Period distribution of generalized discrete Arnold cat map

The generalized discrete Arnold cat map is adopted in various cryptographic and steganographic applications where chaos is employed. In this paper, we analyze the period distribution of this map. A systematic approach for addressing the general period distribution problem for any integer value of the modulus N is outlined, followed by a complete analysis for the case of prime N. The analysis is based on similar techniques studying linear feedback shift register (LFSR) sequences. Together with our previous results when N is a power of a prime [1,2], the period distribution of the cat map is characterized nearly completely for any integer N. Our results are also useful for evaluating the security of the cryptographic and steganographic algorithms based on the cat map as well as computing all unstable periodic orbits of the chaotic Arnold cat map.