Algorithms for the generation of full-length shift- register sequences

Two algorithms are presented for the generation of full-length shift-register cycles, also referred to as de Bruijn sequences. The first algorithm generates
2^{k \cdot g(n,k)
full cycles of length
2^{n}
, using
3n + k \cdot g(n, k)
bits of storage, where
k
is a free parameter in the range
1 \leq k \leq 2^{((n-4)/2)}
, and
g(n, k)
is of the order of
n - 2 \log k
. The second algorithm generates about
2^{n^{2}/4}
full cycles of length
2^{n}
, using about
n^{2}/2
bits of storage. In both algorithms, the time required to produce the next bit from the last
n
bits is close to
n
. A possible application to the construction of stream ciphers is indicated.