Cunningham chain

Abstract: A chain of nearly doubled primes is an ordered set {a 1 , a 2 , ..., aλ} of prime numbers, interlinked by a k =2a k−1 ±1. Chains of length up to 13 have been found. Shorter chains have been counted in some restricted ranges. Some of these counts are compared with the frequencies predicted by a quantitative version of the prime k-tuples conjecture

Abstract: By extending the operations +,× on natural numbers to the operations on finite sets of natural numbers, we founded a new formal system of a second order arithmetic 〈P(N), N,+,×,0,1, ∈〉. We designed a recursive sieve method on residue classes and obtained recursive formulas of a set sequence and its subset sequence of Sophie Germain primes, both the set sequences converge to the set of all Sophie Germain primes. Considering the numbers of elements of this two set sequences, one is strictly monotonically increasing and the other is monotonically increasing, the order topological limits of two cardinal sequences exist and these two limits are equal, we concluded that the counting function of Sophie Germain primes approaches infinity. The cardinal function is sequentially continuous with respect to the order topology, we proved that the cardinality of the set of all Sophie Germain primes is χ0 using modular arithmetical and analytic techniques on the set sequences. Further we extended this result to attack on Twin primes, Cunningham chains and so on.

Abstract: A Cunningham chain of length k is a finite set of primes p 1, p 2,...,p k such that p i+1=2p i +1, or p i+1=2p i−1 for i=1,2,3, ...,k−1. In this paper we present an algorithm that finds Cunningham chains of the form p i+1=2p i+1 for i=2,3 and a prime p 1. Such a chain of primes were recently shown to be cryptographically significant in solving the problem of Auto-Recoverable Auto-Certifiable Cryptosystems [YY98]. For this application, the primes p 1 and p 2 should be large to provide for a secure enough setting for the discrete log problem. We introduce a number of simple but useful speed-up methods, such as what we call trial remaindering and explain a heuristic algorithm to find such chains. We ran our algorithm on a Pentium 166 MHz machine. We found values for p 1, starting at a value which is 512 bits and ending at a value for p 1 which is 1,376 bits in length. We give some of these values in the appendix. The feasibility of efficiently finding such primes, in turn, enables the system in [YY98] which is a software-based public key system with key recovery (note that every cryptosystem which is suggested for actual use must be checked to insure that its computations are feasible).

Abstract: For integers a and b we define the Shanks chain p 1, p 2,..., p k of length k to be a sequence of k primes such that \(p_{i+1} = ap_{i}^{2} -- b\) for i = 1,2,..., k − 1. While for Cunningham chains it is conjectured that arbitrarily long chains exist, this is, in general, not true for Shanks chains. In fact, with s = ab we show that for all but 56 values of s ≤1000 any corresponding Shanks chain must have bounded length. For this, we study certain properties of functional digraphs of quadratic functions over prime fields, both in theory and practice. We give efficient algorithms to investigate these properties and present a selection of our experimental results.