We consider the solution of «-by-« Toeplitz systems T„x = b by preconditioned conjugate gradient methods. The preconditioner Cn is the T. Chan circulant preconditioner, which is defined to be the circulant matrix that minimizes \\B„ T„\\f over all circulant matrices B„ . For Toeplitz matrices generated by positive In -periodic continuous functions, we have shown earlier that the spectrum of the...

This paper proposes a new, large diffusion layer for the AES block cipher. This new layer replaces the ShiftRows and MixColumns operations by a new involutory matrix in every round. The objective is to provide complete diffusion in a single round, thus sharply improving the overall cipher security. Moreover, the new matrix elements have low Hamming-weight in order to provide equally good perfor...

The goal of this paper is two-fold. We first focus on the problem of deciding whether two monomial rotation symmetric (MRS) Boolean functions are affine equivalent via a permutation. Using a correspondence between such functions and circulant matrices, we give a simple necessary and sufficient condition. We connect this problem with the well known Ádám’s conjecture from graph theory. As applica...

MICHAEL KIERMAIER, ALFRED WASSERMANN michael.kiermaier, [email protected] Mathematical Department, University of Bayreuth, D-95440 Bayreuth, GERMANY Abstract. In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from α-circulant matrices. For a non-trivial ideal I < R we give a method to lift such codes over R/I to codes over R, ...

We obtain a few structural theorems for circulant weighing matrices whose weight is the square of a prime number. Our results provide new schemes to search for these objects. We also establish the existence status of several previously open cases of circulant weighing matrices. More specifically we show their nonexistence for the parameter pairs (n, k) (here n is the order of the matrix and k i...

In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $\delta$, $m\sim \delta^{-4} s\log(N/s\delta)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $\delta$ via an efficient program. We derive this result by proving that...

The recursive algorithm of a (fast) discrete wavelet transform, as well as its generalizations, can be described as repeated applications of block-Toeplitz operators or, in the case of periodized wavelets, multiplications by block circulant matrices. Singular values of a block circulant matrix are the singular values of some matrix trigonometric series evaluated at certain points. The norm of a...

In this paper, A-factor circulant matrices with the structure of a circulant, but with the entries below the diagonal multiplied by the same factor A are introduced. Then the generalized rotation and hyperbolic matrices are defined, using an idea due to Ungar. Considering the exponential property of the generalized rotation and hyperbolic matrices, additive formulae for corresponding matrices a...

In this paper we will give a survey on iterative methods for solving linear equations with Toeplitz matrices. We introduce a new class of Toeplitz matrices for which clustering of eigenvalues and singular values can be proved. We consider optimal (ω)circulant preconditioners as a generalization of the circulant preconditioner. For positive definite Toeplitz matrices, especially in the real case...

It is well-known that for each prime power q and for each d ∈ 2N, there exists a circulant weighing matrix of order q d+1−1 q−1 and weight q . We extend this result to show that there exist φ(d+1) 2 inequivalent circulant weighing matrices of order q d+1−1 q−1 and weight q , where φ is the Euler totient function. Further, we obtain a bound on the magnitude of the values taken by the cross-corre...