The Time Required for Group Multiplication

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Abstract
Winograd has considered the time necessary to perform numerical addition and multiplication and to perform group multiplication by means of logical circuits consisting of elements each having a limited number of input lines and unit delay in computing their outputs. In this paper the same model as he employed is adopted, but a new lower bound is derived for group multiplication—the same as Winograd's for an Abelian group but in general stronger. Also a circuit is given to compute the multiplication which, in contrast to Winograd's, can be used for non-Abelian groups. When the group of interest is Abelian the circuit is at least as fast as his. By paralleling his method of application of his Abelian group circuit, it is possible also to lower the time necessary for numerical addition and multiplication.

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