Analysis of a class of easily computable permutations
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ANALYSIS OF A CLASS OF EASILY COMPUTABLE PERMUTATIONS V. G. Skobeleva and E. Ye. Zaitsevab
UDC 681.3+519.21
Properties of easily computable permutations specified in terms of a finite ring are investigated. It is established that problems of parametric identification for this class of permutations are reduced to the resolution of systems of Diophantine equations of sufficiently high degree and to the solution of systems of problems of taking discrete logarithms. Keywords: stream cipher, fractal, finite ring, easily computable permutations. INTRODUCTION Significant attention is recently given to the application of chaotic dynamic systems to the solution of information protection problems [1], in particular, to the construction of stream ciphers [2]. A general outline of a nonstationary stream cipher is presented in Fig. 1. It is assumed that a number generator belonging to a set N m and a class of algorithms A = {Ai }iÎN m represented in an implicit form are given. In generating a number i Î N m , the enciphering of the next fragment of the initial text is performed by an algorithm Ai . An important characteristic of the wide class of chaotic nonlinear dynamic systems is the presence of strange attractors. Fractals, i.e., mappings generating irregular self-similar structures [3–5], are also among them. In [6], the following scheme of the stream cipher based on the application of fractal mappings is proposed. A message (or an image) is represented by a BMP file that is sequentially processed pixel-by-pixel. In processing each pixel, an iterative process is realized that is determined by the fractal mapping being applied. When some given condition is satisfied (or violated), a permutation over the set of colors is performed that is determined by the number of the last iteration. It is the processed file that contains the obtained ciphertext. The correctness of this approach follows from the invertibility of permutations. The proposed scheme is realized for one of the most investigated fractal mappings, namely, the Mandelbrot mapping [5], i.e., the mapping f : C ® C (C is the set of complex numbers) specified by the equality f ( z ) = z 2 + c ( c is a constant) (Fig. 2a depicts a complex plane, and Fig. 2b depicts a display image). The parameters of the cipher that determine the number of iterations are the radius R of the circle ( R , O ) and the upper bound of the number K of iterations applied to a point of the complex plane C. The condition is as follows: the image being processed overruns the boundary of the circle ( R , O ) after K iterations. A class of families of permutations is specified by the equalities ì rn = ( r0 + n × | a n × a1 - a n × a 2 - a n × a 3 | ) (mod 256), 1 2 3 ï ï n n f n : í g n = ( g 0 + n × | a 2 × a 2 - a1 × a1 - a n3 × a 3 | ) (mod 256) ( n = 1,K, 28 - 1) ï n n n ïî bn = ( b0 + n × | a 3 × a 3 - a1 × a1 - a 2 × a 2 | ) (mod 256).
(1)
Here, r0 , g 0 , and b0 and rn , g n , and bn are, respectively, the initial and transformed components of the color of a
a
Institute of Applied Math
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