Secret Sharing refers to a method through which a secret key K can be shared among a group of authorized participants, such that when they come together later, they can figure out the secret key K to decrypt the encrypted message. Any group which is not authorized cannot determine the secret key K. Some of the important secret schemes are Shamir Threshold Scheme, Monotone Circuit Scheme, and Brickell Vector Space Scheme. Brikell’s vector space secret sharing construction requires the existence of a function from a set of participant P in to vector space Zdp, where p is a prime number and d is a positive number. There is no known algorithm to construct such a function in general. We developed an efficient algorithm to construct function for some special secret sharing scheme. We also give an algorithm to demonstrate how a secret sharing scheme can be used in visual cryptography.

In this paper, we will review two fundamental counting methods: inclusionexclusion and pigeonhole principles. The inclusion-exclusion principle considers the elements of the sets satisfied some conditions, and avoids repeat counting by disjoint sets. We also use the inclusion-exclusion principle to solve the problems of Euler phi function and the number of onto functions in number theory, and derangement and the number of nonnegative integer solutions of equations in combinatorics. We derive the closed-form formula to those problems. For the forbidden positions problems, we use the rook polynomials to simplify the counting process. We also show the form of the inclusion-exclusion principle in probability, and use it to solve some probability problems. The pigeonhole principle is an easy concept. We can establish some sets and use the pigeonhole principle to discuss the extreme value about the number of elements. Choose the pigeons and pigeonholes, properly, and solve problems by the concept of the pigeonhole principle. We also introduce the Ramsey theorem which is an important application of the pigeonhole principle. This theorem provides a method to solve problems by complete graph. Finally, we give some contest problems about the inclusion-exclusion and pigeonhole principles to show how those principles are used.