Efficient Modulo-(2^(2n)+1) Arithmetic via Two Parallel n-bit Residue Channels

Core Concepts

The proposed complex-number moduli (2^n±j) provide efficient modulo-(2^(2n)+1) arithmetic operations in terms of delay, area, and power consumption compared to the traditional modulo-(2^(2n)+1) approach.

Abstract

The paper presents a new approach to performing modulo-(2^(2n)+1) arithmetic operations using a set of complex-number moduli (2^n±j) instead of the traditional modulo-(2^(2n)+1). The key highlights and insights are: The classic moduli set {2^n, 2^n-1, 2^n+1} has a limited dynamic range, covering up to 18-bit integers. The authors propose augmenting this set with the co-prime modulo 2^(2n)+1 to increase the dynamic range by around 70%. However, the double bit-width of the 2^(2n)+1 residue channel is counter-productive and jeopardizes the speed balance across the moduli set. To address this, the authors decompose 2^(2n)+1 into two complex-number n-bit moduli 2^n±j, which preserves the dynamic range and co-primality. The authors design efficient n-bit modulo-(2^n±j) adders and multipliers, which are tested and synthesized on a Spartan 7S100 FPGA. The 6-bit LUT-based realizations are optimized for n=5, providing a dynamic range of 2^25-2^5. Experiments show that the power-of-two channel m1=2^n can be as wide as 12 bits with no harm to the speed balance across the five moduli. The proposed modulo-(2^25±j) arithmetic operations outperform modulo-(2^5±1) and modulo-(2^10+1) in terms of speed, cost, and energy measures. The authors also provide comparative assessments of the proposed moduli set against other moduli sets used in previous DNN hardware accelerator studies, demonstrating the advantages of the complex-number moduli.

Stats

The dynamic range of the proposed 4-moduli set is [0, 2^(4n)-2^n). The dynamic range of the classic 3-moduli set {2^n, 2^n-1, 2^n+1} is [0, 2^(3n)-2^n).

Quotes

"Augmenting the balanced residue number system moduli-set {m1= 2^n, m2= 2^n-1, m3= 2^n+ 1}, with the co-prime modulo m4= 2^(2n)+ 1, increases the dynamic range (DR) by around 70%." "The Mersenne form of product m2 m3 m4= 2^(4n)-1, in the moduli-set {m1, m2, m3, m4}, leads to a very efficient reverse convertor, based on the New Chinese remainder theorem."

Key Insights Distilled From

Modulo-$(2^{2n}+1)$ Arithmetic via Two Parallel n-bit Residue Channels

by Ghassem Jabe... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08228.pdf

Deeper Inquiries

How can the proposed complex-number moduli be extended to support even larger dynamic ranges beyond 32-bit integers

To extend the proposed complex-number moduli to support even larger dynamic ranges beyond 32-bit integers, one approach could be to increase the number of moduli in the set. By introducing additional co-prime moduli that are powers of two or have specific relationships with powers of two, the dynamic range can be expanded further. For example, incorporating moduli like 2^64 or 2^128, along with complex-number moduli, would allow for the representation of much larger numbers. This expansion would require careful consideration of the balance between speed, area utilization, and power consumption in the hardware implementation.

What are the potential challenges and trade-offs in implementing the complex-number modular arithmetic in real-world DNN hardware accelerators

Implementing complex-number modular arithmetic in real-world DNN hardware accelerators presents several challenges and trade-offs. One challenge is the increased complexity of the arithmetic operations, which may impact the speed and efficiency of the accelerator. Additionally, the utilization of complex-number moduli could require specialized hardware components or algorithms, leading to higher development and maintenance costs. Trade-offs may include sacrificing some speed for the ability to handle more complex calculations or accepting higher power consumption for improved accuracy in the computations. Balancing these factors while ensuring compatibility with existing DNN frameworks and algorithms is crucial for successful integration.

Could the complex-number moduli approach be applied to other domains beyond DNN hardware, such as cryptography or signal processing, and what would be the implications

The application of the complex-number moduli approach is not limited to DNN hardware accelerators and can be extended to other domains such as cryptography or signal processing. In cryptography, the use of complex-number moduli could enhance the security and efficiency of encryption and decryption algorithms by introducing additional layers of complexity and randomness. In signal processing, complex-number moduli could enable more advanced and accurate processing of signals, especially in applications like radar systems or wireless communications. The implications of applying complex-number moduli in these domains would include improved performance, enhanced security, and the potential for developing novel algorithms and techniques based on the unique properties of complex numbers.