High Precision Multiplier for RNS {2ⁿ−1,2ⁿ,2ⁿ+1}

• Main Authors: Shang Ma, Shuai Hu, Zeguo Yang, Xuesi Wang, Meiqing Liu, Jianhao Hu
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Electronics
• Subjects: residue number system; scaler; multiplier; Chinese remainder theorem; very large scale integration circuits
• Online Access: https://www.mdpi.com/2079-9292/10/9/1113

The Residue Number System (RNS) is a non-weighted number system. Benefiting from its inherent parallelism, RNS has been widely studied and used in Digital Signal Processing (DSP) systems and cryptography. However, since the dynamic range in RNS has been fixed by its moduli set, it is hard to solve the overflow problem, which can be easily solved in Two’s Complement System (TCS) by expanding the bit-width of it. For the multiplication in RNS, the traditional way to deal with overflow is to scale down the inputs so that the result can fall in its dynamic range. However, it leads to a loss of precision. In this paper, we propose a high-precision RNS multiplier for three-moduli set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="{" close="}"><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>,</mo><msup><mn>2</mn><mi>n</mi></msup><mo>,</mo><msup><mn>2</mn><mi>n</mi></msup><mo>+</mo><mn>1</mn></mfenced></semantics></math></inline-formula>, which is the most used moduli set. The proposed multiplier effectively improves the calculation precision by adding several compensatory items to the result. The compensatory items can be obtained directly from preceding scalers with little extra effort. To the best of our knowledge, we are the first one to propose a high-precision RNS multiplier for the moduli set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="{" close="}"><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo>,</mo><msup><mn>2</mn><mi>n</mi></msup><mo>,</mo><msup><mn>2</mn><mi>n</mi></msup><mo>+</mo><mn>1</mn></mfenced></semantics></math></inline-formula>. Simulation results show that the proposed RNS multiplier can get almost the same calculation precision as the TCS multiplier with respect to Mean Square Error (MSE) and Signal-to-Noise Ratio(SNR), which outperforms the basic scaling RNS multiplier about 2.6–3 times with respect to SNR.