euclid's algorithm
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Author(s):  
Josep M. Brunat ◽  
Joan-C. Lario

AbstractMotivated by the design of satins with draft of period m and step a, we draw our attention to the lattices $$L(m,a)=\langle (1,a),(0,m)\rangle$$ L ( m , a ) = ⟨ ( 1 , a ) , ( 0 , m ) ⟩ where $$1\le a<m$$ 1 ≤ a < m are integers with $$\gcd (m,a)=1$$ gcd ( m , a ) = 1 . We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a.


2021 ◽  
Vol 11 (1) ◽  
pp. 346-363
Author(s):  
Al Cuoco ◽  
Paul Goldenberg

How you think about a phenomenon certainly influences how you create a program to model it. The main point of this essay is that the influence goes both ways: creating programs influences how you think. The programs we are talking about are not just the ones we write for a computer. Programs can be implemented on a computer or with physical devices or in your mind. The implementation can bring your ideas to life. Often, though, the implementation and the ideas develop in tandem, each acting as a mirror on the other. We describe an example of how programming and mathematics come together to inform and shape our interpretation of a classical result in mathematics: Euclid's algorithm that finds the greatest common divisor of two integers.


Author(s):  
Chris Bleakley

Chapter 1 traces the origins of algorithms from ancient Mesopotamia to Greece in the 2th century BC. The oldest known algorithms were inscribed on clay tablets by the Babylonians more than 4,000 years ago. The clay tablets document algorithms ranging from geometry to accountancy. One tablet in particular - YBC 7289 - indicates knowledge of the Pythagorean Theorem thousands of years before its supposed invention by the ancient Greeks. The Greeks made other advances in algorithms. Euclid’s algorithm determines the greatest common divisor of two numbers. The Sieve of Eratosthenes finds prime numbers. Both algorithms proved to be important stepping stones to modern cryptography - the mathematics of secret messages.


Author(s):  
Robin Wilson

‘Multiplying and dividing’ looks at multiples and divisors, focusing on the least common multiple and greatest common divisor of two numbers. We use Euclid’s algorithm as a method for computing the greatest common divisor of two numbers by using the division rule repeatedly. Perfect squares (integers that are the product of two equal integers) feature throughout number theory. Tests are given for divisibility by certain small numbers. An ancient method called ‘casting out nines’, was developed in India in around the year 1000, based on the argument that a number and its digital sum leave the same remainder when divided by 9. We can still use this method to verify the accuracy (or otherwise) of arithmetical calculations.


Author(s):  
Daniel J. Bernstein ◽  
Bo-Yin Yang

This paper introduces streamlined constant-time variants of Euclid’s algorithm, both for polynomial inputs and for integer inputs. As concrete applications, this paper saves time in (1) modular inversion for Curve25519, which was previously believed to be handled much more efficiently by Fermat’s method, and (2) key generation for the ntruhrss701 and sntrup4591761 lattice-based cryptosystems.


2018 ◽  
Vol 26 (2) ◽  
pp. 165-173
Author(s):  
Ievgen Ivanov ◽  
Artur Korniłowicz ◽  
Mykola Nikitchenko

Summary In this paper we present a formalization in the Mizar system [2, 1] of the correctness of the subtraction-based version of Euclid’s algorithm computing the greatest common divisor of natural numbers. The algorithm is written in terms of simple-named complex-valued nominative data [11, 4]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [7]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic with partial pre- and post-conditions [8, 10, 5, 3].


2014 ◽  
Vol 54 ◽  
pp. 27-65 ◽  
Author(s):  
Valérie Berthé ◽  
Hitoshi Nakada ◽  
Rie Natsui ◽  
Brigitte Vallée

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