Q-adic变换再探
Q-adic Transform revisited
摘要：
我们提出了一个算法来执行一个同时模约简的几个残基。该算法适用于快速模多项式乘法。其思想是将模多项式的$X$ -adic表示（其中$X$是不定式）转换为$q$ -adic表示，其中$q$是大于域特征的整数。通过对不同大小的控制，可以直接使用机器整数或浮点执行某些$q$ -adic算术。还取决于所执行的数值运算的数目，然后可以转换回$q$ -adic或$X$ -adic表示，并且最终模出高残差。在本说明中，我们提出了一个新版本的转换：更多的表格和方法，以减少在这个过程中所涉及的部门的数量。然后将多项式乘法应用于小有限域扩展中的算术。
英文摘要：
We present an algorithm to perform a simultaneous modular reduction of several residues. This algorithm is applied fast modular polynomial multiplication. The idea is to convert the $X$-adic representation of modular polynomials, with $X$ an indeterminate, to a $q$-adic representation where $q$ is an integer larger than the field characteristic. With some control on the different involved sizes it is then possible to perform some of the $q$-adic arithmetic directly with machine integers or floating points. Depending also on the number of performed numerical operations one can then convert back to the $q$-adic or $X$-adic representation and eventually mod out high residues. In this note we present a new version of both conversions: more tabulations and a way to reduce the number of divisions involved in the process are presented. The polynomial multiplication is then applied to arithmetic in small finite field extensions.
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