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Randomized OBDDs for the most significant bit of multiplication need exponential size
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
On the OBDD complexity of the most significant bit of integer multiplication
Theoretical Computer Science
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CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
A larger lower bound on the OBDD complexity of the most significant bit of multiplication
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Journal of the ACM (JACM)
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ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We give an O(N • log N • 2O(log*N)) algorithm for multiplying two N-bit integers that improves the O(N • log N • log log N) algorithm by Schönhage-Strassen. Both these algorithms use modular arithmetic. Recently, Fürer gave an O(N • log N • 2O(log*N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürer's algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of Fürer's algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.