We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices (B1, . . . , Bm) and (C1, . . . , Cm), whether there exists an invertible matrix A such that for every i ∈ {1, . . . ,m}, ABiA = Ci. We show that this problem can be solved in randomized polynomial time over finite fields of o...

In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise description of quotients of skew polynomial rings by a left principal ideal, using results relating skew polynomial rings to Azumaya algebras. We use this des...

In this paper we develop lower bounds for the spectral radius of symmetric , skew{symmetric, and arbitrary real matrices. Our approach utilizes the well{known Leverrier{Faddeev algorithm for calculating the co-eecients of the characteristic polynomial of a matrix in conjunction with a theorem by Lucas which states that the critical points of a polynomial lie within the convex hull of its roots....

We describe an attack on the family of Diffie-Hellman and El-Gamal like cryptosystems recently presented at PQ Crypto 2010. We show that the reference hard problem is not hard. 1 Description of the Cryptosystems Skew polynomials are polynomials with a particular noncommutative inner product. Let Fq denote the finite field with q elements, and p be the characteristic of the field. Automorphisms ...

We consider a skew-symmetric n -ary bracket on the polynomial algebra K [ x 1 , … + ] ( ≥ 2 ) over field of characteristic zero defined by { } = Jac C where is fixed element and Jacobian. If then this Poisson if 3 it an -Lie-Poisson . describe center corresponding show that quotient / − λ ideal generated 0 ≠ ∈ simple central homogeneous not proper power any nonzero polynomial. This construction...