Related papers: Number Theoretic Transform and Its Applications in…
The Number Theoretic Transform (NTT) can be regarded as a variant of the Discrete Fourier Transform. NTT has been quite a powerful mathematical tool in developing Post-Quantum Cryptography and Homomorphic Encryption. The Fourier Transform…
Number Theoretic Transform (NTT) is an essential mathematical tool for computing polynomial multiplication in promising lattice-based cryptography. However, costly division operations and complex data dependencies make efficient and…
The Number Theoretic Transform (NTT) is an indispensable tool for computing efficient polynomial multiplications in post-quantum lattice-based cryptography. It has strong resemblance with the Fast Fourier Transform (FFT), which is the most…
Homomorphic encryption (HE) draws huge attention as it provides a way of privacy-preserving computations on encrypted messages. Number Theoretic Transform (NTT), a specialized form of Discrete Fourier Transform (DFT) in the finite field of…
Lattice-based cryptography (LBC) exploiting Learning with Errors (LWE) problems is a promising candidate for post-quantum cryptography. Number theoretic transform (NTT) is the latency- and energy- dominant process in the computation of LWE…
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…
This tutorial aims to establish connections between polynomial modular multiplication over a ring to circular convolution and discrete Fourier transform (DFT). The main goal is to extend the well-known theory of DFT in signal processing…
Polynomial multiplication is one of the fundamental operations in many applications, such as fully homomorphic encryption (FHE). However, the computational inefficiency stemming from polynomials with many large-bit coefficients poses a…
Polynomial multiplication is fundamental in lattice-based cryptography. While the Number Theoretic Transform (NTT) enables fast multiplication, it imposes constraints on the modulus of the coefficient field. Hafiz et al. (2025) addressed…
The Number Theoretic Transform (NTT) is a critical computational bottleneck in many lattice-based postquantum cryptographic (PQC) algorithms. By leveraging the Fast Fourier Transform (FFT) algorithm, the NTT of a polynomial of degree N - 1…
Polynomial multiplication stands out as a highly demanding arithmetic process in the development of post-quantum cryptosystems. The importance of the number-theoretic transform (NTT) extends beyond post-quantum cryptosystems, proving…
A New Number Theoretic Transform(NTT), which is a form of FFT, is introduced, that is faster than FFTs. Also, a multiplication algorithm is introduced that uses this to perform integer multiplication faster than O(n log n). It uses…
Multimedia contents are inherently sensitive signals that must be protected whenever they are outsourced to an untrusted environment. This problem becomes a challenge when the untrusted environment must perform some processing on the…
Large-degree polynomial multiplication is an integral component of post-quantum secure lattice-based cryptographic algorithms like CRYSTALS-Kyber and Dilithium. The computational complexity of large-degree polynomial multiplication can be…
The Number Theoretic Transform (NTT) is a fundamental operation in privacy-preserving technologies, particularly within fully homomorphic encryption (FHE). The efficiency of NTT computation directly impacts the overall performance of FHE,…
This paper presents digital hardware for computing polynomial multiplication using Number Theoretic Transform (NTT), specifically designed for implementation on Field Programmable Gate Arrays (FPGAs). Multiplying two large polynomials…
We demonstrate a multiplication method based on numbers represented as set of polynomial radix 2 indices stored as an integer list. The 'polynomial integer index multiplication' method is a set of algorithms implemented in python code. We…
Decimal multiplication is the task of multiplying two numbers in base $10^N.$ Specifically, we focus on the number-theoretic transform (NTT) family of algorithms. Using only portable techniques, we achieve a 3x-5x speedup over the mpdecimal…
Homomorphic Encryption (HE) enables users to securely outsource both the storage and computation of sensitive data to untrusted servers. Not only does HE offer an attractive solution for security in cloud systems, but lattice-based HE…
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…