Mustafa Atici

Some recursive constructions for perfect hash families

An (n, m, w)-perfect hash family is a set of functions F such that for each f ϵ F, and for any X ⊆ {1,…,n} such that |X| = w, there exists at least one f ϵ F such that f|x is one-to-one. Perfect hash families have been extensively studied by computer scientists for over 15 years, mainly due to their applications in database management. In particular, much attention has been given to finding efficient algorithms to construct perfect hash families. In this article, we study perfect hash families from a combinatorial viewpoint, and describe some new recursive constructions for these objects.

Computational Complexity of Geodetic Set

For two vertices u and v of a graph G , the set H (u,v) consists of all vertices lying on some u m geodesic in G . If S is a set of vertices of G , then H(S) is the union of all sets H(u,v) for u,v ] S . If H(S) = V(G) , then S is a geodetic set for G . GEODETIC SET decision problem is defined and it is shown to be NP-Complete.

On The Edge Geodetic Number Of A Graph

Many graph-theoretic concepts have both vertex and edge versions. Examples are cut-vertex , cut-edge , vertex-integrity ( integrity ), edge-integrity , vertex coloring , edge coloring , and vertex-connectivity , edge-connectivity . Frank Harary et al. defined the geodetic number of a graph for vertices in [Chartrand, G., Harary, F. and Zhang, P. (2002). On the geodetic number of a graph. Networks , 39 , 1-6; Chartrand, G., Harary, F. and Zhang, P. (2000). Geodetic sets in graphs. Discussions Mathematicae Graph Theory , 20 , 129-138; Harary, F., Loukakis, E. and Tsours, C. (1993). The geodetic number of a graph. Mathl. Comput. Modelling , 17 (11), 89-93]. In this study we give a definition of the edge geodetic number for a graph and derive some results.

Secret Sharing Scheme: Vector Space Secret Sharing and F Function

Let P = {P₁, P₂, ..., P_n} be set of participants and Γ = {B_i|B_i ⊂ P, 1 ≤ i ≤ k} be access structure. Vector space secret sharing realizing access structure Γ requires existence of function φ : P → (Z_p)^d, where p is a prime number and d ≥ 2 is an integer, satisfying the following condition (1, 0, 0, ..., 0) =<; φ(P_i) : P_i ∈ B >;⇔ B ∈ Γ = {B₁, B₂,..., B_k}. There is no known algorithm to construct such a function φ in general. Constructions are mainly done by trial and error. In this paper, we developed a polynomial algorithm to construct a φ function for certain type of access structures. Some examples are given to illustrate the algorithms.

New upper bounds for the integrity of cubic graphs

Integrity, a measure of network to reliability, is defined as where G is a graph with vertex set V, and m(G − S) denotes the order of the largest component of G − S. Let V = n. It is known that (n/3) + O(√n) is a general upper bound for the integrity of any cubic graph. In this article, several theorems are shown that improve this general upper bound. For some families of cubic graphs, an upper bound for the integrity of (n/4) + O(√n) can be established using these theorems.

Vector space secret sharing and an efficient algorithm to construct a φ function

The threshold scheme, the monotone circuit construction, and the vector space construction are some of the well-known secret sharing schemes in cryptography. The threshold and monotone circuit secret sharing schemes are fairly easy to construct for any given access structure Γ. The construction of a secret sharing scheme realizing a given access structure Γ with Vector Space Construction requires the existence of a function φ from a set of participants into a vector space, that is, φ: P → (Zp)d. This function φ must satisfy certain conditions. There is no known algorithm to construct such a function φ in general. Constructions are mainly done by trial and error. In this paper, we develop polynomial algorithms to construct φ functions for vector space secret sharing scheme realizing certain types of access structures. Some examples are given to illustrate the algorithms.

Application of Huffman data compression algorithm in hashing computation

Cryptography is the art of protecting information by encrypting the original message into unreadable format. A cryptographic hash function is a hash function which takes an arbitrary length of text message as input and converts that text into a fixed length of encrypted characters which is infeasible to invert. The values returned by hash function are called as message digest or simply hash values. Because of its versatility, hash functions are used in many applications such as message authentication, digital signatures, and password hashing[2]. The purpose of this study is to apply Huffman data compression algorithm to the SHA1 hash function in cryptography. Huffman data compression algorithm is an optimal compression or prefix algorithm where the frequencies of the letters are used to compress the data [1]. An integrated approach is applied to achieve new compressed hash function by integrating Huffman compressed codes in the core functionality of hashing computation of the original hash function.

A New Approach for Modeling with Discrete Fractional Equations

In this paper, we introduce a new class of nonlinear discrete fractional equations to model tumor growth rates in mice. For the data fitting purpose, we develop a new method which can be considered as an improved version of the partial sum method for parameter estimations. We demonstrate the goodness of fit by comparing the models with three statistical measures.

Efficient Algorithm to Construct Perfect Secret Sharing Scheme for a Given Access Structure

The threshold scheme, the monotone circuit construction, and the vector space construction are some of the well-known secret sharing schemes in cryptography. The threshold and monotone circuit secret sharing schemes are fairly easy to construct for any given access structure Γ. The construction of a secret sharing scheme realizing a given access structure Γ with Vector Space Construction requires the existence of a function φ from a set of participants into a vector space, that is, φ: P → (Zp)d. This function φ must satisfy certain conditions in order to recover the secret key. There is no known algorithm to construct such a function φ in general. Constructions are mainly done by trial and error. In this paper, we develop polynomial algorithm to construct such φ function(s) for given access structures. Using the φ function, we also give an algorithm to construct secret sharing scheme for the access structures.

Parameter estimations of sigmoidal models of cancer

Background Tumor growth, a relationship between tumor size and time, is of special interest since growth estimation is very critical in a clinical practice. There are some mathematical models which describe tumor growth and have prediction capabilities. Typically, there are three ways to model non-complex growth behavior: exponential, logistic and sigmoidal. In 1825, Benjamin Gompertz introduced the Gompertz function, a sigmoid function, which is found to be applicable to various growth phenomena, in particular tumor growth. Besides the Gompertz model which includes three parameters, the Weibull and Richards models with four parameters are known as sigmoidal models. The aim of this project is to introduce continuous fractional and discrete fractional models of the tumor growth and also estimate parameters of these models in order to have better data fitting.