Vol. 85, No. 1, 1979

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ISSN: 0030-8730
Symmetric shift registers

Jan Søreng

Vol. 85 (1979), No. 1, 201–229
Abstract

We will study symmetric shift registers over the field GF(2) = {0,1}. The symmetric shift register 𝜃s : {0,1}n →{0,1}n corresponding to a symmetric polynomial S(x2,,xn) is defined by

𝜃S(a1,⋅⋅⋅ ,an) = (a2,⋅⋅⋅ ,an+1) where an+1 = a1 + S (a2,⋅⋅⋅ ,an).

p is a period of A ∈{0,1}n with respect to 𝜃s if 𝜃sP(A) = A. If p is the least period of A, then A 𝜃s(A) 𝜃SP(A) = A is the cycle corresponding to A. This is the first of two papers where we will determine in a constructive way (for each S):

  1. The minimal period for each A ∈{0,1}n.
  2. The possible minimal periods.
  3. The number of cycles corresponding to each minimal period.

Mathematical Subject Classification
Primary: 68D15, 68D15
Milestones
Received: 14 December 1977
Revised: 19 September 1978
Published: 1 November 1979
Authors
Jan Søreng