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Finding length-3 positive Cunningham chains and their cryptographic significance

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Algorithmic Number Theory (ANTS 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1423))

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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).

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Joe P. Buhler

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© 1998 Springer-Verlag Berlin Heidelberg

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Young, A., Yung, M. (1998). Finding length-3 positive Cunningham chains and their cryptographic significance. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg.

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