We estimate the norms of standard Gaussian random Toeplitz and circulant matrices and their inverses, mostly by means of combining some basic techniques of linear algebra. In the case of circulant matrices we obtain sharp probabilistic estimates, which show that these matrices are expected to be very well conditioned. Our probabilistic estimates for the norms of standard Gaussian random Toeplit...

In this paper we study extensively the discrete logarithm problem in the group of non-singular circulant matrices. The emphasis of this study was to find the exact parameters for the group of circulant matrices for a secure implementation. We tabulate these parameters. We also compare the discrete logarithm problem in the group of circulant matrices with the discrete logarithm problem in finite...

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skewcirculant and g-circulant matrices. The entries are product of binomial coefficients with Fibonacci numbers and Lucas numbers, respectively. We obtain identity estimations for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random circulant matrix is shown to be complex normal, and bounds are given for the probability that a circulant sign matrix is singular.

An important aspect of cryptography with matrices is that given N ×N matrix P over a field F find a class of matrices G over F such that the associated doubly circulant matrix Gc is singular in order that the equation AGB = P in circulant matrices A,B has infinitely many solutions. The aim of this note is to present such a class of matrices G. We also present a direct method of finding the inve...

We combine some basic techniques of linear algebra with some expressions for Toeplitz and circulant matrices and the properties of Gaussian random matrices to estimate the norms of Gaussian Toeplitz and circulant random matrices and their inverses. In the case of circulant matrices we obtain sharp probabilistic estimates, which show that the matrices are expected to be very well conditioned. Ou...

This paper presents a new type of circulant matrices. We call it the first and the last difference r-circulant matrix (FLDcircr matrix). We can verify that the linear operation, the matrix product and the inverse matrix of this type of matrices are still FLDcircr matrices. By constructing the basic FLDcircr matrix, we give the discriminance for FLDcircr matrices and the fast algorithm of the in...