A reverse converter for the 4-moduli superset {2ⁿ-1, 2 ⁿ, 2ⁿ+1, 2ⁿ⁺¹+1}

The authors propose an extension to the popular {2ⁿ-1, 2ⁿ, 2ⁿ+1} moduli set by adding a fourth modulus 2ⁿ⁺¹+1. This extension leads to higher parallelism while keeping the forward conversion and modular arithmetic units simple. The main challenge of efficient reverse conversion is met by three techniques described for the first time. Firstly, we reverse convert linear combinations of moduli hence reducing the number of non-zero bits in the Booth encoded multiplicands from n to merely 2. Secondly, it is shown that division by 3, if introduced at the right stage, can be implemented very efficiently and can, in turn, reduce the cost of the converter. To implement VLSI efficient modulo reduction, we propose two techniques-multiple split tables (MST) and a modified division algorithm (MDA). It is shown that the MST can reduce exponential ROM requirements to quadratic ROM requirements while the MDA can reduce these further to linear requirements. As a result of these innovations, the proposed reverse converter uses simple shift and add operations and needs a lookup with only 6 entries. The delay of the converter is approximately 10n+13 full adder delays and the area cost is quadratic in n