Efficient Binary Polynomial Multiplication Based on Optimized Karatsuba Reconstruction

Christophe Negre (1)

1 DALI - Digits, Architectures et Logiciels Informatiques

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.

Mots clés: Binary polynomial multiplication; optimized recursive reconstruction; Karatsuba formula; optimized recursive reconstruction.

Domaines: Arithmétique des ordinateurs; Cryptographie et sécurité [cs.CR]; Calcul formel [cs.SC]

Christophe Negre. Efficient Binary Polynomial Multiplication Based on Optimized Karatsuba Reconstruction. Journal of Cryptographic Engineering, 2014, 4 (2), pp.91--106. ⟨10.1007/s13389-013-0066-2⟩. ⟨hal-00724778⟩