5,895 research outputs found
Roots of bivariate polynomial systems via determinantal representations
We give two determinantal representations for a bivariate polynomial. They
may be used to compute the zeros of a system of two of these polynomials via
the eigenvalues of a two-parameter eigenvalue problem. The first determinantal
representation is suitable for polynomials with scalar or matrix coefficients,
and consists of matrices with asymptotic order , where is the degree
of the polynomial. The second representation is useful for scalar polynomials
and has asymptotic order . The resulting method to compute the roots of
a system of two bivariate polynomials is competitive with some existing methods
for polynomials up to degree 10, as well as for polynomials with a small number
of terms.Comment: 22 pages, 9 figure
Quantum Algorithms for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems
Decision of whether a Boolean equation system has a solution is an NPC
problem and finding a solution is NP hard. In this paper, we present a quantum
algorithm to decide whether a Boolean equation system FS has a solution and
compute one if FS does have solutions with any given success probability. The
runtime complexity of the algorithm is polynomial in the size of FS and the
condition number of FS. As a consequence, we give a polynomial-time quantum
algorithm for solving Boolean equation systems if their condition numbers are
small, say polynomial in the size of FS. We apply our quantum algorithm for
solving Boolean equations to the cryptanalysis of several important
cryptosystems: the stream cipher Trivum, the block cipher AES, the hash
function SHA-3/Keccak, and the multivariate public key cryptosystems, and show
that they are secure under quantum algebraic attack only if the condition
numbers of the corresponding equation systems are large. This leads to a new
criterion for designing cryptosystems that can against the attack of quantum
computers: their corresponding equation systems must have large condition
numbers
A review of Costas arrays
Costas arrays are not only useful in radar engineering, but they
also present many interesting, and still open, mathematical
problems. This work collects in it all important knowledge about
them available today: some history of the subjects, density
results, construction methods, construction algorithms with full
proofs, and open questions. At the same time all the necessary
mathematical background is offered in the simplest possible format
and terms, so that this work can play the role of a reference for
mathematicians and mathematically inclined engineers interested in the field
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