Table of Contents
Elliptic Divisibility Sequences
Summary
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0, 1, 1, 1, 1, 2, 1, 3, 5, 7, 4, 23, 29, 59, 129, 314, 65, 1529, 3689,
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Properties of E.D.S
The Divisibility Property
Definition
?m+n ?mn = ?m+1 ?m1 ?n2  ?n+1 ?n1 ?m2
When Does p  Zm ?
Example : p=5, ?=8, ?=4
Theorem (Morgan Ward)Given a prime p and an elliptic divisibility sequence {Z}, ? integers a,b,c such that for all m,k ?Z we have
And now for something completely different……….or is it?
Elliptic Curves
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Example
Theorem
Working with Elliptic Divisibility Sequences gives us a way to work with Elliptic Curves
E.D.S. Formulae
Theorem Given 4 consecutive terms of the sequence {Z} including the term Zk we can compute the term Z?k in O(log ??) multiplication steps for any given integer ?.
Proof
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Example : ? = 92
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Cryptography
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The {Z} Sequence DLOG Problem
Example : ? = p
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Summary

Author: MCS
Email: map01rs@gold.ac.uk
Home Page: http://homepages.gold.ac.uk/rachel/
