Efficient Binary Polynomial Multiplication Based on Optimized Karatsuba Reconstruction

Christophe Negre 1
1 DALI - Digits, Architectures et Logiciels Informatiques
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier, UPVD - Université de Perpignan Via Domitia
Abstract : At Crypto 2009 Bernstein proposed two optimized Karatsuba formulas for binary polynomial multiplication. Bernstein obtained these optimizations by re-expressing the reconstruction of one or two recursions of the Karatsuba formula. In this paper we present a generalization of these optimizations. Specifically, we optimize the reconstruction of s recursions of the Karatsuba formula for s >= 1. To reach this goal, we express the recursive reconstruction through a tree and re-organize this tree to derive an optimized recursive reconstruction of depth $s$. When we apply this approach to a recursion of depth s=\log_2(n) we obtain a parallel multiplier with a space complexity of 5.25 n^(\log_2(3))+O(n) XOR gates and n^(\log_2(3)) AND gates and with a delay of 2log_2(n) D_X+D_A.
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Christophe Negre. Efficient Binary Polynomial Multiplication Based on Optimized Karatsuba Reconstruction. Journal of Cryptographic Engineering, Springer, 2014, 4 (2), pp.91--106. ⟨10.1007/s13389-013-0066-2⟩. ⟨hal-00724778⟩

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