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# Lucas Chain

A Lucas chain for an integer is an increasing sequence

of integers such that every , , can be written as a sum of smaller elements whose difference is also en element of the sequence or zero (i.e., taking is allowed). The number is called the length of the chain.

For example, is a Lucas chain of length 3 for 5 because , , , , , and . Further examples are sequences of consecutive powers of 2 or the Fibonacci numbers 1, 2, 3, 5, 8, 13, 21, ....

Lucas chains are a special kind of addition chain and can be used to evaluate Lucas functions, which have been proposed for use in public-key cryptography.

Addition Chain, Fibonacci Number, Lucas Sequence

This entry contributed by Martin Kutz

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## References

Kutz, M. "Lower Bounds for Lucas Chains." SIAM J. Comput. 31, 1896-1908, 2002.Montgomery, P. L. "Evaluating Recurrences of Form via Lucas Chains." Unpublished manuscript. ftp://ftp.cwi.nl:/pub/pmontgom/Lucas.ps.gz.Yen, S.-M. and Laih, C.-S. "Fast Algorithms for LUC Digital Signature Computation." IEE Proc.--Computers and Digital Techn. 142, 165-169, Mar. 1995.

Lucas Chain

## Cite this as:

Kutz, Martin. "Lucas Chain." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LucasChain.html