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Some Intersection Theorems for Ordered Sets and Graphs
 IOURNAL OF COMBINATORIAL THEORY, SERIES A 43, 2337
, 1986
"... A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F of a ..."
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Cited by 52 (1 self)
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A classical topic in combinatorics is the study of problems of the following type: What are the maximum families F of subsets of a finite set with the property that the intersection of any two sets in the family satisfies some specified condition? Typical restrictions on the intersections F n F of any F and F ’ in F are: (i) FnF’ # 0, where all FEF have k elements (Erdos, Ko, and Rado (1961)). (ii) IFn F’I> j (Katona (1964)). In this paper, we consider the following general question: For a given family B of subsets of [n] = { 1, 2,..., n}, what is the largest family F of subsets of [n] satsifying F,F’EFFnFzB for some BE B. Of particular interest are those B for which the maximum families consist of socalled “kernel systems, ” i.e., the family of all supersets of some fixed set in B. For example, we show that the set of all (cyclic) translates of a block of consecutive integers in [n] is such a family. It turns out rather unexpectedly that many of the results we obtain here depend strongly on properties of the wellknown entropy function (from information theory).
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
Large Period Nearly deBruijn FCSR Sequences (Extended Abstract)
 In L.C. Guillou and J.J. Quisquater� editors� Advances in Cryptology � Eurocrypt �95
, 1995
"... Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2adic numbers. The sequences generated by these registers have many algebraic properties similar to those generated by linear feedback shift registers. However, it appears to be significantly more di ..."
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Cited by 9 (4 self)
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Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2adic numbers. The sequences generated by these registers have many algebraic properties similar to those generated by linear feedback shift registers. However, it appears to be significantly more difficult to find maximal period FCSR sequences. In this paper we exhibit a technique for easily finding FCSRs that generate nearly maximal period sequences. We further show that these sequence have excellent distributional properties. They are balanced, and nearly have the deBruijn property for distributions of subsequences.
Several generalizations of Weil sums
 J. Number Theory
, 1994
"... We consider several generalizations and variations of the character sum inequalities of Weil and Burgess. A number of incomplete character sum inequalities are proved while further conjectures are formulated. These inequalities are motivated by extremal graph theory with applications to problems in ..."
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Cited by 3 (0 self)
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We consider several generalizations and variations of the character sum inequalities of Weil and Burgess. A number of incomplete character sum inequalities are proved while further conjectures are formulated. These inequalities are motivated by extremal graph theory with applications to problems in computer science. 1 1.