Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers

Details Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers

2009-04-08

arxiv.org/abs/0904.1331v2

Designs, Codes and Cryptography, Vol. 58, No. 2 (2011), pp. 123-134

Mathematics

Combinatorics

Version 2 with some minor changes; to appear in Designs, Codes and Cryptography.

Scientific paper

10.1007/s10623-010-9387-7

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng, Han and He (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 $\sigma$-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 (1995) on the enumeration of splitting subspaces of a given dimension.

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