Primitive polynomials, singer cycles and word-oriented linear feedback shift registers

Authors:
Sudhir R. Ghorpade;Sartaj Ul Hasan;Meena Kumari
Affiliations:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India 400076;Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India 400076 and Scientific Analysis Group, Defense Research and Development Organisation, Delhi, India 110054;Scientific Analysis Group, Defense Research and Development Organisation, Delhi, India 110054
Venue:
Designs, Codes and Cryptography
Year:
2011

Citing 6
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Abstract

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng et al. (Word-Oriented Feedback Shift Register: 驴-LFSR, 2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This conjecture is about the number of primitive 驴-LFSRs of a given order over a finite field, and it generalizes a known formula for the number of primitive LFSRs, which, in turn, is the number of primitive polynomials of a given degree over a finite field. Moreover, this conjecture is intimately related to an open question of Niederreiter (Finite Fields Appl 1:3---30, 1995) on the enumeration of splitting subspaces of a given dimension.