Scalar multiplication is the most important and expensive operation in elliptic curve cryptosystems. In this paper we improve the efficiency of the Elliptic Net algorithm to compute scalar multiplication by using the equivalence of elliptic nets. The proposed method saves f our multiplications in each iteration loop. Experimental results also indicates that our algorithm will be more efficient than the previously known results in this line.
The Weierstrass semigroups and pure gaps can be helpful in constructing codes with better parameters. In this paper, we investigate explicitly the minimal generating set of the Weierstrass semigroups associated with several totally ramified places over arbitrary Kummer extensions. Applying the techniques provided by Matthews in her previous work, we extend the results of specific Kummer extensions studied in the literature. Some examples are included to illustrate our results.
We investigate multi-point algebraic geometric codes defined from curves related to the generalized Hermitian curve introduced by Alp Bassa, Peter Beelen, Arnaldo Garcia, and Henning Stichtenoth. 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 F27 is presented as one of the examples.
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