HT90 and “simplest” number fields

Abstract

A standard formula (1) leads to a proof of HT90, but requires proving the existence of θ such that α = 0, so that β = α/σ(α).We instead impose the condition (M), that taking θ = 1 makes α = 0. Taking n = 3, we recover Shanks's simplest cubic fields. The "simplest" number fields of degrees 3 to 6, Washington's cyclic quartic fields, and a certain family of totally real cyclic extensions of Q(cos(π/4m)) all have defining polynomials whose zeroes satisfy (M).Further investigation of (M) for n = 4 leads to an elementary algebraic construction of a 2-parameter family of octic polynomials with "generic" Galois group 8 T 11 . Imposing an additional algebraic condition on these octics produces a new family of cyclic quartic extensions. This family includes the "simplest" quartic fields and Washington's cyclic quartic fields as special cases.We obtain more detailed results on our octics when the parameters are algebraic integers in a number field. In particular, we identify certain sets of special units, including exceptional sequences of 3 units, and give some of their properties.

A Short-List of Pairing-Friendly Curves Resistant to Special TNFS at the 128-Bit Security Level

“…Pairing-friendly curves have a special characteristic p, given by a polynomial p(x) of small degree evaluated at an integer u. For BLS12 curves, we have p(x) = (x 6 −2x 5 +2x 3 +x+1)/3, and for a 381-bit prime p, u = −(2 63 +2 62 +2 60 + 2 57 + 2 48 + 2 16 ) [7]. Joux and Pierrot introduced a dedicated polynomial selection that takes advantage of the polynomial form p = p(u) [25].…”

“…We recall a variant of the Joux-Pierrot method to obtain a pair of polynomials (f y , g y ) admitting an automorphism, when k is not prime. First select an auxiliary polynomial with automorphism, for example from the list in [16].…”

Improving NFS for the Discrete Logarithm Problem in Non-prime Finite Fields

“…Indeed, the literature, e.g. [11], offers examples of polynomials g 0 , g 1 ∈ Q[x] and rational fractions A(x) ∈ Q(x) such that, for any number field K and any parameter a ∈ K, the polynomial g 0 + ag 1 admits A as a K-automorphism:…”

A generalization of simplest number fields and their integral basis

“…M.N.Gras [9], [10], V.Ennola [2], [3] and A.J.Lazarus [18] investigated the unit group of the simplest fields. K.Foster [4] obtained the simplest parametric polynomials, just by using a special identity of units in cyclic extensions. It shows, that these fields have some other unique and interesting properties.…”