Multiplier Efficiency: A Hidden Strength of Modular Computation

Among the many advantages of modular computation, one of the most overlooked is multiplier efficiency. While often ignored by software developers, this issue is critical for FPGA, ASIC, and custom IC designers, especially when trying to pack as many multipliers as possible onto a single die.

Multipliers are among the most resource-intensive components in hardware design. In binary arithmetic, the complexity of a multiplier grows quadratically with precision. That’s because binary multipliers generate partial products, and the number of these products increases as the square of the input width. For example, building an 18-bit binary multiplier from 9×9 unit multipliers requires four such units. A 36-bit multiplier may require as many as sixteen. This quadratic growth—O(n²)—quickly consumes area and power.

By contrast, residue number system (RNS) multipliers are far more efficient. Since RNS arithmetic eliminates carry propagation between digits, it avoids partial product expansion entirely. Instead, each modular digit multiplier operates independently, and the total multiplier width scales linearly with precision—O(n). For instance, an 8-bit modular multiplier supporting a fractional range might require two 9-bit units, a 16-bit multiplier requires four, and a 32-bit multiplier requires eight. This linear scaling makes it far easier to implement high-throughput designs using modular arithmetic.

Early-generation FPGAs like Intel’s Cyclone-IV and Stratix-IV, which expose their internal 9×9 and 18×18 unit multipliers, clearly highlight the difference in multiplier efficiency between binary and modular arithmetic. Unlike modern FPGAs with highly integrated and opaque DSP blocks, these older platforms allow for direct measurement and comparison of multiplier resource usage. At Maitrix, we leverage these FPGA devices to conduct accurate evaluations of resource requirements for modular versus binary computation, and our findings consistently show that modular multipliers offer superior efficiency in both area and scalability.

The implications go beyond area. Power consumption is also significantly lower in modular designs, thanks to simpler logic and fewer switching events. This means more RNS multipliers can be integrated per chip, delivering greater parallelism and performance per watt than binary architectures.

In short, multiplier efficiency in modular computation isn’t just a bonus—it’s a fundamental architectural win that makes modular arithmetic especially attractive for modern, high-density computing applications.

There’s always more to the story when it comes to modular arithmetic!