Analysis of the Berlekamp-Massey Linear Feedback Shift-Register Synthesis Algorithm

An analysis of the Berlekamp-Massey Linear Feedback Shift-Register (LFSR) Synthesis Algorithm is provided which shows that an input string of length 12 requires C ( i 2 ) multiplication/ addition operations in the underlying field of definition. We also derive the length distribution for digit strings of length n. Results show that, on the average, the encoded length is no greater than n + l . Furthermore, we exhibit a connection between step 1 of the Ling-Palermo algorithm and the LFSR Algorithm, and the LFSR Algorithm turns out to be computationally superior. Introduction The Block-Oriented Information Compression (BOIC) scheme of Ling and Palermo [ 11 is isomorphic to the Linear Feedback Shift-Register (LFSR) Synthesis Algorithm proposed by Massey [2] when one deals with bit strings. The LFSR algorithm, which is essentially the same as the Berlekamp Iterative Algorithm [ 31, was developed for encoding and is more general and computationally faster than the BOIC scheme. This work contains analysis of the LFSR scheme; in particular, we derive the distribution D ( n ; 1) of lengths 1 needed to generate strings of length n. In the binary case, D ( n ; 1) appears when the binary symmetric source is assumed. The distribution D ( n ; 1) can be used to compute the expected value E(n) of a random bit string. It turns out that n i E(n) 5 n + 1 : hence one gets, on the average, data expansion with this method. [The length of n (log,n bits) is not included in this estimate.] A fundamental result in information theory [4, p. 43, Theorem 3.1 I ] states that any encoding must lead, on the average, to data expansion; the result here is quite good in view of that fact. The following section of this paper can be considered a supplement to [2] and [3, Chapter 71. In it we present Massey's algorithm and indicate which steps contribute to the computation cost. We then prove what the minimum, average, and maximum computation costs are in terms of the numbers of multiplications and additions. To first order, these three values are 0, in' (2 p ' ) , and i n z when computations are done in a field with p elements. We determine that the average number of polynomial evaluations is n( 2 p") . This section also contains a derivation of formulas for D ( n ; 1) and E(n) . The next section contains a comparison of step I of 204 the BOIC scheme and the LFSR Algorithm. We show there that steps 2, 3, and 4 of the BOIC scheme compression stage are redundant; i.e., the result they seek is contained in step 1. Then we show that step 1 requires more work than the LFSR Algorithm; step 1 turns out to be C(n") for most sequences. We use the notation of [ I ] and refer to specific equations there. We do not give a parallel evaluation ofthe LFSR Algorithm and the BOlC scheme. However, assuming that we can count n bit operations as one operation, we find that the BOIC scheme can be done in O(n2) operations whereas the LFSR Algorithm requires no more than a( n log,n) operations. Description and specification of the LFSR Algorithm The problem: Given n elements sl, sZ; . ., s, in a finite or an infinite field, find 1 constants c , , c2,. . ., cl in the same field such that