Topics on Register Synthesis Problems

Pseudo-random sequences are ubiquitous in modern electronics and information technology. High speed generators of such sequences play essential roles in various engineering applications, such as stream ciphers, radar systems, multiple access systems, and quasi-Monte-Carlo simulation. Given a short prefix of a sequence, it is undesirable to have an efficient algorithm that can synthesize a generator which can predict the whole sequence. Otherwise, a cryptanalytic attack can be launched against the system based on that given sequence. Linear feedback shift registers (LFSRs) are the most widely studied pseudorandom sequence generators. The LFSR synthesis problem can be solved by the Berlekamp-Massey algorithm, by constructing a system of linear equations, by the extended Euclidean algorithm, or by the continued fraction algorithm. It is shown that the linear complexity is an important security measure for pseudorandom sequences design. So we investigate lower bounds of the linear complexity of different kinds of pseudorandom sequences. Feedback with carry shift registers (FCSRs) were first described by Goresky and Klapper. They have many good algebraic properties similar to those of LFSRs. FCSRs are good candidates as building blocks of stream ciphers. The FCSR synthesis problem has been studied in many literatures but there are no FCSR synthesis algorithms for multi-sequences. Thus one of the main contributions of this dissertation is to adapt an interleaving technique to develop two algorithms to solve the FCSR synthesis problem for multi-sequences. Algebraic feedback shift registers (AFSRs) are generalizations of LFSRs and FCSRs. Based on a choice of an integral domain R and π ∈ R, an AFSR can produce sequences whose elements can be thought of elements of the quotient ring R/(π). A modification of the Berlekamp-Massey algorithm, Xu\u27s algorithm solves the synthesis problem for AFSRs over a pair (R, π) with certain algebraic properties. We propose two register synthesis algorithms for AFSR synthesis problem. One is an extension of lattice approximation approach but based on lattice basis reduction and the other one is based on the extended Euclidean algorithm

Correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences

Sequences with optimal autocorrelation properties play an important role in wireless communication, radar and cryptography. Interleaving is a very important method in constructing the optimal autocorrelation sequence. Tang and Gong gave three different constructions of interleaved sequences (generalized GMW sequences, twin prime sequences and Legendre sequences). Su et al. constructed a series of sequences with optimal autocorrelation magnitude via interleaving Ding-Helleseth-Lam sequences. In this paper we further study the correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences

A convolutional-like approach to p-adic codes

AbstractIn this paper, we will see that codes defined over p-adic fields can be used in the same way as convolutional codes. We will prove some theoretical results concerning their encoders and show that, in practice, they can be encoded and decoded efficiently