Encryption based on Ballot, Stack permutations and Balanced Parentheses using Catalan-keys

This paper examines the possibilities of applying Catalan numbers in cryptography. It also offers the application of appropriate combinatorial problems (Ballot Problem, Stack permutations and Balanced Parentheses) in encryption and decryption of files and plaintext. The paper analyzes the properties of Catalan numbers and their relation to these combined problems. Applied copyright method is related to the decomposition of Catalan numbers in the process of efficient keys generating. Java software solution which enables key generating with the properties of the Catalan numbers is presented at the end of the paper. Java application allows encryption and decryption of plaintext based on the generated key and combinatorial problems.

A Left Weighted Catalan Extension

There are many representations of the Catalan numbers. In this article, we will examine the ballot problem and extend it beyond the standard

Optimal stopping on trajectories and the ballot problem

An urn contains m minus balls and p plus balls, and we draw balls from this urn one at a time randomly without replacement until we wish to stop. Let P n and M n denote the respective numbers of plus balls and minus balls drawn by time n and define Z 0 = 0, Z n = P n - M n , 1 ≤ n ≤ m + p. The main problem of this paper is to stop with maximum probability on the maximum of the trajectory formed by . This problem is closely related to the celebrated ballot problem, so that we obtain some identities concerning the ballot problem and then derive the optimal stopping rule explicitly. Some related modifications are also studied.

Optimal stopping on trajectories and the ballot problem

An urn contains m minus balls and p plus balls, and we draw balls from this urn one at a time randomly without replacement until we wish to stop. Let Pn and Mn denote the respective numbers of plus balls and minus balls drawn by time n and define Z0 = 0, Zn = Pn - Mn, 1 ≤ n ≤ m + p. The main problem of this paper is to stop with maximum probability on the maximum of the trajectory formed by . This problem is closely related to the celebrated ballot problem, so that we obtain some identities concerning the ballot problem and then derive the optimal stopping rule explicitly. Some related modifications are also studied.