For a subgroup \ langle \ gamma \ rangle of prime order \ omega of the multiplicative group GF \ left ( p ^ t \ right ) ^ * of an extension field GF ( p ^ t ) of GF ( p ) for some t, there are now two possible ways to attack the system.
32.
HFE is in fact a family which consists of basic HFE and combinatorial versions of HFE . The HFE family of cryptosystems is based on the hardness of the problem of finding solutions to a system of multivariate quadratic equations ( the so-called MQ problem ) since it uses private affine transformations to hide the extension field and the private polynomials.
33.
This fact also follows from general considerations in Galois theory ( one views the " a " " k " as elements of a base field with roots in an extension field whose Galois group permutes them according to the full symmetric group, and the field fixed under all elements of the Galois group is the base field ).
34.
The most basic example of a fundamental group is ? 1 ( Spec " k " ), the fundamental group of a separably closed extension field " K ", and the fundamental group with respect to that base point identifies with the Galois group Gal ( " K " / " k " ).
35.
7 ) In elliptic curve cryptography, the Weil descent attack uses the Weil restriction to transform a discrete logarithm problem on an elliptic curve over a finite extension field L / K, into a discrete log problem on the Jacobian variety of a hyperelliptic curve over the base field K, that is potentially easier to solve because of K's smaller size.
36.
Davis undertook a major technological overhaul of Alabama Extension field offices, providing each county office with " an addressing machine so that both big and little mailing lists can be used for effective contact through the mails . " Outlying offices also were provided with film-strip and stereopticon machines " so that each county worker may combine vision with sound in presenting facts at meetings ."
37.
Furthermore, the \ ell th power of this non-principal ideal is principal in the base field K, in particular \ mathfrak { j } ^ { \ ell } = \ mathrm { N } _ { L / K } ( A ) \ mathcal { O } _ K \ in \ mathcal { P } _ K, hence the class number of the base field must be divisible by \ ell and the extension field L can be called a class field of K.
38.
Use Hensel's lift to get the factorization over the p-adic integers, find the ring of integers A of the splitting field of the p-adic polynomial, the galois group is abelian so you can take an arbitrary factor of ( p ) as P and then look at A / P ^ r as the " extension field " of Z / p ^ rZ . Make sure that n and p are coprime ( 6 and 2 is naughty ) otherwise the factorization can change.
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