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High-speed long polynomial multiplication is important for applications in homomorphic encryption (HE) and lattice-based cryptosystems. This paper addresses low-latency hardware architectures for long polynomial modular multiplication using the number-theoretic transform (NTT) and inverse NTT (iNTT). Parallel NTT and iNTT architectures are proposed to reduce the number of clock cycles to process the polynomials. Chinese remainder theorem (CRT) is used to decompose the modulus into multiple smaller moduli. Our proposed architecture, namely PaReNTT, makes three novel contributions. First, cascaded parallel NTT and iNTT architectures are proposed such that any buffer requirement for permuting the product of the NTTs before it is input to the iNTT is eliminated. This is achieved by using different folding sets for the NTTs and iNTT. Second, a novel approach to expand the set of feasible special moduli is presented where the moduli can be expressed in terms of a few signed power-of-two terms. Third, novel architectures for pre-processing for computing residual polynomials using the CRT and post-processing for combining the residual polynomials are proposed. These architectures significantly reduce the area consumption of the pre-processing and post-processing steps. The proposed long modular polynomial multiplications are ideal for applications that require low latency and high sample rate such as in the cloud, as these feed-forward architectures can be pipelined at arbitrary levels. Pipelining and latency tradeoffs are also investigated. Compared to a prior design, the proposed architecture reduces latency by a factor of 49.2, and the area-time products (ATP) for the lookup table and DSP, ATP(LUT) and ATP(DSP), respectively, by 89.2% and 92.5%. Specifically, we show that for n =4096 and a 180-bit coefficient, the proposed 2-parallel architecture requires 6.3 Watts of power while operating at 240 MHz, with 6 moduli, each of length 30 bits, using Xilinx Virtex Ultrascale+ FPGA.
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Architectural Tradeoffs for Long Polynomial Modular Multiplication
Polynomial multiplication over the quotient ring is a critical operation in Ring Learning with Errors (Ring-LWE) based cryptosystems that are used for post-quantum cryptography and homomorphic encryption. This operation can be efficiently implemented using number-theoretic transform (NTT)-based architectures. Among these, pipelined parallel NTTbased polynomial multipliers are attractive for cloud computing as these are well suited for high throughput and low latency applications. For a given polynomial length, a pipelined parallel NTT-based multiplier can be designed with varying degrees of parallelism, resulting in different tradeoffs. Higher parallelism reduces latency but increases area and power consumption,and vice versa. In this paper, we develop a predictive model based on synthesized results for pipelined parallel NTT-based polynomial multipliers and analyze design tradeoffs in terms of area, power, energy, area-time product, and area-energy product across parallelism levels up to 128. We predict that, for very long polynomials, area and power differences between designs with varying levels of parallelism become negligible. In contrast, areatime product and energy per polynomial multiplication decrease with increased parallelism. Our findings suggest that, given area and power constraints, the highest feasible level of parallelism optimizes latency, area-time product, and energy per polynomial multiplication.
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- Award ID(s):
- 2243053
- PAR ID:
- 10581266
- Publisher / Repository:
- IEEE
- Date Published:
- ISBN:
- 979-8-3503-5405-8
- Page Range / eLocation ID:
- 1797 to 1801
- Subject(s) / Keyword(s):
- Post-quantum cryptography (PQC) number theoretic transform (NTT) polynomial multiplication predictive model parallel processing pipelining folding
- Format(s):
- Medium: X
- Location:
- Pacific Grove, CA, USA
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/IEEECONF60004.2024.10942651
