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CRYSTAL-Kyber (Kyber) is one of the post-quantum cryptography (PQC) key-encapsulation mechanism (KEM) schemes selected during the standardization process. This paper addresses optimization for Kyber architecture with respect to latency and throughput constraints. Specifically, matrix-vector multiplication and number theoretic transform (NTT)-based polynomial multiplication are critical operations and bottle-necks that require optimization. To address this challenge, we propose an algorithm and hardware co-design approach to systematically optimize matrix-vector multiplication and NTT-based polynomial multiplication by employing a novel sub-structure sharing technique in order to reduce computational complexity, i.e., the number of modular multiplications and modular additions/subtractions consumed. The sub-structure sharing approach is inspired by prior fast parallel approaches based on polyphase decomposition. The proposed efficient feed-forward architecture achieves high speed, low latency, and full utilization of all hardware components, which can significantly enhance the overall efficiency of the Kyber scheme. The FPGA implementation results show that our proposed design, using the fast two-parallel structure, leads to an approximate reduction of 90% in execution time (μs) , along with a 66× improvement in throughput performance.
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This content will become publicly available on August 10, 2026
Low-Complexity NTT and INTT Structures via Twiddle Shifting
Polynomial modular multiplication is an important operation used in post-quantum cryptography and homomorphic encryption, which are based on ring learning with errors (RLWE) problems. For long polynomial lengths, this operation can be efficiently computed using number theoretic transform (NTT) and inverse NTT (INTT). In particular, negative wrapped convolution (NWC) has been proposed to compute this operation where zero padding is eliminated. Low-complexity structures for NTT (LCNTT) and INTT (LC-INTT) have been derived in prior work by using a divide-and-conquer approach. This paper presents an alternate derivation of the LC-NTT and LC-INTT structures from traditional NTT and INTT structures. Specifically, we show that using twiddle factor pushing (pulling) from left to right (right to left), we can derive the prior LC-NTT (LC-INTT) structures. We present systematic algorithms for twiddle factor pushing and pulling to derive the equivalent architectures. The alternate approach may provide opportunities for optimizing hardware implementations of polynomial modular multiplication.
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- Award ID(s):
- 2243053
- PAR ID:
- 10649628
- Publisher / Repository:
- IEEE Midwest Symposium on Circuits and Systems
- Date Published:
- ISSN:
- 1558-3899
- Page Range / eLocation ID:
- 444 to 448
- Subject(s) / Keyword(s):
- Number theoretic transform (NTT) Polynomial multiplication. Twiddle shifting Post-quantum cryptography (PQC) Homomorphic encryption (HE)
- Format(s):
- Medium: X
- Location:
- Lansing, MI
- Sponsoring Org:
- National Science Foundation
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