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.

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 Unpublished manuscript.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.