We present a new solution to the problem of determining the path a packet traversed over the Internet (called the traceback problem) during a denial of service attack. This paper reframes the traceback problem as a polynomial reconstruction problem and uses algebraic techniques from coding theory and learning theory to provide robust methods of transmission and reconstruction. 1

The method of assigning labels to the nodes of the XML tree is called a labeling scheme. Based on the labels only, both ordered and un-ordered queries can be processed without accessing the original XML file. One more important point for the labeling scheme is the label update cost in inserting or deleting a node into or from the XML tree. All the current labeling schemes have high update cost, therefore in this paper we propose a novel quaternary encoding approach for the labeling schemes. Based on this encoding approach, we need not re-label any existing nodes when the update is performed. Extensive experimental results on the XML datasets illustrate that our QED works much better than the existing labeling schemes on the label updates when considering either the number of nodes or the time for re-labeling.

It is important to process the updates when nodes are inserted into or deleted from the XML tree. All the existing labeling schemes have high update cost, thus in this paper we propose a novel Compact Dynamic Binary String (CDBS) encoding to efficiently process the updates. CDBS has two important properties which form the foundations of this paper: (1) CDBS supports that codes can be inserted between any two consecutive CDBS codes with the orders kept and without re-encoding the existing codes; (2) CDBS is orthogonal to specific labeling schemes, thus it can be applied broadly to different labeling schemes or other applications to efficiently process the updates. We report our experimental results to show that our CDBS is superior to previous approaches to process updates in terms of the number of nodes to re-label and the time for updating. 1.

The complete factorization of Chebyshev polynomials, of the rst and sec-ond kind, into irreducible factors over the integers Z is described. Conditions are given for determining when a Chebyshev polynomial is divisible by another. And, if non-zero, the remainder is again a Chebyshev polynomial, up to a sign. Algorithms are also speci ed to nd two in nite sets of elds Zp where a given Chebyshev polynomial factors completely into linear factors and to obtain the factors. The results also lead to the assertion: An odd integer n>0is prime if and only if the Chebyshev polynomial of the rst kind Tn(x), divided byx,is irreducible over the integers.

Abstract. A number of labeling schemes have been designed to facilitate the query of XML, based on which the ancestor-descendant relationship between any two nodes can be determined quickly. Another important feature of XML is that the elements in XML are intrinsically ordered. However the label update cost is high based on the present labeling schemes. They have to re-label the existing nodes or re-calculate some values when inserting an order-sensitive element. Thus it is important to design a scheme that supports order-sensitive queries, yet it has low label update cost. In this paper, we design a binary string prefix scheme which supports order-sensitive update without any re-labeling or re-calculation. Theoretical analysis and experimental results also show that this scheme is compact compared to the existing dynamic labeling schemes, and it provides efficient support to both ordered and un-ordered queries. 1

Algebraic properties of Chebyshev polynomials are presented. The com-plete factorization of Chebyshev polynomials of the rst kind (Tn(x)) and second kind (Un(x)) over the integers are linked directly to divisors of n and n + 1 respectively. For any odd integer n, itisshown that the polynomial Tn(x)=x is irreducible over the integers i n is prime. The result leads to a generalization of Fermat's little theorem and an e ective test for the com-positeness of an integer. Also, factoring of integers is linked directly to the construction of a related Chebyshev polynomial. 1