Another Look at Square Roots (and Other Less Common Operations) in Fields of Even Characteristic
Another Look at Square Roots (and Other Less Common Operations) in Fields of Even Characteristic
复制标题
再看看偶特征域中的平方根（和其他不太常见的运算）
DOI：
--
复制
发表时间：
2007
期刊：
影响因子：
--
通讯作者：
R. Avanzi
中科院分区：
文献类型：
--
作者：
R. Avanzi
我们讨论了一个不可约多项式的家族，可用于加快特征两个领域的平方根提取。他们概括了Fong等人讨论的三项官方。 [20]。我们称这种多项式平方根友好。
主要的应用是将椭圆曲线的减半方法（在较小程度上，也较小的分裂曲线减半方法和配对计算）。
我们注意到，当我们已经知道存在相同程度的不可还原的三项式时，并且在平方根友好多项式术语的程度上表达了一个猜想时，我们注意到给定程度的平方根友好三项元素的存在。在Bluher的类似结果之后，我们还给出了朝向猜想方向的部分结果。
我们还讨论了如何提高求解二次方程的速度。进行模块化减少所需的时间的增加是微不足道的，不会对性能产生不利影响。估计证实了新的多项式曼坦因的承诺。点减半的加速度为20％，标量乘法提高至少11％。
We discuss a family of irreducible polynomials that can be used to speed up square root extraction in fields of characteristic two. They generalize trinomials discussed by Fong et al. [20]. We call such polynomials square root friendly.
The main application is to point halving methods for elliptic curves (and to a lesser extent also divisor halving methods for hyperelliptic curves and pairing computations).
We note the existence of square root friendly trinomials of a given degree when we already know that an irreducible trinomial of the same degree exists, and formulate a conjecture on the degrees of the terms of square root friendly polynomials. Following similar results by Bluher, we also give a partial result that goes in the direction of the conjecture.
We also discuss how to improve the speed of solving quadratic equations. The increase in the time required to perform modular reduction is marginal and does not affect performance adversely. Estimates confirm that the new polynomials mantain their promises. Point halving gets a speed-up of 20% and scalar multiplication is improved by at least 11%.