No abstract uploaded!

No abstract uploaded!

Let$E$ç$S$^{1}be a set with Minkowski dimension$d(E)1$. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions of$E$. The main result of this paper shows that these operators are bounded on$L$_{$rad$}^{$p$}(R^{2}) for$p>1+d(E)$and unbounded when$p<1+d(E)$. We also give some end-point results.

To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions, called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies. We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected. We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification. As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.

The public key of the Oil-Vinegar scheme consists of a set of m quadratic equations in m + n variables over a finite field F_q. Kipnis and Shamir broke the balanced Oil-Vinegar scheme where d = n − m = 0 by finding equivalent keys of the cryptosytem. Later their method was extended by Kipnis et al to attack the unbalanced case where 0 < d < m and d is small with a complexity of O(q^{d−1} m^4). This method uses the matrices associated with the quadratic polynomials in the public key, which needs to be symmetric and invertible. In this paper, we give an optimized search method for Kipnis el al’s attack. Moreover, for the case that the finite field is of characteristic 2, we find the situation becomes very subtle, which, however, was totally neglected in the original work of Kipnis et al. We show that the Kipnis-Shamir method does not work if the field characteristic is 2 and d is a small odd number, and we fix the situation by proposing an alternative method and give an equivalent key recovery attack of complexity O(q^{d+1} m^4). We also prove an important experimental observation by Ding et al for the Kipnis-Shamir attack on balanced Oil-Vinegar schemes in characteristic 2.