An algorithm for computing a shortest linear recurrence relation for a sequence of matrices: general - Information Theory, 1998. Proceedings. 1998 IEEE International Symposium on

The Berlekamp-Massey a lgor i thm (1968) i teratively constructs a shor tes t linear recurrence relation for a finite sequence of numbers. Here we present a system-theoretic explana t ion of t h e algor i t h m as well as an extension that cons t ruc ts a shortest linear recurrence relation for a finite sequence of matrices, r a t h e r than numbers. Originally designed for decoding BCH/Reed-Solomon codes, the Berlekamp-Mawey algorithm [l, 61 is mostly used for cryptographic purposes: as it iteratively constructs a shortest linear recurrence relation, it is ideally suited for the calculation of the linear complexity profile of a secperice of numbers. Recall that a shodest linear recurrence r d a tion for a finite sequence a i , a ~ , . . . ,UN is a pair (L,c(s) = 1 + C I S +. . . + c,:sL) with 0 5 L 5 N, siich that L is as small as possible and aj+c1aj-1+~~0j-a+...+c~aj -~ = 0 for j = L + l , . . ., N Over the past decade, a new approach to system theory has been introduced [8]-[10] which is referred to as the “behavioral” approach. Using this approach, ideas concerning the modeling of data have been developed. The BerlekmnpMassey algorithm can be interpreted a s a special instance of the modeling procedure of [lo, p. 2891-in our view this behavioral interpretation provides a clear and elegant explanation of the workings of the algorithm. The key part of the algorithm consists of a clevcr choice of a 2 x 2 polynomial update matrix at each step, see [3] for more details. In a system-t:heoretic context, tlie algorithm computes a solution to the c:lassical system-theoretic problem of “minimal partial realization” in the single-iiiput-sirigle-oiitpiit case. Using the behaviixal modeling view, the algorithm can he extended to the derivation of a shortest linear reciirrence relation for a sequence of matrices. The generalization of the algorithm to the Inulti-inputmulti-output cast: involves a nontrivial design of a (p + m) x (p + m) polynomial update matrix at each step, as detailed in [4]. In 1991 Feng ;utd Tzeng published a generalization of the Berlekamp-Massey algorithm in their paper 121. Involving “multi-sequence shift registers”, it applies to a finite sequence of 1 x m constant matrices and can be used to decode BCH codes beyond their designed error correcting capability. The algorithm of [4] that we present here, is a further extension of this generalized Elerlekamp-Massey algorithm: it applies to a finite sequence of p x rn constant matrices and coiricicles wit11 the generalized Berlekamp-Milssey algoritlim of [2] for 1) = 1. This author was swmorted bv the Aiist,ralian Research Council. The Netherlands Email j.c.willemsQrnath.rug.111 For m = 1 the algorithm computes a shortest linear recurrence relation for a sequence of p x 1 vectors and essentially coincides with the recent algorithm of Ill]. System-theoretically the iniilti-inpnt-multi-oiitpiit exterision has a natural relevance as it computes the solution of the above-mentioned minimal partial realization problem in the multi-input-multi-outpiit case. In the area of coding theory applications of the algorithm in tlie ~nulti-iriput-sirigle-oiitpiit case (I, = 1) are related to decoding of BCH codes beyond their designed error-correcting capability, see 121. Our behavioral formulation in terms of a (TIL + 1) x (Trr. + 1) polynornial matrix gives rise to conditions in terms of TIZ + 1 row degrees that are necessary and srifficient for the algorithm to produce the error locator polynomial. It is still a subject of investigation (see [5]) to determine how these conditions cornpare to the Roos bo1ind ([7]). Further applicatioris in the area of coding theory arid cryptography are u~ider irivestigatiori. R.EFEH.EN(:ES E.R. Berlekamp, Algebrraic Curling T l r ~ r y , New York, McGrawHill, 196X. G-L. Feng and K.K. Tzerrg, “A generalization of the BerlekampMassey algorithni fnr multisecpence shift,-register synt,hesis with applications to decoding cyclic codes”, IEEE T7nns. Infoniz. Theory, vol. 37, pp. 1274-1287, 1991. M. Kiiijper and .I.(;. Willeins, “On constriicting a shortest linear recurrence relation”, IEEE Tmns. Aut. Control, vol. 42, no. 11, pp. 1554-1558, 19!)7. M. Kuijper, “An algorithm for constructing a minimal partial realization in t,lre mult,ivariahle case”, Systerirs €4 Control LetM. I. 225-233, 1!3!)7. [IO] .I,<:. Willenis, “F‘aradigiiis and piizzles in the theory of dynamical syst,rms”, IEEE T7rrri.s. Ant. Gorrtrul, vnl. 36, ~ j p . 259-294, 1991. [11] ,I.(:. Willenis, “Fit,t,ing t1at.a secpencm t o linear syst,enis”, i n .Sy.st,:ri~s a r i d Coritrd 171 the Twenty-First Century (invited papers ~ , r w r u t , r r l at. t.lw 12th MTNS, St. Louis, . J u r i e 1336, e&: (:.I. Byrnw, B.N. Datta, C.F. Mart,iii arid D.S. Gilliam, BirkliHriser, Boston, pp. 405-416), 19!j7. 0-7803-5000-6/98/$10.00