An Explicit Similarity Transform between Cellular Automata and LFSR Matrices*

Transcription

1 FINITE FIELDS AND THEIR APPLICATIONS 4, (1998) ARTICLE NO FF An Explicit Similarity Transform between Cellular Automata and LFSR Matrices* Kevin Cattell and Jon C Muzio Department of Computer Science, University of Victoria, Victoria British Columbia, Canada V8W 3P6 kcattell@csrcscuvicca Communicated by Peter Jau-Shyong Shiue Received August 30, 1996; revised February 9, 1998 This paper demonstrates a similarity transform between the tridiagonal matrices of one-dimensional linear hybrid cellular automata and the companion matrices of linear feedback shift registers Such a transform is of interest to the VLSI design community, as it provides an explicit mapping between the states of these two linear finite state machines 1998 Academic Press A one-dimensional linear hybrid cellular automata (CA) is a linear finite state machine used in VLSI for test pattern generation and signature analysis As well as their practical applications, these machines have proved to have fascinating theoretical properties, one of which is the relationship between a CA and its characteristic polynomial One facet of this relationship is the similarity transform between a CA and its corresponding linear feedback shift register (LFSR) Given a CA, it is easy to find its (unique) characteristic polynomial The reverse problem of finding a CA for a given polynomial was open for several years [1, 7], and was solved (for GF(2) only) independently for irreducible polynomials by [2, 9] Both authors give a proof of existence, a proof of uniqueness (up to rule reversal), and an algorithm (the algorithms are quite different but have the same order of complexity) The transform presented here does not solve any of the above problems, as it relies on knowing the CA in advance Rather, given the above results, it * This work was supported in part by Research Grants and Postgraduate Scholarships from the Natural Sciences and Engineering Research Council of Canada /98 $2500 Copyright 1998 by Academic Press All rights of reproduction in any form reserved

2 240 CATTELL AND MUZIO shows that the similarity transform between a CA and its corresponding LFSR (the latter being in trivial correspondence with the characteristic polynomial) has an interesting structure Also, a CA and LFSR with the same characteristic polynomial are different realisations of the same linear operator, and the transform provides a direct mapping between the labels of the states for these two realisations The first section introduces the necessary background material on CA and similarity transforms, and the second demonstrates the transforms For further details on finite fields, see [5, 6]; for the background on linear finite state machines, see [8]; for the CA background see [3, 4, 10] 1 BACKGROUND A null-boundary linear hybrid cellular automata is a linear finite state machine (LFSM), composed of a one-dimensional array of n cells Each cell consists of a single memory element capable of storing an element of GF(q), and a next-state computation function Communication between cells is nearest-neighbour, meaning that each cell is connected to only its left and right neighbours Figure 1 shows the interconnection structure of a CA In an LFSM, time evolves in discrete steps At each time step t, each cell i has a state s, an element of GF(q) For time step t#1, each cell i computes its new state s, using its cell rule f For a CA, the requirements of linearity and irreducibility imply that each cell uses a rule of the form s"c s #d s#b s, with c and b all nonzero The leftmost and rightmost cells behave as though their left and right neighbours, respectively, are always in state 0, making the CA null-boundary (that is, s "s "0 for all t) The nearest-neighbour communication to which a CA is restricted has the consequence that the transition matrix A is tridiagonal: d b c d b 0 0 c d 0 d b c d

3 SIMILARITY BETWEEN CA AND LFSRs 241 FIG 1 Null-boundary CA interconnection structure The characteristic polynomial of CA (in general, of any LFSM) is defined by its transition matrix as A!xI, where ) is the determinant The characteristic polynomial of a CA can be calculated efficiently from the rule vector using the recurrence relation: "0 "1 "(x!d )!b c, k*1 The characteristic polynomial, denoted, is the final term of the recurrence, EXAMPLE 1 Consider the 5-cell CA with [d,2, d ]"[1, 1, 1, 1, 0] (note that in GF(2), b "c "1 for all i) The transition matrix for this CA is A" The calculation of the via the recurrence is "0 "1 "x#1 "x "x#x#x#1 "x#x#1 "x#x#1

4 242 CATTELL AND MUZIO FIG 2 Five-cell LFSR interconnection structure An LFSR (type-1) is also a form of LFSM The interconnection structure of 5-cell LFSRs is shown in Fig 2 A multiplier p, being nonzero, denotes the presence of a feedback tap in front of cell i (the tap multiplies its input by!p ) An LFSR, its characteristic polynomial, and its transition matrix have a simple relationship: the feedback taps are p, p,2, p, the characteristic polynomial is x#p x#2#p x#p, and the!p form the last column of the transition matrix C: !p C" 1 0 0!p !p 0 1 0!p !p A matrix in this form is called a companion form matrix For example, the LFSR (over GF(2)) with transition matrix C" has characteristic polynomial x#x#1 (the same as the CA above) and is shown in Fig 3 The transition matrix ¹ of an LFSM is a representation or realisation of a linear operator Since ¹ has elements from GF(q), is a linear operator on the vector space (GF(q)) A linear operator is uniquely determined by

5 SIMILARITY BETWEEN CA AND LFSRs 243 FIG 3 A 5-cell LFSR its minimal polynomial, which is defined to be the minimal polynomial of any matrix representing Hence two matrices ¹ and ¹ represent the same linear operator if and only if they have the same minimal polynomial This property is called similarity The minimal polynomial of a matrix divides the characteristic polynomial, and, thus, if the characteristic polynomial is irreducible, it equals the minimal polynomial In this case, two matrices are similar if and only if they have the same characteristic polynomial This allows us to work with the characteristic polynomial in determining similarity Alternatively, similarity can be defined as follows DEFINITION 1 Two matrices ¹ and ¹ are similar if and only if there exists a nonsingular matrix P such that P¹P"¹ The matrix P in Definition 1 is called a similarity transform from ¹ to ¹ The similarity transform provides a mapping between the states of LFSMs with transition matrices ¹ and ¹ that preserves the next-state function: s"¹s s"p¹ps Ps"¹Ps That is, if s is the successor of s under ¹, then Ps is the successor of Ps under ¹ Let M be an n-cell CA with transition matrix A and irreducible characteristic polynomial, and let be the linear operator represented by A There are two natural matrix representations of : the companion matrix of (or of ), and the diagonal matrix of (or of ) The companion matrix of is the transition matrix of the LFSR that has characteristic polynomial The companion matrix of a polynomial always exists, since it is formed with negatives of the coefficients of the degree 0 through n!1 terms in the last column of the matrix, and 1 s in the subdiagonal

6 244 CATTELL AND MUZIO To analyse the diagonal matrix of, must be considered as a linear operator on the vector space (GF(q)) Importantly, is uniquely and well defined on (GF(q)) by its definition on (GF(q)) (see [6]) DEFINITION 2 A linear operator is diagonalisable if it can be represented by a diagonal matrix; that is, if the matrix A represents, then A is similar to a diagonal matrix A linear operator is diagonalisable if and only if the minimal polynomial of can be written as a product of distinct linear factors If is diagonalisable, then the entries on the diagonal are the roots of the minimal polynomial of The following classic result is used to show the invertibility of the similarity transform matrices in the following section THEOREM 1 [6, Corollary 238] ¹he elements α, α,2, α in GF(q) form a basis of GF(q) over GF(q) if and only if α α 2 α α α 2 α O0 α α 2 α 2 THE SIMILARITY TRANSFORM The transform is derived as the composition of three transforms The first is a transform between the CA form and a tridiagonal unit-upper-diagonal (tri-uud) form; the second is a transform between the tri-uud form and diagonal form; the third is a transform between diagonal form and companion form Consider now the characteristic polynomial of the CA M As is assumed to be irreducible, it has a root α in GF(q) It follows that all n roots of lie in GF(q), the roots are distinct and are given by α, α, α,2, α This means that can be factored in GF(q) as (x!α)(x!α)(x!α )2(x!α ), a product of nonrepeated linear factors Hence if is irreducible over GF(q), then is diagonalisable over GF(q)

7 SIMILARITY BETWEEN CA AND LFSRs 245 The diagonal form of has the roots of the characteristic polynomial on the main diagonal As these roots can appear in any order, has multiple diagonal forms (n!, as the roots are distinct) Without loss of generality, the following diagonal form is used: D"α α α α In the computation of the characteristic polynomial of A, a sequence of polynomials 1,,,2,, " is formed Each of these polynomials is mapped to an element of GF(q) by the usual process of evaluation at the root of an irreducible polynomial in GF(q)[x] The root α of is used, and the result of this mapping is denoted m : m " (α), i"0,2, n (evaluation) Note that m " (α)"1, and m " (α)"0 Initially, we transform the CA matrix to a matrix in tri-uud form where P AP "A, A" d b c d b c d 0 d b c d (1) It is easily verified that the required transform is the invertible diagonal matrix

8 246 CATTELL AND MUZIO 1 P " b (b b ) (b b 2b ) (recall that b, c O0 for all i) The second transform is between tri-uud form and diagonal form with P given by P DP "A, P " m m m 2 m m m m 2 m m m m 2 m m m m 2 m We now show that P is the required transform LEMMA 1 P is nonsingular Proof By Theorem 1, the determinant P is not zero if and only if 1, m, m,2, m is a basis of GF(q) over GF(q) To show that the elements 1, m, m,2, m form such a basis, it suffices to show that no nontrivial linear combination of the m is zero Since m is the image of a degree i polynomial (under the mapping evaluation at α ), m has the form α#c α#2#c α#c, where the c,0)j(i, are the coefficients of the polynomial (note that the are monic) Hence, a linear combination z #z m #z m #2#z m (2) has a term α, where i is the greatest index such that z O0 Therefore (2) is not zero, so long as not all z are zero

9 SIMILARITY BETWEEN CA AND LFSRs 247 THEOREM 2 P is a similarity transform between A and D ¹hat is, P DP "A Proof It is shown that P D"AP and, since P is nonsingular, the theorem follows The product of the ith row of P and the jth column of D is (P D) "[m, m,2, m ] ) [0, 0, 2, α,2,0] "α m "(αm ) The product of the ith row of A (io1, n) and the jth column of P is (AP ) "[0, 0,2, b c, d,1,2,0])[1, m, 2, m ] "b c ) m #d ) m #1 ) m "(b c m #d ) m #m ) (3) As the CA polynomials satisfy the recurrence "(x!d )!b c, 1)i)n, in GF(q), m "(α!d )m!b c m "αm!d m!b c m Thus, (3) is (AP ) "(αm ), 2)i)n!1, 1)j)n The boundary cases for i"1 and i"n are handled separately For i"1 and using m "(α!d )m and m "1, (AP ) "[d,1,0,2,0])[1, m, 2, m ] "d #m "(d #m ), "(d #α!d ), "(α), "(αm )

10 248 CATTELL AND MUZIO For i"n and using the fact that m is 0 ( evaluated at its own root), Hence, (AP ) "[0,2,0,b c, d ] ) [1, m, 2, m ] "b c m #d m "(b c m #d m ), "(b c m #d m #m ), "(αm ) (AP ) "(αm ) "(P D) and P D"AP The third transform is between the diagonal form and companion form,!p P 1!p 1!p 1!p P "D, where P is given by P " α α α 2 α α (α) (α) 2 (α) α (α) (α) 2 (α) α (α) (α) 2 (α) It is straightforward to verify that P is the required transform Composing the transforms, A"P P P CP P P "(P P P )C(P P P )

11 SIMILARITY BETWEEN CA AND LFSRs 249 The entries of the composed transform P P P are given by (P P P ) " b Tr(m α) EXAMPLE 2 Consider the 5-cell CA with rule vector [1, 1, 1, 1, 0] over GF(2) The characteristic polynomial of this CA is "x#x#1, which is irreducible The calculation of the characteristic polynomial yields the polynomials,,2, which evaluated at α (a root of ) are, m "1 m "α#1 m "α "α "α m "α#α#α#1 "α m "α#α#1 "α m "α#α#1 "0 Note that over GF(2), P is the identity matrix The product P DP is α α α α α α α α α α α α α α α α α α α α ) α α α α α ) α α α α α α α α α α α α α α α α α α α α α α α α α " , which is the transition matrix A of the CA The inverse matrix P can be obtained from P by using elementary row operations (ie, transforming [P I]P[IP]) Obtaining the companion form C for, the product P CP is

12 250 CATTELL AND MUZIO α α α α α α α α α α α α α α α ) α α α α α α α α α α α α α α α α α α α α α 0 α ) α α α α α " 0 0 α 0 0, α α α α α α 0 α α α α α α which is the diagonal matrix D of 3 CONCLUSION This paper demonstrates a general form for similarity transforms between a CA transition matrix and companion form for a CA with an irreducible characteristic polynomial The transform is the composition of three transforms, using a tridiagonal matrix with upper diagonal all 1, and a diagonal matrix as intermediate forms These transforms provide explicit mappings between the states of a CA and an LFSR that are similar REFERENCES 1 P H Bardell, Analysis of cellular automata used as pseudorandom pattern generators, in Proceedings of IEEE International Test Conference, 1990, pp K Cattell and J C Muzio, Synthesis of one-dimensional linear hybrid cellular automata, IEEE ¹rans Computer-Aided Design 15, No 3 (1996), A K Das, A Ganguly, A Dasgupta, S Bhawmik, and P P Chaudhuri, Efficient characterisation of cellular automata, IEE Proc E Comput Digital ¹ech 137 (1990), P D Hortensius, R D McLeod, and H C Card, Cellular automata-based signature analysis for built-in self-test, IEEE ¹rans Comput 39 (1990), D Jungnickel, Finite Fields: Structures and Arithmetics, B I Wissenschaftsverlag, Mannheim, R Lidl and H Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ Press, Cambridge, 1986

13 SIMILARITY BETWEEN CA AND LFSRs M Serra, T Slater, J C Muzio, and D M Miller, The analysis of one dimensional linear cellular automata and their aliasing properties, IEEE ¹rans Computer-Aided Design 9 (1990), H S Stone, Discrete Mathematical Structures and Their Applications, Sci Res Assoc, Chicago, T Shu and M Fushimi, A method of designing cellular automata as pseudo-random number generators for built-in self-test for VLSI, Contemp Math 168 (1994), S Wolfram, Random sequence generation by cellular automata, Adv in Appl Math 7 (1986),