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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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Cited by 50 (7 self)
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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 ground states of the Bernasconi model
 J. Phys. A: Math. Gen
, 1997
"... The ground states of the Bernasconi model are binary sequences of length N with low autocorrelations. We introduce the notion of perfect sequences, binary sequences with onevalued offpeak correlations of minimum amount. If they exist, they are ground states. Using results from the mathematical ..."
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Cited by 6 (0 self)
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The ground states of the Bernasconi model are binary sequences of length N with low autocorrelations. We introduce the notion of perfect sequences, binary sequences with onevalued offpeak correlations of minimum amount. If they exist, they are ground states. Using results from the mathematical theory of cyclic difference sets, we specify all values of N for which perfect sequences exist and how to construct them. For other values of N , we investigate almost perfect sequences, i.e. sequences with twovalued offpeak correlations of minimum amount. Numerical and analytical results support the conjecture that almost perfect sequences exist for all values of N , but that they are not always ground states. We present a construction for lowenergy configurations that works if N is the product of two odd primes.
Interpretation of the LempelZiv Complexity Measure in the Context of Biomedical Signal Analysis
 IEEE Transactions on Biomedical Engineering
"... Abstract—LempelZiv complexity (LZ) and derived LZ algorithms have been extensively used to solve information theoretic problems such as coding and lossless data compression. In recent years, LZ has been widely used in biomedical applications to estimate the complexity of discretetime signals. Desp ..."
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Cited by 6 (2 self)
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Abstract—LempelZiv complexity (LZ) and derived LZ algorithms have been extensively used to solve information theoretic problems such as coding and lossless data compression. In recent years, LZ has been widely used in biomedical applications to estimate the complexity of discretetime signals. Despite its popularity as a complexity measure for biosignal analysis, the question of LZ interpretability and its relationship to other signal parameters and to other metrics has not been previously addressed. We have carried out an investigation aimed at gaining a better understanding of the LZ complexity itself, especially regarding its interpretability as a biomedical signal analysis technique. Our results indicate that LZ is particularly useful as a scalar metric to estimate the bandwidth of random processes and the harmonic variability in quasiperiodic signals. Index Terms—Complex analysis, LempelZiv complexity (LZ), nonlinear analysis.
Multiplicative characters, the Weil bound, and polyphase sequence families with low correlation
 University of Waterloo
, 2009
"... Power residue and Sidelnikov sequences are polyphase sequences with low correlation and variable alphabet sizes, represented by multiplicative characters. In this paper, sequence families constructed from the shift and addition of the polyphase sequences are revisited. Initially, ψ(0) = 1 is assume ..."
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Cited by 4 (4 self)
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Power residue and Sidelnikov sequences are polyphase sequences with low correlation and variable alphabet sizes, represented by multiplicative characters. In this paper, sequence families constructed from the shift and addition of the polyphase sequences are revisited. Initially, ψ(0) = 1 is assumed for multiplicative characters ψ to represent power residue and Sidelnikov sequences in a simple form. The Weil bound on multiplicative character sums is refined for the assumption, where the character sums are equivalent to the correlations of sequences represented by multiplicative characters. General constructions of polyphase sequence families that produce some of known families as the special cases are then presented. The refined Weil bound enables the efficient proofs on the maximum correlation magnitudes of the sequence families. From the constructions, it is shown that Mary known sequence families with large size can be partitioned into (M + 1) disjoint subsequence families with smaller maximum correlation magnitudes. More generalized constructions are also considered by the addition of multiple cyclic shifts of power residue and Sidelnikov sequences.
New Construction of Mary Sequence Families With Low Correlation From the Structure of Sidelnikov Sequences
"... The main topics of this paper are the structure of Sidelnikov sequences and new construction of Mary sequence families from the structure. For prime p and a positive integer m, it is shown that Mary Sidelnikov sequences of period p2m − 1, if M  pm − 1, can be equivalently generated by the operati ..."
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Cited by 4 (3 self)
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The main topics of this paper are the structure of Sidelnikov sequences and new construction of Mary sequence families from the structure. For prime p and a positive integer m, it is shown that Mary Sidelnikov sequences of period p2m − 1, if M  pm − 1, can be equivalently generated by the operation of elements in a finite field GF(pm), including a pmary msequence. The equivalent representation over GF(pm) requires low complexity for implementing the Sidelnikov sequences of period p2m − 1. From the (pm − 1) × (pm + 1) array structure of the sequences, it is then found that a half of the column sequences and their constant multiples have low correlation enough to construct new Mary sequence families of period pm − 1. In particular, new Mary sequence families of period pm − 1 are constructed from the combination of the column sequence families and known Sidelnikovbased sequence families, where the new families have larger family sizes than the known ones with the same maximum correlation magnitudes. Finally, it is shown that the new Mary sequence family of period pm −1 and the maximum correlation magnitude 2 √ pm +6 asymptotically achieves √ 2 times the equality of the Sidelnikov’s lower bound when M = pm − 1 for odd prime p.
New Binary Sequences with Optimal Autocorrelation Magnitude
, 2006
"... New binary sequences of period � � for even � � are found. These sequences can be described by a � interleaved structure. The new sequences are almost balanced and have fourvalued autocorrelation, i.e., � � � ��, which is optimal with respect to autocorrelation magnitude. Complete autocorrelati ..."
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Cited by 3 (0 self)
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New binary sequences of period � � for even � � are found. These sequences can be described by a � interleaved structure. The new sequences are almost balanced and have fourvalued autocorrelation, i.e., � � � ��, which is optimal with respect to autocorrelation magnitude. Complete autocorrelation distribution and exact linear complexity of the sequences are mathematically derived. From the simple implementation with a small number of shift registers and a connector, the sequences have a benefit of obtaining large linear complexity.
Character Sums and Polyphase Sequence Families with Low Correlation, DFT and Ambiguity
"... We present a survey on the current status of the constructions of polyphase sequences with low correlation, discrete Fourier transform (DFT), and ambiguity in both time and phase domain, including some new insights and results. Firstly, we systematically introduce the concepts of phaseshift operato ..."
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We present a survey on the current status of the constructions of polyphase sequences with low correlation, discrete Fourier transform (DFT), and ambiguity in both time and phase domain, including some new insights and results. Firstly, we systematically introduce the concepts of phaseshift operators and ambiguity functions of sequences, and give a new construction of polyphase sequences from combinations of different indexing field elements and hybrid characters. We then present the constructions, some known and some new, of polyphase sequences with low degree polynomials, for their low correlation, DFT and ambiguity can be bounded by directly applying the Weil bounds. Thirdly, we introduce the Hadamard equivalence, restate the conjectured new ternary 2level autocorrelation sequences, and present their Hadamard equivalence relations. Some open problems are presented.