The Complexity of Testing Monomials in Multivariate Polynomials

The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. The complexity aspects of this problem and its variants are investigated with two folds of objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those problems. A series of results about $\Pi\Sigma\Pi$ and $\Pi\Sigma$ polynomials are obtained in this paper, laying a basis for further study along this line

Monomial Testing and Applications

In this paper, we devise two algorithms for the problem of testing $q$-monomials of degree $k$ in any multivariate polynomial represented by a circuit, regardless of the primality of $q$. One is an $O^*(2^k)$ time randomized algorithm. The other is an $O^*(12.8^k)$ time deterministic algorithm for the same $q$-monomial testing problem but requiring the polynomials to be represented by tree-like circuits. Several applications of $q$-monomial testing are also given, including a deterministic $O^*(12.8^{mk})$ upper bound for the $m$-set $k$-packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note: substantial text overlap with arXiv:1302.5898; and text overlap with arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author

Intractability of Integration and Derivative for Multivariate Polynomial and Trigonometric Function

We study the hardness of some basic linear operators which involve high dimension integration or derivative. For a multivariate polynomial ��(��1, ⋯ , ����) which has format ∏ ∑ ��1, we show that there is no any factor polynomial time approximation for the integration of ��(��1, ⋯ , ����) in the unit cube [0,1] �� unless P = NP. In addition to polynomials, we extend the discussion to nonlinear function. For a trigonometric function ��(��1, ⋯ , ����) of format ∏ ∑ �������� ∗ , we show that it is #P-hard to compute derivative ����(��) (��1,⋯,����) ����1⋯������ at the origin point (��1, ⋯ , ����) = (0, ⋯ ,0). Consider the linear operator ��(��) = ∫ �� ���� (��1, ⋯ , ����)��−��(��1,⋯,����) ����1 ⋯ ������, we show that it is NP-hard to compute ��(��) for a ∏ ∑ �������� trigonometric function with the range �� = [0, ��]. And there is no any factor approximation to compute ��(��) for the ∏ ∑ �������� trigonometric function with the range �� = [0, ��]

Improving Key Recovery to 784 and 799 rounds of Trivium using Optimized Cube Attacks

Dinur and Shamir have described cube attacks at EUROCRYPT ’09 and they have shown how efficient they are on the stream cipher Trivium up to 767 rounds. These attacks have been extended to distinguishers but since this seminal work, no better results on the complexity of key recovery attacks on Trivium have been presented. It appears that the time complexity to compute cubes is expensive and the discovery of linear superpoly also requires the computation of many cubes. In this paper, we increase the number of attacked initialization rounds by improving the time complexity of computing cube and we show attacks that go beyond this bound. We were able to find linear superpoly up to 784 rounds, which leads to an attack requiring $2^{39}$ queries. Using quadratic superpoly, we were also able to provide another attack up to 799 rounds which complexity is $2^{40}$ queries and $2^{62}$ for the exhaustive search part. To achieve such results, we find a way to reduce the density of the polynomials, we look for quadratic relations and we extensively use the Moebius transform to speed up computations for various purposes