Efficient parallel exponentiation in GF(2n) using normal basis representations

Vonzur Gathen proposed an efficient parallel exponentiation algorithm in finite fields using normal basis representations. In this paper we present a processor-efficient parallel exponentiation algorithm in <i>GF</i>(2<sup> <i>n</i> </sup>) which improves upon von zur Gathen's algorithm. We also show that exponentiation in <i>GF</i>(2<sup> <i>n</i> </sup>) can be done in <i>&Ogr;</i>(log <i>n</i>) time using <i>n</i>/(log <i>n</i>)<sup>2</sup> processors. Hence we get processor x time bound <i>&Ogr;</i>(<i>n</i>/log <i>n</i>), which is optimal. Finally, we present an on-line processor assignment scheme which was missing in von zur Gathen's algorithm, and show that its time complexity is negligible.

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