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Paper Abstract and Keywords
Presentation 2023-06-09 14:30
An Extension of the Karatsuba Algorithm in Case that the Bit Widths of the Multiplicand and Multiplier are Different
Susumu Yamazaki (Univ. of Kitakyushu) RECONF2023-11
Abstract (in Japanese) (See Japanese page) 
(in English) The order of the number of multiplications required for bignum multiplication is O(n2) for the usual method and O(n log_2 3) for the Karatsuba algorithm.However, the conventional Karatsuba algorithm requires that the bit widths of the multiplicand and the multiplier be the same. On the other hand, DSP slices such as Xilinx DSP48E1 have different bit widths, with a multiplicand of 25 bits and a multiplier of 18 bits. Therefore, it was not possible to utilize all of those effective bits.

In this research, we propose an extension the Karatsuba algorithm to be able to use a DSP slice multiplier such as DSP48E1 where the bit widths of the multiplicand and multiplier are different. The proposed method makes it possible to reduce the number of multiplications to three when the larger number of bits of the multiplicand and multiplier of integer multiplication is less than or equal to the number of bits obtained by subtracting 2 from the sum of the multiplicand and multiplier of the DSP slice, and when the smaller number of bits of the multiplicand and the multiplier for integer multiplication is less than or equal to twice the value obtained by subtracting 1 from the smaller number of bits of the multiplicand and multiplierof the DSP slice. Moreover, by applying the proposed method recursively, it can be applied to any bignum multiplication. Our evaluation shows that the proposed method can reduce the number of multiplications by about 4 percent compared to the Karatsuba method when the multiplicand equals the multiplier. Also it shows that the proposed method can reduce the number of multiplications by half compared to the Karatsuba method for the most efficient combination of multiplicand and multiplier.
Keyword (in Japanese) (See Japanese page) 
(in English) Integer multiplication / Optimization of number of multiplications / The Karatsuba algorithm / / / / /  
Reference Info. IEICE Tech. Rep., vol. 123, no. 71, RECONF2023-11, pp. 58-61, June 2023.
Paper # RECONF2023-11 
Date of Issue 2023-06-01 (RECONF) 
ISSN Online edition: ISSN 2432-6380
Copyright
and
reproduction		 All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034)
Notes on Review This article is a technical report without peer review, and its polished version will be published elsewhere.
Download PDF RECONF2023-11

Conference Information
Committee RECONF  
Conference Date 2023-06-08 - 2023-06-09 
Place (in Japanese) (See Japanese page) 
Place (in English) Eikokuji Campus, Kochi University of Technology 
Topics (in Japanese) (See Japanese page) 
Topics (in English) Reconfigurable system, etc. 
Paper Information
Registration To RECONF 
Conference Code 2023-06-RECONF 
Language Japanese 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) An Extension of the Karatsuba Algorithm in Case that the Bit Widths of the Multiplicand and Multiplier are Different 
Sub Title (in English)  
Keyword(1) Integer multiplication  
Keyword(2) Optimization of number of multiplications  
Keyword(3) The Karatsuba algorithm  
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1st Author's Name Susumu Yamazaki  
1st Author's Affiliation University of Kitakyushu (Univ. of Kitakyushu)
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Speaker Author-1 
Date Time 2023-06-09 14:30:00 
Presentation Time 25 minutes 
Registration for RECONF 
Paper # RECONF2023-11 
Volume (vol) vol.123 
Number (no) no.71 
Page pp.58-61 
#Pages 4 
Date of Issue 2023-06-01 (RECONF) 

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