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In this paper, the authors describe the addition law for a new form for elliptic curves over fields of characteristic. Specifically, it presents explicit formulae for adding two different points and for doubling points. The case of differential point addition (that is, point addition with a known difference) is also addressed. Finally, this paper presents unified point addition formulae; i.e., point addition formulae that can be used for doublings. Applications to cryptographic implementations are discussed.