Complete addition formulas on the level four theta model of elliptic curves

摘要

The addition formula provided for the recently discovered level four theta model of elliptic curve (Diao and Fouotsa in Afr Math 26(3):283-301, 2015) is not complete; meaning that it does not work for all inputs. In this work we provide three alternative addition formulas to solve this problem. The obtained formulas are unified in the sense that they can be used for both addition of points and doubling. These formulas are also valid both in odd and even characteristics. In particular our new formula in binary fields improves the one previously obtained on this curve (Diao and Fouotsa, 2015). We compute the cost of each formula. We also provide a sage and magma codes (Fouotsa and Diao in Sage and magma code for the verification of addition formulas, algorithm for cost computation and completeness of addition formulas. , 2018) to ensure the correctness of all formulas and algorithms in this work.