ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 2, pp. 149–165.
Pleiades Publishing, Inc., 2011.
Original Russian Text
V.R. Sidorenko, G. Schmidt, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 2, pp. 72–89.
A Linear Algebraic Approach to Multisequence
V. R. Sidorenko
and G. Schmidt
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Ulm University, Germany
Esslingen University of Applied Sciences, Germany
Received March 31, 2008; in ﬁnal form, December 29, 2010
Abstract—An eﬃcient algorithm which synthesizes all shortest linear-feedback shift regis-
ters generating K given sequences with possibly diﬀerent lengths over a ﬁeld is derived, and
its correctness is proved. The proposed algorithm generalizes the Berlekamp–Massey and
Feng–Tzeng algorithms and is based on Massey’s ideas. The time complexity of the algo-
rithm is O(KλN) O(KN
), where N is the length of a longest sequence and λ is the linear
complexity of the sequences.
Problem statement. Consider K sequences s
, k =1,...,K,overan
arbitrary ﬁeld F with possibly diﬀerent lengths N
. The problem of shift register synthesis for
multiple sequences can be stated as follows.
Problem 1. Let s
be sequences over a ﬁeld F with lengths N
respectively. Find the smallest nonnegative integer λ for which there is a vector of coeﬃcients
) over F such that for k =1, 2,...,K
− ...− σ
for n = λ +1,...,N
Moreover, ﬁnd a vector of coeﬃcients σ which fulﬁlls (1).
We are interested in a low complexity algorithm solving Problem 1 (i.e., ﬁnding λ and σ). A found
integer λ,0≤ λ ≤ N, is called the linear complexity of the sequences s
is the length of a longest sequence. In addition, we would like to know whether a found vector σ
is a unique solution of Problem 1 and, if not, we would like to ﬁnd all solutions σ.
Let us write a solution of Problem 1 as a polynomial
x + ...+ σ
deg σ(x) ≤ λ. (3)
For every k the sequence s
can be generated by a linear-feedback shift register of length λ
shown in the ﬁgure provided that the λ memory elements of the shift register are initially loaded
Supported in part by the German Research Council (DFG), project nos. BO 867/14 and BO 867/21-1.