Computing cube root of a positive number

Proposed here is a new algorithm to compute the cube root of large positive integer. The algorithm is based on the implementation of long division method also known as manual method we usually use to find the square root of a number. To implement the long division method, the given number is first represented in a radix-10 representa and then Bino’s Model of Multiplication is used to systematically implement the long division method. A representa is a special array to represent a number in the form of an array so as to enable us to treat the representas in the same way as we treat numbers. This simplifies the difficulty of dealing large numbers in a computer. Also, at the same time it simplifies the implementation of long division method to find the cube root of positive number, ranging from single digit number to arbitrarily large positive number such as RSA challenge numbers. The algorithm can be used to compute cube root of a non-perfect cube number up to desired precision and each computed digit of cube root gives the best precision. Cube root of 2, 5, 10 up to 30 digits and integer parts of cube roots of first few and last few RSA challenge numbers are also provided in the experimental result to show that the algorithm works perfectly to compute the cube root of any positive integer, however small or large it may be.Keywords:Bino‘s Model of Multiplication, Convolution, Cube of a large number, Large number manipulation, Long division method, RSA challenge numbers, Representa, Cube root computatio

Finding Connected Components in a Gray Scale Image

Abstract—Finding connected components are well defined for binary images. The concept of connected components can be extended for gray level image. But the problem is the criteria based on which a connected component would be defined. A gray level image is an image having 256 different pixel intensity levels. If we consider connected regions having only a particular pixel values, the number of connected components would not be meaningful and the purpose of finding connected components would be lost. So, we define a connected component in a gray scale image based on range of pixel mapping and new method to find connected components in a gray scale image is proposed. Three different types of pixel range mapping are introduced, using connected components in a gray level image can be successfully found. Connected components in a gray level image are the segments of image having the same range of pixel values. Different regions or segments of image can be found easily.Keywords—Connected component, gray scale labelling, pixel range mapping, linear mapping, logarithmic mapping, square root mapping.(Article history: Received 1 November 2016 and accepted 30 December 2016

Image Encryption Using Meitei Lock Sequence Generated from Hash Functions

Proposed here is a secure image encryption scheme based on generalized Vigenere cipher and Meitei Lock Sequence (MLS) generated from standard hash functions. MLS is a unique random sequence of any length greater than 2 generated from a non-negative array having two or more elements. It is unique in the sense that no two arrays can generate the same sequence however close or similar the two arrays are. In other words, when there is any slight change in any of the input array, the generated MLS’s are drastically different. Also, the length of the sequence can be as infinitely long. These properties make MLS a good key string for a secure encryption scheme. SHA(Secure Hashing Algorithm) or any hash code generator has desirable feature which can be used for generation of MLS. In a hash code generator, it produces unique fixed length sequence from any input string, if there is any slight change in the input, the generated output will be totally different. This feature is made used of  in generating an MLS of any desired length for use in the proposed image  encryption scheme. Experimental results show that the proposed encryption scheme is a secure encryption scheme. The correlation coefficient between the original image and encrypted images are negligibly small indicating that there is no trace of original image information in the encrypted image. Also, the correlation coefficients between the original image and decrypted images with wrong passwords which are close to the encryption password are also negligibly small. These show the tightness of the key system in the encryption scheme. 

An Analysis on Extracting Square and Cube Roots by Aryabhata’s Methods

Abstract— Finding accurate root of a number is still considered as challenging in computer science community. The only popularly known method to compute root is the long division method of finding square root. Aryabhata contributed two methods similar to long division method to compute square root and cube root. In recent years, his methods have also been studied, explained and tried to implement as computer algorithms.  The explained methods and proposed algorithms fail to give correct results. Some analyses have been made on these methods in order to ascertain why the algorithms fail. Improved algorithms have been provided to give correct result while computing square root or cube root using Aryabhata’s methods.    .Keywords—cube root, square root, Bino’s Model of multiplication, Large number manipulation, Long division method, Aryabhata’s methods.  (Article history: Received 12 November 2016 and accepted 30 December 2016

Performing Binary Logical Operation in Decimal Arithmetic

Proposed here are the new ways of performing binary logical operations using decimal arithmetic. Usually, binary logical operations such as AND, OR, XOR etc are performed in binary arithmetic which is time consuming and quite erroneous when performed manually. We are familiar with decimal arithmetic, so if we can perform binary logical operations in decimal arithmetic, it will be easier for us and at the same time less error prone and help us understanding the logical operations in a better way. These logical operations are basic building block of digital devices and systems in the modern digital world. Understanding different ways of performing these operations may altogether bring a change in how the digital devices are designed. A deeper study has been made on these operations and a new logical operation named YOR has also been introduced

Computing square root of a large Positive Integer

Dealing in large number is of interest in asymmetric key cryptography using RSA. The security of RSA is solely based on difficulty of factorization of a large number. Factoring a number requires finding all divisible primes less than or equal to the square root of the number. Proposed here is a new algorithm to compute the square root of large positive integer. The algorithm is based on the implementation of long division method also known as manual method we usually use to find the square root of a number. To implement the long division method, the given number is first represented in a radix-10 representa and then Bino’s Model of Multiplication is used to systematically implement the long division method. A representa is a special array to represent a number in the form of an array so as to enable us to treat the representas in the same way as we treat numbers. This simplifies the difficulty of dealing large numbers in a computer. The proposed algorithm is applied to the RSA–challenge numbers for factorization. The square roots of the challenge numbers can be computed easily in less than a second. The square roots of first few challenge number and last few challenge number are also provided, which may be used for factorization of corresponding challenge number.Key words: Asymmetric key cryptogra phy, Bino’s Model of Multiplication, Large number manipulation, Long division method, Prime factorization, RSA challenge numbers, Representa, Square root computatio

Incorporating Edge Information in Digital Halftoning

Digital halftoning is the process of generating a binary image preserving gray shade information so as to make the binary image appears visually similar to gray image. It was used in printing machines and display devices to produce binary images having gray shades. Ordered dithering and error diffusion methods are two most popular methods to generate halftone image. Generally, in a halftone image, the edges become blurred or loses its sharpness. Edges carry significant information of the foreground objects in an image and increase visual clarity by distinguishing the objects from background. A method is proposed to generate edge sharpened halftone images using a strong unsharp masks. Such edge sharpened halftone images are visually more pleasing and informative as compared with the normal halftone images. The proposed method is found to be better than Xin-Li’s edge-adaptive method of generating halftone images.Keywords: Digital halftoning, edge enhancement, unsharp masking, error diffusion, Sierra’s Filter*Cite as: Yumnam Kirani Singh, “Incorporating Edge Information in Digital Halftoning†ADBU J.Engg Tech., 1(2014) 0011403(6pp

A New Approach to Generating Textured Binary Images Using VK Transforms

Proposed here is a new method of generating textured binary images using Varied Kernel (VK) transforms. The varied kernel transforms are perfectly invertible real kernel orthogonal transforms. A property of VK transform is that when an image is decomposed by a kernel and is reconstructed by a slightly different inverse kernel and binarized, the resulted binary image is a textured image. The generated textured binary image carries shade information as in dithered or halftone image and hence is more meaningful and artistic than normal binary image. Textured binary images of different qualities can be achieved by varying the types of the kernels used.Key words: Textured binary image, dithering, VK transforms, real kernel, orthogonal transform