  We follow two main objectives in this article. On the one hand, we introduce
a security model called LORBACPA$^+$ for self-synchronized stream ciphers which
is stronger than the blockwise LOR-IND-CPA, where we show that standard
constructions as delayed CBC or similar existing self-synchronized modes of
operation are not secure in this stronger model. Then, on the other hand,
following contributions of G.~Mill\'{e}rioux et.al., we introduce a new
self-synchronized stream cipher and prove its security in LORBACPA$^+$ model.
