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The LLL algorithm described a method for finding simultaneous Diophantine approximations to rational numbers using a lattice with real number elements [32]. In [33] (Appendix A), a comparable solution for a lattice with integer elements is provided. Theorem 3 ([33]). There is a polynomial ...
The technique in Boneh and Freeman, 2011 established on the worst-case hardness assumption of ‘finding short vectors’ from the integer lattices. This scheme can also be used for authenticating signed data's linear transformations (mean, Fourier, network coding). Its an example of Homomorphic ...
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By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation (nk)=(ml)+d for −3≤d≤3 and (k,l)∈{(2,3),(2,4),(2,5),(2,6),(2,8),(3,4),(3,6),(4,6),(4,8)}. Moreover, we present some other observations ...
More generally, for any non-zero scaling parameter , the function defined by solves (3). As an “extra challenge” posed in an exercise in one of my books (Exercise 2.28, to be precise), I asked the reader to use the embeddings (or more generally ) to explicitly connect together the...
Learn the concept of the axis of symmetry of a geometrical shape. Understand the formula for finding the equations of the axis of symmetry & vertex...
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The Euclidean algorithm provides a good method for finding solutions of these equations. This result is one of the most useful in the whole of number theory. II.A.2 Pell's Equation and Quadratic Forms Suppose d is a positive integer and not square, then the equation x2−dy2=1 has ...
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