“…When using chaos in digital ciphers, many researchers have found dynamical degradation of digital chaotic systems and such degradation reduces the security of the designed chaotic ciphers [Erdmann & Murphy, 1992;Li et al, 2001aLi et al, , 2003aMasuda & Aihara, 2002b;Sang et al, 1998a,b;Wheeler & Matthews, 1991;. Actually, motivated by various "strange" phenomena of chaos observed on digital computers and in numerical simulations, pathologies of digital chaotic systems have been observed and extensively studied in the field of chaos theory [Arrowsmith & Vivaldi, 1994;Beck & Roepstorff, 1987;Benettin et al, 1978;Binder, 1992;Binder & Jensen, 1986;Blank, 1994Blank, , 1997Borcherds & McCauley, 1993;Bosioand & Vivaldi, 2000;Chambers, 1999;Chirkikov & Vivaldi, 1999;Diamond et al, 1994Diamond et al, , 1995Earn & Tremaine, 1992;Fryska & Zohdy, 1992;Góra & Boyarsku, 1988;Grebogi et al, 1988;Hogg & Huberman, 1985;Huberman, 1986;Kaneko, 1988;Karney, 1983;Keating, 1991;Levy, 1982;Li et al, 2001a;Lowenstein & Vivaldi, 1998;Masuda & Aihara, 2002b;McCauley & Palmore, 1986;Palmore & Herring, 1990;Palmore & McCauley, 1987;…”