Inversion of two new circulant matrices over Zm

In this paper we consider the problem of inverting an n×n RSFMLR circulant matrix with entries over Zm. We present two different algorithms. Our algorithms require different degrees of knowledge of m and n, and their costs range, from n log n loglog n to n log2 n loglog n log m operations over Zm. Moreover, for each algorithm we give the cost in terms of bit operations. Finally, the extended algorithms is used to solve the problem of inverting RSFMLR circulant matrices over Zm.


1
  x x n -circulant matrices, which are called RSFMLR circulant matrices, are better than those of the general ) (x f -circulant matrices, so there are good algorithms for finding the inverse of the RSFMLR circulant matrices. In this paper we consider the problem of inverting RSFMLR circulant with entries over the ring m Z .
In this paper we describe two algorithms for inverting an n n  RSFMLR circulant matrix over  Obviously, the RSFMLR circulant matrix over [9], and that is neither the extension of circulant matrix over m Z [3] nor its special case and they are two different families of patterned matrices.
We define It is easily verified that   both the minimal polynomial and the characteristic polynomial of the matrix . In addition to the algebraic properties that can be easily derived from the representation (3), we mention that RSFMLR circulant matrices have very nice structure. The product of two RSFMLR circulant matrices is a RSFMLR circulant matrix and 1 is a commutative ring with the matrix addition and multiplication.
The congruence modulo . Hence, the problem of inverting a RSFMLR circulant matrix is equivalent to inversion in the ring The following theorem states a necessary and sufficient condition for the invertibility of a RSFMLR circulant matrix over m Z .

Theorem 1 Let
as claimed. The proof that the above condition is sufficient for invertibility is constructive and will be given in Section 2 (Lemmas 2 and 3). Review of bit complexity results [3]. In the following we will give the cost of each algorithm in terms of number of bit operations. In our analysis we use the following well-known results (see for example [15] or [16] [16], Theorem 1.7.1). Therefore, the asymptotic cost of polynomial multiplication is Given two polynomials The same algorithm also returns . The bound (7) follows by a straightforward modification of the polynomial gcd algorithm described in [15] (Section 8.9: the term p p n log log ) (log  comes from the fact that we must compute the inverse of

Factorization of m Known
In this section we consider the problem of computing the inverse of a RSFMLR circulant matrix over , and we show that we can compute the inverse by combining known techniques (Chinese remaindering, the extended Euclidean algorithm, and Newton-Hensel lifting). We start by showing that it suffices to find the inverse of Proof The proof is constructive. Since or, equivalently,  (7) and (6) respectively.
known as Newton-Hensel lifting. It is straightforward to verify by induction that using Newton-Hensel lifting (Lemma 3); 6. else 7.

Algorithm 1 Inversion in
. Factorization of m known.

A General Inversion Algorithm in
The algorithm described in Section 2 relies on the fact that the factorization of the modulus m is known. If this is not the case and the factorization must be computed beforehand, the increase in the running time may be significant since the fastest known factorization algorithms require time . Note that we must force the gcd algorithm to return a monic polynomial.
If the computation of The computation of the factors 2 1 , m m is done by procedure GetFactors whose correctness is proven by Lemmas 4 and 5. Combining these procedures together we get Algorithm 2. __________________________________________________________________________________ Factorization of m unknown. The following Lemma 4 and Lemma 5 proved in [3]. Proof One can easily prove the correctness of the algorithm by induction on m , the base on the induction being the case in which m is prime where the inverse is computed by the gcd algorithm.
To prove the bound on the number of bit operations we first consider the cost of the single steps. By (7) we know that computing In addition, by Lemma 1 (i) (ii), Algorithm 1 and Algorithm 2, it is easily to get two algorithms for inverting RSLMFL circulant matrices over m Z , respectively.

Conclusion
In this paper, the problem of inverting an n n  RSFMLR circulant matrix with entries over m Z is studied. Two different algorithms are presented. Furthermore, for each algorithm the cost in terms of bit operation are given. Finally, the extended algorithms is used to solve the problem of inverting RSFMLR circulant matrices over m Z .