On Periods of Arnold-type Transformations
LI Luo-luo · Acta Scientiarum Naturalium Universitatis Sunyatseni · 2005
Arnold transformation is one of the techniques for scrambling digital images for the sake of communication security. Under certain conditions Arnold transformation A modulo N possesses periodicity by which the encoder and decoder can control randomly the number of the transformations in the image transportation. To study periodicity of Arnold-type transformations, basic modular arithmetic properties of integer matrices are firstly established. A theorem is secondly obtained to show that the periodicity of A modulo N equals the least common multiple of all those moduloes mutually prime factors of N. Thus the periodicity analysis can be reduced to transformation modulo prime numbers (and their powers). Finally formulas for periodicity of transformation modulo powers of prime number are proved and the concept of period character code is introduced by which an elegant and concise method to calculate the period of Arnold-type scrambling transformation is obtained.