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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…

Number theoretic transform (NTT) is the most efficient method for multiplying two polynomials of high degree with integer coefficients, due to its series of advantages in terms of algorithm and implementation, and is consequently…

The hard mathematical problems that assure the security of our current public-key cryptography (RSA, ECC) are broken if and when a quantum computer appears rendering them ineffective for use in the quantum era. Lattice based cryptography is…

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…

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…

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…

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…

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…

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…

Multiplication is an essential step in a lot of calculations. In this paper we look at multiplication of 2 binary polynomials of degree at most $n-1$, modulo an irreducible polynomial of degree $n$ with $2n$ input and $n$ output qubits,…

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…

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…

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…

We set new speed records for multiplying long polynomials over finite fields of characteristic two. Our multiplication algorithm is based on an additive FFT (Fast Fourier Transform) by Lin, Chung, and Huang in 2014 comparing to previously…

While the Karatsuba algorithm reduces the complexity of large integer multiplication, the extra additions required minimize its benefits for smaller integers of more commonly-used bitwidths. In this work, we propose the extension of the…

We present COPSIM a parallel implementation of standard integer multiplication for the distributed memory setting, and COPK a parallel implementation of Karatsuba's fast integer multiplication algorithm for a distributed memory setting.…

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…

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…

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,…

Code-based cryptography is one of the main propositions for the post-quantum cryptographic context, and several protocols of this kind have been submitted on the NIST platform. Among them, BIKE and HQC are part of the five alternate…