An algorithm for computing minimal bidirectional linear recurrence relations

Sălăgean, Ana

doi:10.1109/isit.2008.4595287

Cited by 8 publications

(20 citation statements)

References 3 publications

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“…This then implies some additional results on bidirectional shortest feedback shift registers. For the field case these results imply findings in [25].…”

Section: Discussionsupporting

confidence: 74%

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An iterative algorithm for parametrization of shortest length linear shift registers over finite chain rings

Kuijper

Pinto

2016

Des. Codes Cryptogr.

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The construction of shortest feedback shift registers for a finite sequence S1, . . . , SN is considered over the finite ring Zpr . A novel algorithm is presented that yields a parametrization of all shortest feedback shift registers for the sequence of numbers S1, . . . , SN , thus solving an open problem in the literature. The algorithm iteratively processes each number, starting with S1, and constructs at each step a particular type of minimal Gröbner basis. The construction involves a simple update rule at each step which leads to computational efficiency. It is shown that the algorithm simultaneously computes a similar parametrization for the reciprocal sequence SN , . . . , S1. * M. Kuijper is with the

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“…This then implies some additional results on bidirectional shortest feedback shift registers. For the field case these results imply findings in [25].…”

Section: Discussionsupporting

confidence: 74%

“…, S 1 . This connection with the reciprocal sequence leads us to results on bidirectionality which is relevant for cryptographic applications, see [25].…”

Section: Introductionmentioning

confidence: 93%

An iterative algorithm for parametrization of shortest length linear shift registers over finite chain rings

Kuijper

Pinto

2016

Des. Codes Cryptogr.

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“…is referred to as the characteristic polynomial of (2). We will say that the sequence s satisfies the recursion f , or that s is an f -sequence, if (2) holds for all n ∈ Z, and we will write L F (f ) to denote the collection of all f -sequences s over F. We say that a sequence satisfies a recursion f where f [17,18,20].…”

Section: Preliminariesmentioning

confidence: 99%

Some basic results on finite linear recurring sequence subgroups

Hollmann¹,

Zhanbulatuly²

2021

Preprint

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An f -subgroup is a linear recurring sequence subgroup, a multiplicative subgroup of a field whose elements can be generated (without repetition) by a linear recurrence relation, with characteristic polynomial f . It is called non-standard if it can be generated in a non-cyclic way (that is, not in the order α i , α i+1 , α i+2 . . . for a zero α of f ), and standard otherwise. We will show that a finite f -subgroup is necessarily generated by a subset of the zeros of f . We use this result to improve on a recent theorem of Brison and Nogueira. A old question by Brison and Nogueira asks if there exist automatically non-standard f -subgroups, f -subgroups that cannot be generated by a zero of f . We answer that question affirmatively by constructing infinitely many examples.

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“…Algorithm 2.2 of [4] also computes a minimal polynomial of s. We show that these two algorithms are closely related: a straightforward rewrite of the former using the notation of [4] yields the latter, except that we initialise a polynomial to 0 instead of 1. Further, the rewrite uses fewer variables and is simpler.…”

Section: Introductionmentioning

confidence: 98%