Scalar multiplication is the most important and expensive operation in elliptic curve cryptosystems. In this study, the authors improve the efficiency of the elliptic net algorithm to compute scalar multiplication by using the equivalence of elliptic nets. The proposed method saves four multiplications by a constant in each iteration loop. Experimental results also indicate that the proposed algorithm will be more efficient than the previously known results on this line while it is still slower than the state‐of‐the‐art algorithm to compute scalar multiplication.

We present multi-point algebraic geometric codes overstepping the Gilbert-Varshamov bound. The construction is based on a curve in the tower introduced by Bassa et al. These codes are described in detail by constructing generator matrices. It turns out that these codes have nice properties similar to those of Hermitian codes. The duals are also such codes and can be expressed as an explicit formula.

We investigate multi-point algebraic geometric codes defined from curves related to the generalized Hermitian curve introduced by Bassa et al. Our main result is to find a basis of the Riemann-Roch space of a series of divisors, which can be used to construct multi-point codes explicitly. These codes turn out to have nice properties similar to those of Hermitian codes, for example, they are easy to describe, to encode and decode. It is shown that the duals are also such codes and an explicit formula is given. In particular, this formula enables one to calculate the parameters of these codes. Finally, we apply our results to obtain linear codes attaining new records on the parameters. A new record-giving [234, 141,≥ 59]-code over F27is presented as one of the examples.