Further on the Construction of Feedback Shift Registers with Maximum Strong Linear Complexity

In this paper, we present the more accurate definition of strong linear complexity of feedback shift registers based on Boolean algebraic than before, and analyze the bound of strong linear complexity by the fixed feedback function. Furthermore, the feedback shift registers with maximum strong linear complexity are constructed, whose feedback functions require the least number of monomials. We also show that the conclusions provide particular ideas and criteria for the design of feedback shift registers

Modified Alternating Step Generators

Irregular clocking of feedback shift registers is a popular technique to improve parameters of keystream generators in stream ciphers. Another technique is to implement nonlinear functions. We join these techniques and propose Modified Alternating Step Generators built with linear and nonlinear feedback shift registers. Adequate nonlinear Boolean functions are used as feedbacks or as filtering functions of shift registers in order to increase complexity of sequences produced by individual registers and the whole generator. We investigate basic parameters of proposed keystream generators, such as period, linear complexity and randomness

Extended class of linear feedback shift registers

Shift registers with linear feedback are frequently used. They owe their popularity to very well developed theoretical base. Registers with feedback of prime polynomials are of particular practical importance. They are willingly applied as test sequence generators and test response compactors. The article presents an attempt to extend the class of registers with linear feedback. Basing on the formal description of the register, the algorithms of register transformation are proposed. It allows to obtain the registers with equivalent graphs.[1] I. Gosciniak, “Linear Registers with Mixed Feedback, in Polish; Rejestry liniowe z mieszanym sprzȩżeniem zwrotnym,” Pomiary Automatyka Kontrola, no. 1, pp. 4–6, 1996.[2] K. Iwasaki, “Analysis and proposal of signature circuits for LSI testing,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 7, no. 1, pp. 84–90, 1988.[3] L.-T. Wang, N. Touba, R. Brent, H. Xu, and H. Wang, “On Designing Transformed Linear Feedback Shift Registers with Minimum Hardware Cost – Technical Report,” Computer Engineering Research Center Department of Electrical & Computer Engineering The University of Texas at Austin, 2011.[4] J. Rajski, J. Tyszer, M. Kassab, and N. Mukherjee, “Method for Synthesizing Linear Finite State Machines,” U.S. Patent, No. 6,353,842, 2002.[5] I. Gosciniak, “Equivalent Form of Linear Feedback Shift Registers,” in XIXth National Conference Circuit Theory and Eletronic Networks, 1996, pp. 115–120.[6] L. Alaus, D. Noguet, and J. Palicot, “A Reconfigurable LFSR for Tristandard SDR Transceiver, Architecture and Complexity Analysis,” in Digital System Design Architectures, Methods and Tools, 2008. DSD ’08. 11th EUROMICRO Conference on. IEEE Computer Society, 2008, pp. 61–67.[7] R. Ash, Information Theory. John Wiley & Sons, 1967.[8] M. Kopec, “Can Nonlinear Compactors Be Better than Linear Ones?” IEEE Trans. Comput., no. 11, pp. 1275–1282, 1995.[9] A. Gucha and L. Kinney, “Relating the Cyclic Behaviour of Linear Intrainverted Feedback shift Registers,” IEEE Transactions on Computers, vol. 41, no. 9, pp. 1088–1100, 1992

Stream Cipher Analysis Based on FCSRs

Cryptosystems are used to provide security in communications and data transmissions. Stream ciphers are private key systems that are often used to transform large volumn data. In order to have security, key streams used in stream ciphers must be fully analyzed so that they do not contain specific patterns, statistical infomation and structures with which attackers are able to quickly recover the entire key streams and then break down the systems. Based on different schemes to generate sequences and different ways to represent them, there are a variety of stream cipher analyses. The most important one is the linear analysis based on linear feedback shift registers (LFSRs) which have been extensively studied since the 1960\u27s. Every sequence over a finite field has a well defined linear complexity. If a sequence has small linear complexity, it can be efficiently recoverd by Berlekamp-Messay algorithm. Therefore, key streams must have large linear complexities. A lot of work have been done to generate and analyze sequences that have large linear complexities. In the early 1990\u27s, Klapper and Goresky discovered feedback with carry shift registers over Z/(p) (p-FCSRS), p is prime. Based on p-FCSRs, they developed a stream cipher analysis that has similar properties to linear analysis. For instance, every sequence over Z/(p) has a well defined p-adic complexity and key streams of small p-adic complexity are not secure for use in stream ciphers. This disstation focuses on stream cipher analysis based on feedback with carry shift registers. The first objective is to develop a stream cipher analysis based on feedback with carry shift registers over Z/(N) (N-FCSRs), N is any integer greater than 1, not necessary prime. The core of the analysis is a new rational approximation algorithm that can be used to efficiently compute rational representations of eventually periodic N-adic sequences. This algorithm is different from that used in $p$-adic sequence analysis which was given by Klapper and Goresky. Their algorithm is a modification of De Weger\u27s rational approximation algorithm. The second objective is to generalize feedback with carry shift register architecture to more general algebraic settings which are called algebraic feedback shift registers (AFSRs). By using algebraic operations and structures on certain rings, we are able to not only construct feedback with carry shift registers, but also develop rational approximation algorithms which create new analyses of stream ciphers. The cryptographic implication of the current work is that any sequences used in stream ciphers must have large N-adic complexities and large AFSR-based complexities as well as large linear complexities

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