Elliptic Divisibility Sequence - EDS Over Finite Fields

EDS Over Finite Fields

An EDS over a finite field Fq, or more generally over any field, is a sequence of elements of that field satisfying the EDS recursion. An EDS over a finite field is always periodic, and thus has a rank of apparition r. The period of an EDS over Fq then has the form rt, where r and t satisfy

r \le \left(\sqrt q+1\right)^2 \quad\text{and}\quad t \mid q-1.

More precisely, there are elements A and B in Fq* such that

W_{ri+j} = W_j\cdot A^{ij} \cdot B^{j^2} \quad\text{for all}~i \ge 0~\text{and all}~j \ge 1.

The values of A and B are related to the Tate pairing of the point on the associated elliptic curve.

Read more about this topic:  Elliptic Divisibility Sequence

Famous quotes containing the words finite and/or fields:

    “Any language is necessarily a finite system applied with different degrees of creativity to an infinite variety of situations, and most of the words and phrases we use are “prefabricated” in the sense that we don’t coin new ones every time we speak.”
    —David Lodge (b. 1935)

    “And sweet it was to dream of Fatherland,
    Of child, and wife, and slave; but evermore
    Most weary seemed the sea, weary the oar,
    Weary the wandering fields of barren foam.”
    —Alfred Tennyson (1809–1892)