Related papers: Explicit formulas for efficient multiplication in …
Matrix multiplication (hereafter we use the acronym MM) is among the most fundamental operations of modern computations. The efficiency of its performance depends on various factors, in particular vectorization, data movement and arithmetic…
In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P in GF(q)[x] of degree <n at all n-th roots of unity in GF($q^d$) can essentially be computed d-time faster than evaluating Q in GF($q^d$)[x] at all these roots,…
Large neural networks spend most computation on floating point tensor multiplications. In this work, we find that a floating point multiplier can be approximated by one integer adder with high precision. We propose the linear-complexity…
In this study, we propose a simple method for fault-tolerant Strassen-like matrix multiplications. The proposed method is based on using two distinct Strassen-like algorithms instead of replicating a given one. We have realized that using…
This paper will describe a simulator developed by the authors to explore the design of Fourier transform based multiplication using optics. Then it will demonstrate an application to the problem of constructing an all-optical modular…
It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3 x 3-matrices. It is not known whether this can also be done with fewer multiplications. However, there are several mutually…
Current LLM structured pruning methods typically involve two steps: (1) compression with calibration data and (2) costly continued pretraining on billions of tokens to recover lost performance. This second step is necessary as the first…
Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two polynomials in F_p[X] of degree less than n. For n large compared to p, we establish the bound M_p(n) = O(n log n 8^(log^* n) log p), where log^* is the iterated…
The rapid scaling of large language models demands more efficient hardware. Quantization offers a promising trade-off between efficiency and performance. With ultra-low-bit quantization, there are abundant opportunities for results reuse,…
Tensor decomposition methodologies are proposed to reduce the memory requirement of translation operator tensors arising in the fast multipole method-fast Fourier transform (FMM-FFT)-accelerated surface integral equation (SIE) simulators.…
The plane wave method is most widely used for solving the Kohn-Sham equations in first-principles materials science computations. In this procedure, the three-dimensional (3-dim) trial wave functions' fast Fourier transform (FFT) is a…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes…
An algorithm is given to factor an integer with $N$ digits in $\ln^m N$ steps, with $m$ approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a…
Efficient multi-party secure matrix multiplication is crucial for privacy-preserving machine learning, but existing mixed-protocol frameworks often face challenges in balancing security, efficiency, and accuracy. This paper presents an…
The currently fastest algorithm for regular expression pattern matching and membership improves the classical O(nm) time algorithm by a factor of about log^{3/2}n. Instead of focussing on general patterns we analyse homogeneous patterns of…
Moosbauer and Poole have recently shown that the multiplication of two $5\times 5$ matrices requires no more than 93 multiplications in the (possibly non-commutative) coefficient ring, and that the multiplication of two $6\times 6$ matrices…
We show that multiplication can be done in polynomial time on a three counter machine that receives its input as the contents of two counters. The technique is generalized to functions of two variables computable by deterministic Turing…
This paper describes several new improvements of modular arithmetic and how to exploit them in order to gain more efficient implementations of commonly used algorithms, especially in cryptographic applications. We further present a new…
Cyclotomic fast Fourier transforms (CFFTs) are efficient implementations of discrete Fourier transforms over finite fields, which have widespread applications in cryptography and error control codes. They are of great interest because of…