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arXiv · 1605.06760

Space-Efficient Karatsuba Multiplication for Multi-Precision Integers

Yiping Cheng
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Abstract

The traditional Karatsuba algorithm for the multiplication of polynomials and multi-precision integers has a time complexity of $O(n^{1.59})$ and a space complexity of $O(n)$. Roche proposed an improved algorithm with the same $O(n^{1.59})$ time complexity but with a much reduced $O(\log n)$ space complexity. In Roche's paper details were provided for multiplication of polynomials, but not for multi-precision integers. Multi-precision integers differ from polynomials by the presence of carries, which poses difficulties in implementing Roche's scheme in multi-precision integers. This paper provides a detailed solution to these difficulties. Finally, numerical comparisons between the schoolbook, traditional Karatsuba, and space-efficient Karatsuba algorithms are provided.

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Yiping Cheng. 2016-05-22. Space-Efficient Karatsuba Multiplication for Multi-Precision Integers. https://arxiv.org/abs/1605.06760

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