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Revisiting Square Roots with a Fast Estimator Scan to know paper details and author's profile Revisiting Square Roots with a Fast Estimator Dr. Anthony Overmars, Dr. Sitalakshmi Venkatraman & Dr. Sazia Parvin ABSTRACT The square root of prime numbers is an important mathematical primitive with wide applications and its computational complexity has drawn much attention among researchers. In particular, principal square root of 2 is known to be irrational. The Silver Ratio δS = 1 + √2 is also irrational. These can be approximated by a ratio of rational numbers. This paper revisits some important applications of square root and proposes a new method to compute square root of 2. The proposed estimator determines the infinite series ratio of rational numbers with infinite precision. The optimised method isolates the simple fractional part and expresses this as a ratio of two rational numbers, such that δS − 2 = √2 − 1 = r1/r2 . We compare the new method with the well known Babylonian method. The experimental results show that our proposed method is efficient and outperforms the Babylonian method in both accuracy and speed. Keywords: square root, applications, silver ratio, Babylonian method, infinite series, optimisation, efficiency. ​ ​ Classification: D.1.0 ​ Language: English ​ LJP Copyright ID: 975711 Print ISSN: 2514-863X London Online ISSN: 2514-8648 LJP Journals Press London Journal of Research in Computer Science and Technology 24UK Volume 18 | Issue 1 | Compilation 1.0 © 2018. Dr. Anthony Overmars, Dr. Sitalakshmi Venkatraman & Dr. Sazia Parvin. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 4.0 Unported License http://creativecommons.org/licenses/by-nc/4.0/), permitting all noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Revisiting Square Roots with a Fast Estimator α σ ρ Dr. Anthony Overmars ,​ Dr. Sitalakshmi Venkatraman ​ & Dr. Sazia Parvin ​ ​ ​ ____________________________________________ I. ABSTRACT roots to the basic operations [3]. In particular, square root of 2 and its associated Silver ratio The square root of prime numbers is an have fascinated mathematicians since ancient important mathematical primitive with wide times due to their computational complexity and applications and its computational complexity applications in many real-world scenarios has drawn much attention among researchers. including architecture, modern printing, nature, In particular, principal square root of 2 is known music and even today's information system to be irrational. The Silver Ratio δ = 1 + √2 is S applications such as security and big data [4]. also irrational. These can be approximated by a ratio of rational numbers. It is a classical hard problem to get an accurate result for computing square root and methods are This paper revisits some important applications available in literature such as Rough estimation, of square root and proposes a new method to Babylonian method, Taylor-series expansion compute square root of 2. The proposed algorithm and Newton-Raphson method [5]. estimator determines the infinite series ratio of However, in this information age with mountains rational numbers with infinite precision. The of data, applications require improved methods optimised method isolates the simple fractional that are sufficiently fast with high precision [6]. part and expresses this as a ratio of two rational Various applications make use of square root of numbers, such that δS − 2 = √2 − 1 = r1/r2 . We different numbers and understanding their compare the new method with the well known requirements would help in making use of s Babylonian method. The experimental results suitable square root estimator. This paper show that our proposed method is efficient and proposes a new method that we derive for outperforms the Babylonian method in both estimating square root with high accuracy. In of Research in Computer Science and Technology accuracy and speed. particular, we show how square root of 2 can be calculated very fast and compare with the Keywords: square root, applications, silver ratio, ​ well-known Babylonian method. The associated Babylonian method, infinite series, optimisation, Silver Ratio is also estimated and discussed. efficiency. The remaining paper is organized as follows. London Journal Author α σ ρ: School of Engineering, Construction & ​ Section III provides an overview of the various Design, Melbourne Polytechnic, Victoria, Australia. real-life applications square root. The proposed method is derived in Sections IV and optimised in II. INTRODUCTION Section V. The Babylonian method is described in Square root is one of the most useful and Section VI. Experimental results for frequently used operations in many different real benchmarking our proposed method are world applications such as computer graphics, summarized in Section VII. Finally, we present multimedia, data processing, cryptosystems and our conclusions in Section VIII. many scientific calculation applications [1] [2]. III. SQUARE ROOT APPLICATIONS Several mathematical studies have determined that the theoretical complexity of such scientific Several real world applications require computing calculations can be reduced by adding square square root and square root of two [7] [1]. The © 2018 London Journals Press Volume 18 | Issue 1 | Compilation 1.0 45 most common application of square root in ratio of square root application is even more everyday life is found in standard paper sizes. The prominent in today's multimedia world of graphic base A0 size of paper is defined as having an area designs and web designs. The reciprocal of square of 1 m2 and a side ratio of 1 by √2,. Successive root is also an important operation with ​ paper sizes in the series A1, A2, A3, and the applications in three-dimensional graphics or frequently used A4 paper sizes have a nice scientific computations. property that they are derived by halving the preceding one along the larger dimension. Square root is commonly used to transform data Similarly, folded brochures of any size can be which has applications in computer processors, made by using sheets of the next larger size. For statistics, big data and information security. It is example, A5 sixe brochures can be made by possible to reduce the theoretical complexity of folding A4 sheets along the length of the paper. certain problems by adding square roots to the The standard system of basing a paper size upon basic operations in the computer processor. There an aspect ratio of √2 has several advantages of not exists a number of situations in which square only scaling from one size to but also in fast roots are used as common transformations, for processing of office photocopiers. e.g. square roots are computed to be faster than divides. Similarly, while both the logarithm and Another real life example of square root is to square root transformations are commonly used calculate the interest of saving account. The factor for power transformations of positive data, the by which the account grows is exponential. This square root is preferred because there is no means that if it grows by some factor in a given requirement to use special treatment for zeros. time, it will grow by the square of that factor in Power transformations have wide applications in double the time. So, to compute the growth in half big data and information security. The square the time, square root is used. roots of small integers are used in both the SHA-1 and SHA-2 hash function designs in Square roots have been traditionally used to cryptosystems to achieve information security [8] calculate time and distance, such as in the [9]. Square roots are used in decryption in order applications of object falling and land area to compute the encrypted text called the cipher triangulation measurements. For falling objects, text. If the cipher text c is modulo of the primes p square root can be used to calculate the time t it of Research in Computer Science and Technology and q, then square roots can be more easily takes for something to fall x distance when it is calculated by choosing p≡ q ≡3 mod 4 in order to dropped. From Newton's laws with g as ​ acceleration due to gravity, we have, decrypt the message. Hence, the basic step of using square roots in public-key cryptosystems is 2x in decryption where if the cipher text is given, one t = g London Journal √ can extract the keys using square root. The Similarly. let us consider calculating the period T metallic numbers, namely Golden ratio and Silver of a swing that is safe for children amusement ratio have been used extensively in cryptography park without a fall. The period of a swing with [10] [11]. length L is calculated by the following equation: In summary, square root is an important L mathematical primitive having many applications. T = 2π g √ Due its computational complexity there are some In land areas, in order to measure the diagonal of difficulties to ensure its security and efficiency in any rectangular space when the dimensions of the distributed computing [12]. Hence, several sides are known, Pythagorean theorem is usually algorithms have evolved, such as Goldschmidt’s applied to measure it as the square root of sum of algorithm using linear approximation, Manuel square of the two sides. In this context, the aspect Liedel algorithm through the fixed-point Revisiting Square Roots with a Fast Estimator 46 Volume 18 | Issue 1 | Compilation 1.0 © 2018 London Journals Press arithmetic framework of Catrina/Saxena. This is consistent with Fig. 1. From previous work Goldschmidt’s algorithm for square root is [13] [14] it can be shown that sides (a, b, c) can be preferred over the traditional Newton-Raphson expressed as three Diophantine equations in iterations due to faster computation as iteration terms of (m, n) : P (a, b, c) = P (m, n) where contains fewer dependent multiplications with the same computational complexity. In this paper we a = 2n2 + 2n(2m − 1) (2) address the problem of computational complexity 2 by proposing a faster optimised method with high b = 2n(2m − 1) + (2m − 1) (3) precision.
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