TO FIND APPROXIMATE SQUARE ROOTS

Square Roots Modulo p

The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster. In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic fiel...

Ancient Indian Square Roots

Our modern system of positional decimal notation with zero, together with efficient algorithms for computation, which were discovered in India some time prior to 500 CE, certainly must rank among the most significant achievements of all time. And it was not easy. As Pierre-Simon Laplace (1923) explained, “the difficulty of inventing it will be better appreciated if we consider that it escaped t...

Roots of Square: Cryptanalysis of Double-Layer Square and Square+

Square is a multivariate quadratic encryption scheme proposed in 2009. It is a specialization of Hidden Field Equations by using only odd characteristic elds and also X as its central map. In addition, it uses embedding to reduce the number of variables in the public key. However, the system was broken at Asiacrypt 2009 using a di erential attack. At PQCrypto 2010 Clough and Ding proposed two n...

On group Elements Having Square Roots

Archimedes’ calculations of square roots

We reconsider Archimedes’ evaluations of several square roots in ‘Measurement of a Circle’. We show that several methods proposed over the last century or so for his evaluations fail one or more criteria of plausibility. We also provide internal evidence that he probably used an interpolation technique. The conclusions are relevant to the precise calculations by which he obtained upper and lowe...