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

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