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

Abstract

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.

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

Cited by in corpus (7)

• The Splitting Subspace Conjecture
• Block Companion Singer Cycles, Primitive Recursive Vector Sequences, and Coprime Polynomial Pairs over Finite Fields
• A note on the multiple-recursive matrix method for generating pseudorandom vectors
• An asymptotic formula for the number of irreducible transformation shift registers
• Enumeration of Linear Transformation Shift Registers
• A group law on the projective plane with applications in Public Key Cryptography
• Primitive transformation shift registers over finite fields