A Generalization of Addition Chains and Fast Inversions in Binary Fields

In this paper, we study a generalization of addition chains where <inline-formula><tex-math>$k$</tex-math> <alternatives><inline-graphic xlink:type="simple" xlink:href="jarvinen-ieq1-2375182.gif"/></alternatives></inline-formula> previous values are summed together on each step instead of only two values as in traditional addition chains. Such chains are called <inline-formula><tex-math>$k$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="jarvinen-ieq2-2375182.gif"/> </alternatives></inline-formula>-chains and we show that they have applications in finding efficient parallelizations in problems that are known to be difficult to parallelize. In particular, 3-chains improve computations of inversions in finite fields using hybrid-double multipliers. Recently, it was shown that this operation can be efficiently computed using a ternary algorithm but we show that 3-chains provide a significantly more efficient solution.

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