Maximum Likelihood Decoding Algorithm for some Goppa and BCH Codes: Application to the Matrix Encoding Method for Steganography
Résumé
The idea of "Matrix encoding" was introduced in steganography by Crandall in 1998. The implementation was then proposed by Westfeld with steganography algorithm F5. Matrix encoding using linear codes (syndrome coding) is a general approach to improving embedding efficiency of steganographic schemes. The covering radius of the code corresponds to the maximal number of embedding changes needed to embed any message. Steganographers, however, are more interested in the average number of embedding changes rather than the worst case. In fact, the concept of embedding efficiency - the average number of bits embedded per embedding change - has been frequently used in steganography to compare and evaluate performance of steganographic schemes. The aim of this paper is to transform some algebraic decoding algorithms up to the error correcting capacity into a maximum likelihood decoder by the use of a limited exhaustive search. This algorithm is directly inspired from those proposed by N. Courtois, M. Finiasz, and N. Sendrier in the context of electronic signature. It remains exponential, however it becomes practicable for for some small BCH and Goppa codes (typically, with an error correcting capacity until 4).