ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 3, pp. 289–294.
⃝ Pleiades Publishing, Inc., 2009.
Original Russian Text
⃝ F. Galand, G.A. Kabatiansky, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 3, pp. 106–111.
Coverings, Centered Codes, and
IRISA, Rennes, France
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received June 5, 2009; in ﬁnal form, June 22, 2009
Abstract—It is shown that steganography with a given distortion criteria, which we call com-
binatorial steganography, is equivalent to coverings of Hamming spaces or to so-called centered
error-correcting codes, depending on whether an opponent is passive or active, respectively.
A construction of centered error-correcting codes based on Reed–Solomon and algebraic geom-
etry codes is proposed.
Steganography (which literally means concealed writing in Greek) is masking messages that con-
tain conﬁdential information as messages that do not contain such information. The ﬁrst theoretical
model of steganography was proposed in . As usual in informal descriptions of cryptographic
problems, information is transmitted between Alice and Bob, and there is a third participant, their
opponent Eve, who tries to obstruct such information transmission. In the model in question, Alice
and Bob have became prisoners (hence the title of the paper ), and Eve is their warder. Alice and
Bob are going to escape and want to coordinate their actions. They can exchange messages, but
only through Eve. If Eve becomes suspicious that messages contain some secret information, she
stops the message exchange. Therefore, messages with secret information and messages without it
must be practically indistinguishable.
In a probabilistic model proposed in , information is transmitted based on a discrete source
of messages in such a way that statistics of messages containing secret information is (almost)
the same as statistics of . Fore more details about this model and a new asymptotically optimal
method of information transmission, see .
In another model, which we call combinatorial steganography, Alice and Bob are allowed to
“slightly” change an original message. Formally, this means that messages are considered as words
over some ﬁnite alphabet, and the term slightly is deﬁned via the Hamming distance between an
original word and a modiﬁed one, containing secret information. In this problem setting, Eve has
got a degree of freedom since, when “carrying” a message from Alice to Bob (or in reverse direc-
tion), Eve can also make small changes in the message. This model is called active, in contrast to
the passive one where Eve simply transmits messages without distorting them.
It is shown in Section 2 that the problem of passive combinatorial steganography is essentially
equivalent to the problem of constructing covering codes for the Hamming space. This result is
Supported in part by the Russian Foundation for Basic Research, project nos. 09-01-00536 and 08-07-