Enumeration of linear transformation shift registers

Abstract

Abstract. We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field. IntroductionA linear feedback shift register (LFSR) is a mechanism for generating a sequence in a finite field. LFSRs have a plethora of practical applications and are frequently used in generating pseudorandom numbers, fast digital counters and stream ciphers. A generalization of LFSR called word-oriented feedback shift register (σ-LFSR) was considered by Zeng, Han and He [20]. For LFSRs as well as σ-LFSRs, those that are primitive (i.e., for which the corresponding infinite sequence is of maximal possible period) are of particular interest. The following conjecture was proposed in the binary case in [20] and was extended to the q-ary case in [8]:Conjecture 1.1. For positive integers m and n, the number of primitive σ-LFSRs of order n over F q m is given byThe notion of σ-LFSR is essentially equivalent to that of a splitting subspace previously defined by Niederreiter [16]: Given positive integers m, n and α ∈ F q mn , an m-dimensional F q -linear subspace of F q mn is said to be α-splitting if

An asymptotic formula for the number of irreducible transformation shift registers

“…It follows from (9) and [18,Theorem 3] that f (X) ∈ Ψ I (TSRI(m, n, q)) if and only if f (X) is irreducible and can be uniquely expressed in the form (10) g(X) m h X n g(X)…”

“…In view of Theorem 3.2, it is sufficient to enumerate the polynomials in the set Ψ I (TSRI(m, n, q)) to find the number of irreducible TSRs. In fact, Ram [18] enumerates TSRs of order two. Moreover, he re-derives a theorem of Carlitz [3] about the number of self reciprocal irreducible monic polynomials of a given degree over a finite field.…”

“…As it has been mentioned earlier, Ram [18] gives a formula for the number of irreducible TSRs of order two. In Section 3 we give a short proof of Ram's result using a variant of a theorem of Carlitz recently proved [1].…”

“…We refer to [10] and [11] for recent progress on this conjecture and to [4] for a proof of this conjecture. It is also known from [11] and [4], see also [18], that the number of irreducible σ-LFSRs is (4) 1 mn q m(m−1)(n−1)…”

“…Motivated by this fact and in an attempt to obtain a nice formula like (2) and (4), we consider here the problem of enumerating irreducible TSRs. In fact, this problem was first considered in [18] where the author gives a formula for the number of irreducible TSRs of order two. Moreover, in [18], as a consequence of this result, a new proof of a theorem of Carlitz about the number of the self reciprocal irreducible monic polynomials of a given degree over a finite field is deduced.…”