On the number of Linear Feedback Shift Registers with a special structure Srinivasan Krishnaswamy*, H. K. Pillai Abstract— Given a linear recurring relation whose character- istic polynomial is primitive, we find out the number of possible realisations using Linear Feedback Shift Registers (LFSRs) with 2-input 2-output delay elements. We show the equivalence be- tween each realisation and a matrix having a special structure. Further, the number of realisations is computed by calculating the number of these structured matrices. Index Terms— LFSR, LRR, primitive polynomial, canonical- 1, canonical-2. I. I NTRODUCTION A sequence s 0 ,s 1 ,s 2 ,... in a finite field F q is called periodic if there exists an integer r such that s t+r = s t for all t. The smallest such r is called as the period of the sequence. For a periodic sequence we can always give a relation between the elements. Such a relation would be like s t+k = a k−1 s t+k−1 + a k−2 s t+k−2 + ··· + a0st where ai ∈ Fq (1) This is called a Linear Recurring Relation (LRR). The integer k is called the degree of the LRR. Given a periodic sequence in F q there exists a minimum degree LRR associ- ated with it. Given an LRR as in equation 1, we can associate with it a polynomial p(x)= x k − a k−1 x k−1 − a k−2 x k−2 − ···−a 0 . Given an LRR of degree k, one can associate several sequences with it. The maximum possible period of such sequences is q k − 1 (see [1]). If the polynomial associated with the LRR is a primitive polynomial over F q [x] then all the associated sequences have maximum period, q k − 1.(See [2, Theorem 6.33]). LRRs are implemented using electronic switching circuits called as Linear Feedback Shift Registers(LFSRs). These circuits contain unit delay elements, constant multipliers and adders. The gains of the multipliers are the coefficients of the LRR. For example the LFSR for the LRR s t+k = a k−1 s t+k−1 + a k−2 s t+k−2 + ··· + a 0 s t is as in Figure 1. where the D i s are delay elements LFSRs can be viewed as elementary state machines. The collection of outputs of all the delay elements taken together forms the state of this machine. Note that the state machine implementing an LRR of degree k has k delay elements and therefore the state of the system can be viewed as a vector in F k q . As in all state machines one can write a state transition matrix for this state machine which tells us what would be the next state of the machine, given the current state. This state transition matrix is a matrix in F k×k q . Note that the * Corresponding Author Srinivasan Krishnaswamy and H. K. Pillai are with the Depart- ment of Electrical Engineering, Indian Institute of Bombay, Mum- bai, India, 400076. E-mail ids: srinikris@ee.iitb.ac.in, hp@ee.iitb.ac.in characteristic polynomial of this state transition matrix is the same as the polynomial associated with the LRR that was implemented. Given a degree n, we can find LRRs such that all the as- sociated sequences have maximum period. These sequences are statistically independent and uniformly distributed. They have been shown to exhibit statistical properties of ran- domness (see [3]). As a result LFSRs find applications in cryptography (see [4]), error correcting codes (see [5]) and spread spectrum communication (see [6]). LFSRs with single input single output delay blocks output only one character at a time. This puts a restriction on the rate of communication. To overcome this restriction LFSRs with multiple input multiple output delay blocks were proposed in [7]. In the case of LFSRs with m-input m-output delay ele- ments, the feedback gains would be m × m matrices. The LRR of such an LFSR is given by s t+k = A k−1 s t+k−1 + A k−2 s t+k−2 + ··· + A 0 s t (2) where each s i ∈ F m q and each A i ∈ F m×m q . The state vector is obtained by stacking the outputs of the delay elements to get a vector in F mk q . In this paper, we look at the special case of LFSRs constructed with delay elements having two inputs and two outputs and compute the total number of such constructions with maximum period for a given primitive polynomial. II. PRELIMINARIES OF FINITE FIELDS We shall denote a finite field of cardinality q by F q , where q is a prime power. We shall denote the ring of polynomials in x with coefficients from F q by F q [x]. A finite field F q n with cardinality q n , is the smallest field that contains the roots of an irreducible polynomial of degree n in F q [x]. F q n is isomorphic to the vector space of polynomials of degree less than or equal to n − 1, over F q . Any element of F q n can identified with an n tuple of elements of F q . A vector with 1 in the i−th position and 0s in the remaining positions is denoted by e i , which can be identified with the polynomial x i−1 . The nonzero elements of F q n form a multiplicative group (F ∗ q n ). Thus, using the above identification the nonzero elements of the vector space F n q form a multiplicative group. An element of F ∗ q n whose powers generate all the elements of the multiplicative group is called as a primitive element of the group. Irreducible polynomials of degree n whose roots are primitive elements of F ∗ q n are defined as primitive polynomials over F q . We shall henceforth call a matrix, whose characteristic polynomial is a primitive polynomial, Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 • 5–9 July, 2010 • Budapest, Hungary ISBN 978-963-311-370-7 1281