Period of arnold transformation and its application in image scrambling

Abstract

With the security problem of image information as the background, some more properties of the period of Arnold transformation of two-dimension were studied by means of introducing a integer sequence. Some new results are obtained. Two interesting conjectures on the period of Arnold transformation are given. When making digital images scrambling by Arnold transformation, it is important to know the period of the transformation for the image. As the application of the theory, a new method for computing the periods at last are proposed.

Keywords

digital image / period / dynamic system / Arnold transformation / scrambling transformation

References

[1] QiDong-xu, ZouJian-cheng, HanXiao-you. A new class of scrambling transformation and its application in the image information covering [J]. Science in China (Series E), 2000, 43(3): 304-312

[2] DingWei, QiDong-xu. Digital image transformation and information hiding and disguising technology [J]. J Computers, 1998, 21(9): 838-838(in Chinese)

[3] QiDong-xu. Matrix transformation and its application to image hiding [J]. J North China Univ Tech, 1999, 11(1): 24-28(in Chinese)

[4] DysonF J, FalkH. Period of a discrete cat mapping [J]. The Amer Math Monthly, 1992, 99: 603-614

[5] ZouJian-cheng, TieXiao-yun. Arnold transformation of digital image with two dimension and its periodicity [J]. J North China Univ Tech, 2000, 12(1): 10-14(in Chinese)

[6] DingWei, QiDong-xu. A new kind of digital image transformation in information disguising [A]. Information Sciences and Microelectronic Technology[C], 1998, Beijing, Academic Press: 309-311(in Chinese)

[7] SunWei. The periodicity of Arnold transformation [J]. J North China Univ Tech, 1999, 11(1): 29-32(in Chinese)

[8] ZhaoHui. Arnold transformation of n dimension and its periodicity [J]. J North China Univ Tech, 2002, 14(1): 21-25(in Chinese)

[9] KongTao, ZhangDan. A new anti-arnold transformation algorithm [J]. J Software, 2004, 15(10): 1558-1564(in Chinese)

[10] LiBing, XuJia-wei. On the periods of Arnold Transformations and some applications [J]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2004, 43(S2): 139-142(in Chinese)

[11] Lakshminarayan A, Balazs N L. On the quantum cat and sawtooth maps-return to generic behaviour [EB/OL]. http://arxiv:org/pdf/chao-dyn/9307005, 1993. 7.

[12] Barash L, Shchur L N. Periodic orbits of the ensemble of cat maps and pseudorandom number generation [EB/OL]. http://arxiv.org/PS_cache/physics/pdf/0409/0409069.pdf, 2004. 9.

[13] ArnoldV I, AvezA. Ergodic problems of classical mechanics [A]. Mathematical Physics Monograph Series[C], 1968, New York, W. A. Benjamin

[14] ZhouChi-zhongThe Fibonacci-Lucas sequences [M], 1993, Changsha, Hunan Science and Tech Press(in Chinese)

[15] PanCheng-dong, PanCheng-biaoAn introduction to the theory of numbers [M], 1998, Beijing, Beijing University Press(in Chinese)