The2-ADIC Properties Of Binary Periodic Sequences

Crypt analytic attack and crypt design are the important parts of the cryptol ogy, and their confrontations improve the development of cryptography. For the stream cipher, the no-linear sequences, which based on the Linear Feedback S hift Register (LFSR) were well known in the1960s, faded out the research fie Id for the rise of Algebraic attack and Correlation attack. Now, the hots-pot of it is the study of the no-linear feedback shift registers (NFSRs). The NFSRs have good Cryptography features which can resistance the Algebraic attack and Correlation attack, but for the research many Cryptography features of NFSR are not understood systematically.The Feedback Carry Shift Register (FCSR) is one of the class of nonline ar shift registers, which has extensive research, and its theory tool of FCSR is the2-adic theory which is different from LFSR. This paper uses the the res ults of the analysis of FCSR and the2-adic theory tools to study the2-adic properties of the binary periodic sequences. In addition, it also resear ch the no-linear stream register over Galois ring Z/(pe). There are strong corr elations between Z/(pe) and the ZUC stream register, and the sequences over Z/(pe) also have good2-adic properties.The main achievements of this paper as follow:1In the construction of a no-linear key-stream generator, self-shrinking is an established way of getting the binary pseudo-random periodic sequences in cryptography design. Using the theoretical analysis study the self-shrinking sequ ence based on the1-sequence, and the theoretical results reflect its good crypto graphy properties accurately, such that it has the last period when T is an odd number, and the expected value of its autocorrelation belongs {0,1/T} and th e variance is O(T/ln4T).2. The2-adic complexity of a sequence is the least number of cell since the basic feed-back with carry shift register and it is well known the one-to-one correspondence between the2-adic integers and periodic binary sequences. We find that the2-adic complexity of the self-shrinking sequence based on the1-sequence is large enough to resist the Rational Approximation attack. And for the self-shrinking sequence based on the m-sequence, it is shown that its2-adic complexity has a bigger lower bound under some circumstances.3. Furthermore, we use the exponential function to give a new approach to study the2-adic properties of binary balanced sequence and presents a relationship with2-adic integer, the length of period,2-adic complexity.4. The Legendre symbol has been used to construct-sequences with ideal cross-correlation, but it was never used in the arithmetic cross-correlation. A new class of generalized Legendre sequences are described and analyzed with respect to their period, distributional, arithmetic cross-correlation and distinctness properties. This analysis gives a new approach to study the connection between the Legendre symbol and the arithmetic cross-correlation.5. The arithmetic correlation has been studied in the research of the2-adic property of binary sequences, but it has not been analyzed extensively in Boolean functions. A new class of no-linear boo-lean functions are described and their arithmetic correlation are analyzed. This research gives a new approach to construct the functions with optimal arithmetic correlation.