ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 4, pp. 398–408.
Pleiades Publishing, Inc., 2015.
Original Russian Text
V.M. Deundyak, S.A. Yevpak, V.V. Mkrtichyan, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 4,
Analysis of Properties of q-ary Reed–Muller
Error-Correcting Codes Viewed as Codes
for Copyright Protection
V. M. Deundyak
, and V. V. Mkrtichyan
Southern Federal University, Rostov-on-Don, Russia
e-mail: email@example.com, firstname.lastname@example.org
Research Institute “Spetsvuzavtomatika”, Rostov-on-Don, Russia
Received October 20, 2014; in ﬁnal form, September 15, 2015
Abstract—We consider a data protection scheme where error-correcting codes can be used for
eﬃcient protection against unauthorized copying organized by coalitions of cardinality c ∈ N
of malicious users. We ﬁnd limits of range of the parameters of prospective q-ary Reed–Muller
codes for which these codes are c-TA and c-FP codes for copyright protection.
1. INTRODUCTION AND PROBLEM STATEMENT
In [1–3] a prospective method employing error-correcting codes as codes for copyright protection
was considered, in particular, for the use in broadcast encryption systems. In data transmission
schemes where employing such codes is possible, users apply code vector keys for legal access
to the data. However, some legal users can form pirate coalitions of cardinality c ∈ N in order to
produce pirate keys, which in turn can be used to acquire unauthorized access to the data, resulting
in various copyright violations. The protection system assumes detecting malicious users from a
coalition given their pirate key (see [4, 5]).
For the use in such systems, the classes of so-called c-TA and c-FP codes for copyright protection
are widely studied and exploited [1–3,6–12]. The c-TA class consists of the codes for which applying
any maximum likelihood decoder to a pirate key guarantees ﬁnding a key of a malicious user entering
a coalition of cardinality c.Awiderc-FP class consists of the codes for which a pirate key produced
by a coalition of cardinality c cannot be a key of a user that does not enter the coalition, which
makes direct compromising of users impossible. Of interest is the problem of ﬁnding new error-
correcting c-TA and c-FP codes for further use in protection systems to ensure the detection of
malicious users and to protect legal users from being compromised directly, respectively.
In  the possibility for using some Reed–Solomon codes as c-TAcodeswasshown,andin
limits of range of parameters of the Reed–Solomon codes were studied for which they are c-TA
or c-FP codes. In  it was proved that binary Reed–Muller codes cannot be used as codes for
copyright protection, and the possibility for using some q-ary Reed–Muller codes as c-TA codes
was shown. A mathematical model of an eﬃcient special broadcast encryption scheme using the
properties of c-TA and c-FP codes was constructed in [4,5] for Reed–Solomon codes and in  for
q-ary Reed–Muller codes.
In the present paper we compute limits of range of parameters of q-ary Reed–Muller codes for
which they are c-TA and c-FP codes. Below in Section 2 we describe q-ary Reed–Muller codes