  In this paper we study recent reaction attacks against QC-LDPC and QC-MDPC
code-based cryptosystems, which allow an opponent to recover the private
parity-check matrix through its distance spectrum by observing a sufficiently
high number of decryption failures. We consider a special class of codes, known
as monomial codes, to form private keys with the desirable property of having a
unique and complete distance spectrum. We verify that for these codes the
problem of recovering the secret key from the distance spectrum is equivalent
to that of finding cliques in a graph, and use this equivalence to prove that
current reaction attacks are not applicable when codes of this type are used in
the McEliece cryptosystem.
