Cunningham chain. In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes. A Cunningham chain of the first kind of length n is a sequence of prime numbers (p1, ..., pn) such that for all 1 ≤ i < n, pi+1 = 2pi + 1. It follows that Or, by setting (the number is not part of the sequence and need not be a prime number), we have Similarly, a Cunningham chain of the second kind of length n is a sequence of prime numbers (p1,...
It follows that the general term is Now, by setting , we have Cunningham chains are also sometimes generalized to sequences of prime numbers (p1, ..., pn) such that for all 1 ≤ i ≤ n, pi+1 = api + b for fixed coprime integers a, b; the resulting chains are called generalized Cunningham chains. A Cunningham chain is called complete if it cannot be further extended, i.e., if the previous or next term in the chain would not be a prime number anymore. . . . . .
Cunningham Chain records. Diffie–Hellman key exchange. The scheme was first published by Whitfield Diffie and Martin Hellman in 1976.[2] By 1975, James H. Ellis,[3] Clifford Cocks and Malcolm J. Williamson within GCHQ, the British signals intelligence agency, had also shown how public-key cryptography could be achieved; however, their work was kept secret until 1997.[4] Although Diffie–Hellman key agreement itself is an anonymous (non-authenticated) key-agreement protocol, it provides the basis for a variety of authenticated protocols, and is used to provide perfect forward secrecy in Transport Layer Security's ephemeral modes (referred to as EDH or DHE depending on the cipher suite). U.S. Patent 4,200,770,[5] from 1977, is now expired and describes the now public domain algorithm. Name[edit] In 2002, Hellman suggested the algorithm be called Diffie–Hellman–Merkle key exchange in recognition of Ralph Merkle's contribution to the invention of public-key cryptography (Hellman, 2002), writing: Description[edit] Cryptographic explanation[edit] , and.
Modulo operation. Quotient (red) and remainder (green) functions using different algorithms Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of n is 0 to n − 1. (n mod 1 is always 0; n mod 0 is undefined, possibly resulting in a "Division by zero" error in computer programming languages) See modular arithmetic for an older and related convention applied in number theory. When either a or n is negative, the naive definition breaks down and programming languages differ in how these values are defined. Remainder calculation for the modulo operation[edit] In mathematics the result of the modulo operation is the remainder of the Euclidean division.
In nearly all computing systems, the quotient q and the remainder r satisfy Knuth[6] described floored division where the quotient is defined by the floor function q=floor(a/n) and the remainder r is Raymond T. Two corollaries are that or, equivalently, Natural number. Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.
History of natural numbers and the status of zero[edit] The most primitive method of representing a natural number is to put down a dot for each object. The first major advance in abstraction was the use of numerals to represent numbers. The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes.
Independent studies also occurred at around the same time in India, China, and Mesoamerica. as a rule includes zero, because (e.g.) Notation[edit] Prime number. A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6.
The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 × 3, 1 × 1 × 3, etc. are all valid factorizations of 3. The property of being prime (or not) is called primality. . Definition and examples 5 is again prime: none of the numbers 2, 3, or 4 divide 5. Hence, 6 is not prime. 2, 3, ..., n − 1 divides n (without remainder). N = a · b. History.