Estimating the new Initial Value of Trial Division Algorithm for Balanced Modulus to Decrease Computation Loops

Abstract

The aim of this paper is to propose the new initial value of Trial Division Algorithm (TDA) which is suitable for balanced modulus. The traditional initial value is ordinary $\lfloor\sqrt{n}\rfloor$ where $n$ is represented as modulus but the proposed value is always smaller than one. Moreover, it is still larger than a smaller prime factor. That means some integers between the new initial value and $\lfloor\sqrt{n}\rfloor$ are deleted from the computation. Therefore, the computation time is also decreased. In fact, the key to implement process is the way to choose both of initial values of Fermat's Factorization Algorithm (FFA) in order to apply with the proposed method. The reason is that half of the addition between both of them is always less than the higher prime factor. Hence, the first trial divisor from dividing $n$ by the result is greater than the smaller factor. Therefore, it can be assigned as the new initial value for the implementation. The experimental results show that most of $n$ can be factored with decreased loops whenever the new initial value is estimated. However, loops are stable for some values of $n$ when the higher initial value of FFA is not changed.