### Simple verification of completeness of two addition formulas on twisted Edwards curves

#### Abstract

Daniel Bernstein and Tanja Lange proved that

two given addition formulas on twisted Edwards elliptic curves

ax^2 + y^2 = 1 + dxy are complete (i.e. the sum of any two points

on a curve can be computed using one of these formulas). In

this paper we give other simple verification of completeness

of these formulas using for example Groebner bases and an ¨

algorithm implemented in Magma, which is based on the fact that

completeness means that some systems of polynomial equations

have no solutions. This method may be also applied to verify

completeness of additions formulas on other models of elliptic

curves.

two given addition formulas on twisted Edwards elliptic curves

ax^2 + y^2 = 1 + dxy are complete (i.e. the sum of any two points

on a curve can be computed using one of these formulas). In

this paper we give other simple verification of completeness

of these formulas using for example Groebner bases and an ¨

algorithm implemented in Magma, which is based on the fact that

completeness means that some systems of polynomial equations

have no solutions. This method may be also applied to verify

completeness of additions formulas on other models of elliptic

curves.

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