Related papers: Four primality testing algorithms
Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). The recovered period for the sequence of the first 10000 prime numbers is equal to…
Many democratic political parties hold primary elections, which nicely reflects their democratic nature and promote, among other things, the democratic value of inclusiveness. However, the methods currently used for holding such primary…
A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.
An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…
Two prominent methods for integer factorization are those based on general integer sieve and elliptic curve. The general integer sieve method can be specialized to quadratic integer sieve method. In this paper, a probability analysis for…
Permutation is the different arrangements that can be made with a given number of things taking some or all of them at a time. The notation P(n,r) is used to denote the number of permutations of n things taken r at a time. Permutation is…
The best deterministic unconditionally proven integer factorization algorithms have exponential running time complexities of O(N^(1/4)) arithmetic operations, and conditional on the Riemann hypothesis, there is a deterministic algorithm of…
We formalize the problem of selecting the optimal set of options for planning as that of computing the smallest set of options so that planning converges in less than a given maximum of value-iteration passes. We first show that the problem…
Encryption study basically deals with three levels of algorithms. The first algorithm deals with encryption mechanism, second deals with decryption Mechanism and the third discusses about the generation of keys and sub keys used in the…
Testing probabilistic programs is non-trivial due to their stochastic nature. Given an input, the program may produce different outcomes depending on the underlying stochastic choices in the program. This means testing the expected outcomes…
We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small…
We study the query complexity of testing for properties defined by read once formulas, as instances of {\em massively parametrized properties}, and prove several testability and non-testability results. First we prove the testability of any…
In a previous paper, we have shown that any Boolean formula can be encoded as a linear programming problem in the framework of Bayesian probability theory. When applied to NP-complete algorithms, this leads to the fundamental conclusion…
This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…
In this paper we present four necessary and sufficient conditions for a natural number to be prime.
Numerical analysis has no satisfactory method for the more realistic optimization models. However, with constraint programming one can compute a cover for the solution set to arbitrarily close approximation. Because the use of constraint…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…
We remark that the AKS primality testing algorithm [Annals of Mathematics 160 (2), 2004] needs about 1,000,000,000 G (gigabyte) storage space for a number of 1024 bits. The requirement is very hard to meet. The complexity class P which…
Linear-time computational techniques have been developed for combining evidence which is available on a number of contending hypotheses. They offer a means of making the computation-intensive calculations involved more efficient in certain…
Test functions are important to validate new optimization algorithms and to compare the performance of various algorithms. There are many test functions in the literature, but there is no standard list or set of test functions one has to…