Table lookup structures for multiplicative inverses modulo 2k

We introduce an inheritance property and related table lookup structures applicable to simplified evaluation of the modular operations "multiplicative inverse", "discrete log", and "exponential residue" in the particular modulus 2^k. Regarding applications, we describe an integer representation system of Benschop for transforming integer multiplications into additions which benefits from our table lookup function evaluation procedures. We focus herein on the multiplicative inverse modulo 2^k to exhibit simplifications in hardware implementations realized from the inheritance property. A table lookup structure given by a bit string that can be interpreted with reference to a binary tree is described and analyzed. Using observed symmetries, the lookup structure size is reduced allowing a novel direct lookup process for multiplicative inverses for all 16-bit odd integers to be obtained from a table of size less than two KBytes. The 16-bit multiplicative inverse operation is also applicable for providing a seed inverse for obtaining 32/64-bit multiplicative inverses by one/two iterations of a known quadratic refinement algorithm.