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 = 2 pi + 1. (Hence each term of such a chain except the last one is a Sophie Germain prime, and each term except the first is a safe prime).
It follows that, ..., .
Similarly, a Cunningham chain of the second kind of length n is a sequence of prime numbers (p1,...,pn) such that for all 1 ≤ i < n, pi+1 = 2 pi - 1.
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 chains are now considered useful in cryptographic systems since "they provide two concurrent suitable settings for the ElGamal cryptosystem ... can be implemented in any field where the discrete logarithm problem is difficult."
Read more about Cunningham Chain: Largest Known Cunningham Chains, Congruences of Cunningham Chains
Famous quotes containing the words cunningham and/or chain:
“Like an old brandy after a long rain,
Distinguished, and familiar, and aloof.”
—James Vincent Cunningham (1911–1985)
“The name of the town isn’t important. It’s the one that’s just twenty-eight minutes from the big city. Twenty-three if you catch the morning express. It’s on a river and it’s got houses and stores and churches. And a main street. Nothing fancy like Broadway or Market, just plain Broadway. Drug, dry good, shoes. Those horrible little chain stores that breed like rabbits.”
—Joseph L. Mankiewicz (1909–1993)