RCU- 2^<i>m</i>: A VLSI Radix- 2^<i>m</i> Cubic Unit

Cubic operations are among the most used arithmetic operations in many applications that demand higher order simultaneous operand computation, such as cryptography and bicubic polynomial interpolation. This article proposes a novel VLSI radix-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula> cubic unit (RCU-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula>) capable of processing cubic operations at m bits simultaneously, with m values of 2 (RCU-4), 3 (RCU-8), and 4 (RCU-16). RCU-16 emerges as the most area-efficient configuration, surpassing RCU-8 and notably outperforming RCU-4. In the 8-bit scenario, RCU-16 achieves remarkable area savings, surpassing the literature’s proposed cubic unit by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$11.58\times $ </tex-math></inline-formula>. Across all configurations, RCU-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula> consistently outperforms the automatically selected cube unit, with energy savings ranging from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.04\times $ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\times $ </tex-math></inline-formula>. In application specific integrated circuit (ASIC) and field-programmable gate array (FPGA)-based analyses, RCU-16 consistently exhibits superior performance in both area and energy savings compared with RCU-4, RCU-8, and solutions from the literature. These findings emphasize the importance of adopting radix-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m}$ </tex-math></inline-formula> configurations, particularly RCU-16, for optimal energy-constrained VLSI applications.