Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields

An application of index forms in cryptography

“…The methods developed for the solution of "arbitrary" index form equations at the recent moment make it possible to solve index form equations where α has degree at most 6. For results concerning the general cubic case see [9], for the quartic case [8] and for the quintic case [7]. A long computation of about one month on a computer with six parallel processors of 1GH under linux made possible for Bilu, Gaál and Győry to solve an index form equation over a sextic field with Galois group S 6 (see [3]).…”

Computing all power integral bases in orders of totally real cyclic sextic number fields

“…Finally, it turned out [16] that also in this case it is possible to reduce the problem of resolution of index form equations to the resolution of cubic and quartic Thue equations.…”

Computing all monogeneous mixed dihedral quartic extensions of a quadratic field

“…avec l'algorithme de Gaál, de Pethö et de Pohst [6], [7]. On applique la méthode à M = Q(~2), Q(~3), Q(~5).…”

“…We give a fast algorithm for determining all dihedral quartic fields K with mixed signature having power integral bases and containing M as a subfield. We also determine all generators of power integral bases in K. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for M = Q (~2), Q (~3), Q(~5).…”