  We give efficient algorithms for ranking Lyndon words of length $n$ over an
alphabet of size $\sigma$. The rank of a Lyndon word is its position in the
sequence of lexicographically ordered Lyndon words of the same length. The
outputs are integers of exponential size, and complexity of arithmetic
operations on such large integers cannot be ignored. Our model of computations
is the word-RAM, in which basic arithmetic operations on (large) numbers of
size at most $\sigma^n$ take $O(n)$ time. Our algorithm for ranking Lyndon
words makes $O(n^2)$ arithmetic operations (this would imply directly cubic
time on word-RAM). However, using an algebraic approach we are able to reduce
the total time complexity on the word-RAM to $O(n^2 \log \sigma)$. We also
present an $O(n^3 \log^2 \sigma)$-time algorithm that generates the Lyndon word
of a given length and rank in lexicographic order. Finally we use the
connections between Lyndon words and lexicographically minimal de Bruijn
sequences (theorem of Fredricksen and Maiorana) to develop the first
polynomial-time algorithm for decoding minimal de Bruijn sequence of any rank
$n$ (it determines the position of an arbitrary word of length $n$ within the
de Bruijn sequence).
