Simultaneous Conversions with the Residue Number System Using Linear Algebra

Abstract

We present an algorithm for simultaneous conversion between a given set of integers and their Residue Number System representations based on linear algebra. We provide a highly optimized implementation of the algorithm that exploits the computational features of modern processors. The main application of our algorithm is matrix multiplication over integers. Our speed-up of the conversions to and from the Residue Number System significantly improves the overall running time of matrix multiplication.

Faster Arbitrary-Precision Dot Product and Matrix Multiplication

“…In addition to the ideas for algorithmic improvements already noted in this paper, we point out that Arb would benefit from faster integer matrix multiplication in FLINT. More than a factor two can be gained with better residue conversion code and use of BLAS [3], [6]. BLAS could also be used for the radius matrix multiplications in Arb (we currently use simple C code since the FLINT multiplications are the bottleneck).…”

On the Complexity of Symbolic Computation

“…There exist numerous important schemes: fast Fourier transforms [25], Karatsuba and Toom-Cook transforms [83,116], Chinese remaindering [26,61], Nussbaumer (or Schönhage-Strassen) polynomial transforms [99,113], truncated Fourier transforms [55], Chudnovsky 2 evaluations and interpolations [21], and so on.…”

Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

“…• O(n ω d + n 2 d log(d)) if K supports FFT in degree 2d. We also mention a polynomial analogue of an integer matrix multiplication algorithm from [10] which uses evaluation/interpolation, done plainly via multiplication by (inverse) Vandermonde matrices. Then, the corresponding part of the cost (e.g.…”

“…For FFT primes, we use evaluation/interpolation at roots of unity. For general primes, we use either evaluation/interpolation at geometric progressions (if such points exist in F p ), or our adaptation of the algorithm of [10], or 3-primes multiplication (as for polynomials, we lift the product from F p [x] to Z[x], where it is done modulo up to 3 FFT primes). No single variant outperformed or underperformed all others for all sizes and degrees, so thresholds were experimentally determined to switch between these options, with different values for small (less than 23 bits) and for large primes.…”

“…For FFT primes, we use evaluation/interpolation at roots of unity. For general primes, we use either evaluation/interpolation at geometric progressions (if such points exist in F p ), or our adaptation of the algorithm of [10], or 3-primes multiplication (as for polynomials, we lift the product from…”