Duality between Multiplication and Modular Reduction

This paper presents a duality between the classical optimally speeded up multiplication algorithm and some "fast" reduction algo- rithm. For this, the multiplier is represented by the unique signed digit representation with minimal Hamming weight using Reitwiesner's mul- tiplier recoding algorithm. In fact, the present paper proves that this optimal multiplier recoding technique naturally translates into a canoni- cal modular reduction technique. Thus, the resulting reduction algorithm is optimal with respect to its average-time complexity as well. Besides these two new results, our proof of the transfer-theorem serves another interesting purpose: The reason that the considered reduction algorithm from (Sed) is so unknown might lie in the fact that it is rather un-intuitive and no proper understanding was available so far. Therefore, our proper mathematical derivation/explanation solves this lack of understanding.