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17
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 51 (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
Shiftregister synthesis (modulo m)
 SIAM J. Computing
, 1985
"... The BerlekampMassey algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. In this paper we extend the algorithm to the case when the elements of the sequence are integers modulo m, where m is ..."
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Cited by 15 (0 self)
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The BerlekampMassey algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. In this paper we extend the algorithm to the case when the elements of the sequence are integers modulo m, where m is an arbitrary integer with known prime decomposition.
Some Aspects of Hankel Matrices in Coding Theory and Combinatorics
 J. Comb
, 2001
"... Hankel matrices consisting of Catalan numbers have been analyzed by various authors. DesainteCatherine and Viennot found their determinant to be # 1#i#j#k i+j+2n i+j and related them to the Bender  Knuth conjecture. The similar determinant formula # 1#i#j#k i+j1+2n i+j1 can be shown to ho ..."
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Cited by 11 (0 self)
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Hankel matrices consisting of Catalan numbers have been analyzed by various authors. DesainteCatherine and Viennot found their determinant to be # 1#i#j#k i+j+2n i+j and related them to the Bender  Knuth conjecture. The similar determinant formula # 1#i#j#k i+j1+2n i+j1 can be shown to hold for Hankel matrices whose entries are successive middle binomial coe#cients # 2m+1 m # . Generalizing the Catalan numbers in a di#erent direction, it can be shown that determinants of Hankel matrices consisting of numbers 1 3m+1 # 3m+1 m # yield an alternate expression of two Mills  Robbins  Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan  like numbers. The well  known relation of Hankel matrices to orthogonal polynomials further yields a combinatorial application of the famous Berlekamp  Massey algorithm in Coding Theory, which can be applied in order to calculate the coe#cients in the three  term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices.
On Shortest Linear Recurrences.
 J. Symbolic Computation
, 2001
"... This is an expository account of a constructive theorem on shortest linear recurrences over an arbitrary integral domain R. A generalisation of rational approximation, which we call 'realization', plays a key role throughout the paper. We also give the associated 'minimal realization& ..."
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Cited by 6 (3 self)
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This is an expository account of a constructive theorem on shortest linear recurrences over an arbitrary integral domain R. A generalisation of rational approximation, which we call 'realization', plays a key role throughout the paper. We also give the associated 'minimal realization' algorithm, which has a simple control structure and is divisionfree. It is easy to show that the number of Rmultiplications required is O(n 2 ), where n is the length of the input sequence. Our approach is algebraic and independent of any particular application. We view a linear recurring sequence as a torsion element in a natural R[X]module. The standard R[X]module of Laurent polynomials over R underlies our approach to finite sequences. The prerequisites are nominal and we use short Fibonacci sequences as running examples.
The BerlekampMassey Algorithm revisited
"... We propose a slight modification of the BerlekampMassey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation. ..."
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Cited by 4 (0 self)
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We propose a slight modification of the BerlekampMassey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation.
Efficient prediction of MarsagliaZaman random number generators
 IEEE Transactions on Information Theory
, 1993
"... Abstract—We show that the random number generator of Marsaglia and Zaman produces the successive digits of a rationaladic number. (Theadic number system generalizesadic numbers to an arbitrary integer base.) Using continued fractions, we derive an efficient prediction algorithm for this generator ..."
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Cited by 3 (0 self)
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Abstract—We show that the random number generator of Marsaglia and Zaman produces the successive digits of a rationaladic number. (Theadic number system generalizesadic numbers to an arbitrary integer base.) Using continued fractions, we derive an efficient prediction algorithm for this generator. Index Terms — Continued fractions, inductive inference,adic numbers, pseudorandom sequences.
On nDimensional Sequences. I
, 2001
"... Let R be a commutative ring and let n >= 1: We study (s), the generating function and Ann(s), the ideal of characteristic polynomials of s, an ndimensional sequence over R. We express ..."
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Cited by 2 (0 self)
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Let R be a commutative ring and let n >= 1: We study (s), the generating function and Ann(s), the ideal of characteristic polynomials of s, an ndimensional sequence over R. We express
On the Minimal Realizations of a Finite Sequence.
, 2001
"... We develop a theory of minimal realizations of a finite sequence over an integral domain R, from first principles. Our notion of a minimal realization is closely related to that of a linear recurring sequence and of a partial realization (as in Mathematical Systems Theory). From this theory, we deri ..."
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Cited by 2 (0 self)
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We develop a theory of minimal realizations of a finite sequence over an integral domain R, from first principles. Our notion of a minimal realization is closely related to that of a linear recurring sequence and of a partial realization (as in Mathematical Systems Theory). From this theory, we derive Algorithm MR which computes a minimal realization of a sequence of L elements using at most L(5L + 1)=2 R{multiplications. We also characterize all minimal realizations of a given sequence in terms of the computed minimal realization. This algorithm computes the linear complexity of an R sequence, solves nonsingular linear systems over R (extending Wiedemann's method), computes the minimal polynomial of an Rmatrix, transfer/growth functions and symbolic Padé approximations. There are also a number of applications to Coding Theory. We thus provide a common framework for solving some wellknown problems in Systems Theory, Symbolic/Algebraic Computation and Coding Theory.
Continued Fraction Expansion as Isometry: The Law of the Iterated logarithm for Linear, Jump, and 2–Adic Complexity
 IEEE Trans. Inform. Th. Preprint: arxiv.org/CS/0511089
"... Abstract — In the cryptanalysis of stream ciphers and pseudorandom sequences, the notions of linear, jump, and 2–adic complexity arise naturally to measure the (non)randomness of a given that is the precise equivastring. We define an isometry K on F ∞ q lent to Euclid’s algorithm over the reals to ..."
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Cited by 1 (1 self)
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Abstract — In the cryptanalysis of stream ciphers and pseudorandom sequences, the notions of linear, jump, and 2–adic complexity arise naturally to measure the (non)randomness of a given that is the precise equivastring. We define an isometry K on F ∞ q lent to Euclid’s algorithm over the reals to calculate the continued fraction expansion of a formal power series. The continued fraction expansion allows to deduce the linear and jump complexity profiles of the input sequence. Since K is an isometry, the resulting F ∞ q –sequence is i.i.d. for i.i.d. input. Hence the linear and jump complexity profiles may be modelled via Bernoulli experiments (for F2: coin tossing), and we can apply the very precise bounds as collected by Révész, among others the Law of the Iterated Logarithm. The second topic is the 2–adic span and complexity, as defined by Goresky and Klapper. We derive again an isometry, this time on the dyadic integers Z2 which induces an isometry A on F2 ∞. The corresponding jump complexity behaves on average exactly like coin tossing. Index terms — Formal power series, isometry, linear complexity, jump complexity, 2–adic complexity, 2–adic span, law of the iterated logarithm, Lévy classes, stream ciphers, pseudorandom sequences 1 Supported by Project FONDECYT 2001, No. 1010533 of CONICYT, Chile