Balanced Modular Addition for the Moduli Set $ \{2^{q},2^{q}\mp 1,2^{2q}+1\}${2q,2q∓1,22q+1} via Moduli-($ 2^{q}\mp \sqrt{-1}$2q∓-1) Adders

Abstract

Moduli-set $ \mathbf{\tau}=\{2^{\boldsymbol{q}},2^{\boldsymbol{q}}\pm 1\}$ is often the base of choice for realization of digital computations via residue number systems. The optimum arithmetic performance in parallel residue channels, is generally achieved via equal bit-width residues (e.g., $ \boldsymbol{q}~ \mathbf{i}\mathbf{n}~ \mathbf{\tau}$ ) that usually leads to equal computation speed within all the residue channels. However, the commonly difficult and costly task of reverse conversion (RC) is often eased in the existence of conjugate moduli. For example, $ 2^{\boldsymbol{q}}\mp 1\in \mathbf{\tau}$ , lead to the efficient modulo-( $ 2^{2\boldsymbol{q}}-1$ ) addition, as the bulk of $ \mathbf{\tau}$ -RC, via the New-CRT reverse conversion method. Nevertheless, for additional dynamic range, $ \mathbf{\tau}$ is augmented with other moduli. In particular, $ \mathbf{\phi}=\mathbf{\tau}\cup \{2^{2\boldsymbol{q}}+1\}$ , leads to efficient RC, where the added modulo is conjugate with the product $ 2^{2\boldsymbol{q}}-1$ of $ 2^{\boldsymbol{q}}\mp 1\in \mathbf{\tau}$ . Therefore, the final step of $ \mathbf{\phi}$ -RC would be fast and low cost/power modulo-( $ 2^{4\boldsymbol{q}}-1$ ) addition. However, the $ 2\boldsymbol{q}$ -bit channel-width jeopardizes the existing delay-balance in $ \mathbf{\tau}$ . As a remedial solution, given that $ 2^{2\boldsymbol{q}}+1=\left(2^{\boldsymbol{q}}-\boldsymbol{j}\right)\left(2^{\boldsymbol{q}}+\boldsymbol{j}\right)$ , with $ \boldsymbol{j}=\sqrt{-1}$ , we design and implement modulo-( $ 2^{2\boldsymbol{q}}+1$ ) adders via two parallel $ \boldsymbol{q}$ -bit moduli-( $ 2^{\boldsymbol{q}}\mp \boldsymbol{j}$ ) adders. The analytical and synthesis based evaluations of the proposed modulo-( $ 2^{\boldsymbol{q}}\mp \boldsymbol{j}$ ) adders show that the delay-balance of $ \mathbf{\tau}$ is preserved with no cost overhead vs. $ \mathbf{\phi}$ . In particular, the binary-to-complex and complex-to-binary convertors are merely cost-free and immediate.