Hybrid Montgomery Reduction

In this article, we present a hybrid method to improve the performance of the Montgomery reduction by taking advantage of the Karatsuba technique. We divide the Montgomery reduction into two sub-parts, including one for the conventional Montgomery reduction and the other one for Karatsuba-aided multiplication. This approach reduces the multiplication complexity of <i>n</i>-limb Montgomery reduction from θ(<i>n</i><sup>2</sup> + <i>n</i>) to asymptotic complexity θ (7<i>n</i><sup>2</sup>/8 +<i>n</i>). Our practical implementation results over an 8-bit microcontroller also show performance enhancements by 11%.

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