Z <sub> Pe Of </ Sub> On The Primitive Sequence Highest Power Sequence Element Distribution And Z <sub> 4 </ Sub> Cyclic Shift Register

Information age has raised higher requirements on information security and privacy communication, where stream ciphers are a class of the most widely used encryption algorithms. Constructing new and good pseudorandom sequences has always been the key research subject in theory of stream ciphers. With the rapid development of the VLSI,it is feasible to construct newer and better pseudorandom sequences using microprocessors.In the first part of this thesis, we get the estimate the distributions of elements i(i = 0,1, ? ? ? ,p - 1) in the highest level sequence of primitive sequences over Z_p^e. It has been shown that the highest level sequence of primitive sequences over Galois rings have large period and linear complexity. So it is of great cryptographic significance to investigate the random properties of a_e-1.Using the estimates of the character sums over Galois rings, the trace representations of primitive sequences over Z_p^e and the discrete Fourier transformation techniques , we obtain that for any given p, e , the frequency of occurrences of an element i(i = 0,1, ? ? ? ,p - 1) in the highest level sequence of primitive sequences over Z_p^e is asymptotically 1/p+ O(p^-n/2) . This means when n is large enough , the distribution of elements is balanced. The main result is as follows: for i = 0,1,2, ?? ?,p-1 , let f_i denote the proportion of elements of i of the highest level sequence a_e-1 in one period, we have:where In the second part of this thesis , we introduce a new class of feedback shift registers over ring, which combine the basic computer instruction cycle shift with LFSR, are highly efficient in software implementation and have desirable cryptographic properties. We obtain a criterion for irregular σ-Feedback Shift Register and a criterion for σ-Feedback Shift Register over Z₄ to have the maximal period in some conditions . Although the cycle shift operator puts the most significant bit to the lowest one, it overcomes two shortcomings of linear recurring sequence over Z_2^e: the linear character of the lowest bit and the output of the most significant bit has no influence to the lowest one. So the cryptographic property of the σ-Feedback Shift Register is greatly improved. The simulation indicates a-Feedback Shift Register has long period, high complexity and good stability.