable primality test. "Succinct" usually means that the proof should be at most polynomially larger than the number of digits in the number itself (for example, if the number has ''b'' bits, the proof might contain roughly ''b''2 bits). Primality certificates lead directly to proofs that problems such as primality testing and the complement of integer factorization lie in NP, the class of problems verifiable in polynomial time given a solution. These problems already trivially lie in co-NP. This was the first strong evidence that these problems are not NP-complete, since if they were, it would imply that NP is subset of co-NP, a result widely believed to be false; in fact, this was the first demonstration of a problem in NP intersect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lucas Primality Test
In computational number theory, the Lucas test is a primality test for a natural number ''n''; it requires that the prime factors of ''n'' − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification that ''n'' is prime. Concepts Let ''n'' be a positive integer. If there exists an integer ''a'', 1 < ''a'' < ''n'', such that : and for every prime factor ''q'' of ''n'' − 1 : then ''n'' is prime. If no such number ''a'' exists, then ''n'' is either 1, 2, or composite. The reason for the correctness of this claim is as follows: if the first equivalence holds for ''a'', we can deduce that ''a'' and ''n'' are |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest common divisor of and is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, . 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, 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 integers and , at least one of which is nonzero, is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fermat Primality Test
The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. Concept Fermat's little theorem states that if ''p'' is prime and ''a'' is not divisible by ''p'', then :a^ \equiv 1 \pmod. If one wants to test whether ''p'' is prime, then we can pick random integers ''a'' not divisible by ''p'' and see whether the congruence holds. If it does not hold for a value of ''a'', then ''p'' is composite. This congruence is unlikely to hold for a random ''a'' if ''p'' is composite. Therefore, if the equality does hold for one or more values of ''a'', then we say that ''p'' is probably prime. However, note that the above congruence holds trivially for a \equiv 1 \pmod, because the congruence relation is compatible with exponentiation. It also holds trivially for a \equiv -1 \pmod if ''p'' is odd, for the same reason. That is why one usually chooses a random ''a'' in the interval 1 < a < p - 1. Any ''a'' such that : |
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] [Amazon] |
Tanja Lange
Tanja Lange is a German cryptographer and number theorist at the Eindhoven University of Technology. She is known for her research on post-quantum cryptography. Education and career Lange earned a diploma in mathematics in 1998 from the Technical University of Braunschweig. She completed her Ph.D. in 2001 at the Universität Duisburg-Essen. Her dissertation, jointly supervised by Gerhard Frey and YoungJu Choie, concerned ''Efficient Arithmetic on Hyperelliptic Curves''. After postdoctoral studies at Ruhr University Bochum, she became an associate professor at the Technical University of Denmark in 2005. She moved to the Eindhoven University of Technology as a full professor in 2007. At Eindhoven, she chairs the coding theory and cryptology group and is scientific director of the Eindhoven Institute for the Protection of Systems and Information. She is also the coordinator of PQCRYPTO, a European multi-university consortium to make electronic communications future-proof against ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Order (group Theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
