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Feedback shift registers, 2adic span, and combiners with memory
 Journal of Cryptology
, 1997
"... Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presen ..."
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Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presented, including a synthesis algorithm (analogous to the BerlekampMassey algorithm for LFSR’s) which, for any pseudorandom sequence, constructs the smallest FCSR which will generate the sequence. These techniques are used to attack the summation cipher. This analysis gives a unified approach to the study of pseudorandom sequences, arithmetic codes, combiners with memory, and the MarsagliaZaman random number generator. Possible variations on the FCSR architecture are indicated at the end. Index Terms – Binary sequence, shift register, stream cipher, combiner with memory, cryptanalysis, 2adic numbers, arithmetic code, 1/q sequence, linear span. 1
On the Quadratic Span of Binary Sequences
, 2000
"... The length of the shortest FSR that generates a sequence is called the span of the sequence. If the feedback function is linear, then the BerlekampMassey algorithm can be used to efficiently determine the the length of the shortest linear FSR that generate the sequence and its associated linear fee ..."
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Cited by 1 (0 self)
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The length of the shortest FSR that generates a sequence is called the span of the sequence. If the feedback function is linear, then the BerlekampMassey algorithm can be used to efficiently determine the the length of the shortest linear FSR that generate the sequence and its associated linear feedback function. However, for a general nonlinear feedback function, determining the span and an associated feedback function efficiently is difficult because of the nonlinearities involved. Because of its tractability, most of the current research has focused on studying the linear span of a sequence. However, a sequence with a large linear span may be generated by a much shorter feedback shift register with nonlinear feedback function. In this paper we study the quadratic span of binary sequences. We prove that (i) If the quadratic span of the sequence s 0 ; s 1 ; \Delta \Delta \Delta ; s n\Gamma1 is ? n=2, then the quadratic span of the sequence s 0 ; s 1 ; \Delta \Delta \Delta ; s n rema...
Signal Processing Techniques in Cryptography
"... Abstract. Security of cryptographic symmetric primitives is studied in this thesis. Pseudorandomness characteristics of cryptographic sequences are analyzed, resulting in new methods for constructing sequences with high linear complexity. Connections between nonlinear complexity and other cryptograp ..."
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Abstract. Security of cryptographic symmetric primitives is studied in this thesis. Pseudorandomness characteristics of cryptographic sequences are analyzed, resulting in new methods for constructing sequences with high linear complexity. Connections between nonlinear complexity and other cryptographic criteria are also established, whereas a new recursive algorithm for efficiently computing the minimal feedback shift register which generates a given sequence is provided. Furthermore, security issues of cryptographic Boolean functions that are used in cryptographic systems as components of sequence generators are studied; on this direction, new efficient formulas for determining best quadratic approximations of several classes of Boolean functions are derived, leading to new design principles that should be considered in the construction of secure cryptosystems.
Study on Pseudorandom Sequences with Applications in Cryptography and Telecommunications
"... Abstract. Pseudorandom sequences have many applications in cryptography and spread spectrum communications. In this dissertation, on one hand we develop tools for assessing the randomness of a sequence, and on the other hand we propose new constructions of pseudorandom sequences. More precisely, we ..."
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Abstract. Pseudorandom sequences have many applications in cryptography and spread spectrum communications. In this dissertation, on one hand we develop tools for assessing the randomness of a sequence, and on the other hand we propose new constructions of pseudorandom sequences. More precisely, we develop tools for computing the first order approximation of a binary sequence with the minimum linear complexity, we propose two efficient algorithms for computing the second order complexity (quadratic span) of a binary sequence, and we consider and solve the problem of computing the maximum nonlinear complexity (span) of a sequence. Finally, we investigate the properties of a family of sequences constructed as the direct sum of two sequences with ideal autocorrelation, like the GMW sequences. 1
AListofMaximumPeriodNLFSRs
"... Abstract. NonLinear Feedback Shift Registers (NLFSRs) are a generalization of Linear Feedback Shift Registers (LFSRs) in which a current state is a nonlinear function of the previous state. While the theory behind LFSRs is wellunderstood, many fundamental problems related to NLFSRs remain open. Pro ..."
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Abstract. NonLinear Feedback Shift Registers (NLFSRs) are a generalization of Linear Feedback Shift Registers (LFSRs) in which a current state is a nonlinear function of the previous state. While the theory behind LFSRs is wellunderstood, many fundamental problems related to NLFSRs remain open. Probably the most important one is finding a systematic procedure for constructing NLFSRs with a guaranteed long period. Available algorithms either consider some special cases, or are applicable to small NLFSRs only. In this paper, we present a complete list ofnbit NLFSRs with the period 2 n − 1,n < 25, for three different types of feedback functions with algebraic degree two. We hope that the presented experimental data might help analysing feedback functions of maximumperiod NLFSRs and finding a supporting theory characterizing them. 1