Fast Modular Squaring Method for Public-Key Cryptosystems

Abstract

平方演算法在大整數的運算，扮演很重要的角色。標準的平方演算法眾所週知，但有“錯誤進位”的疑慮發生。Guajardo與Paar學者提出的平方演算法修正了這項缺點，但是又延生出“錯誤索引”的問題。在本篇論文中，我們提出一個有效的平方方法，不僅可以解決以上所述兩項問題，亦可改進Yang、Hseih與Laih三位學者所提出的演算法。對於基底b而言，xi * xj 的乘積可以事先計算並儲存之，即1*2, 1*3, …, (b-1)(b-1)可以在實際運算前，事先儲存之，進而加速平方演算法的執行效率。本文所提出的演算法與Yang、Hseih與Laih三位學者所提出的演算法相較之下，快了1.77倍，當然這個演算法比標準的平方法亦快的多。

The squaring algorithm acts an important role in large integer arithmetic. The standard squaring algorithm is quite well-known, but there is an improper carry handling bug in it. The Guajardo-Paar’s squaring algorithm fixes the carry handling bug, but generates error-indexing bug. In this paper, we propose a novel efficient squaring algorithm that not only avoids the bugs between the standard squaring algorithm and the Guajardo-Paar`s squaring algorithm but also improves the performance in squaring computation for Yang-Hseih-Laih squaring algorithm. For base b, the products of xi * xj can be pre-computed on-line, that is, 1*2, 1*3, …, (b-1)(b-1) are pre-computed. Some results will be determined and stored in a look-up table before the computation and we can speed up the performance of squaring algorithm. Our proposed algorithm is about 1.77 times faster in comparison with the Yang-Hseih-Laih’s algorithm, and also faster than the standard squaring algorithm.