Novel Polynomial Basis and Its Application to Reed-Solomon Erasure Codes

Abstract

Abstract-In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that h-point polynomial evaluation can be computed in O(h log 2 (h)) finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from O(h log 2 (h) log 2 log 2 (h)) to O(h log 2 (h)). Based on this basis, we then develop the encoding and erasure decoding algorithms for the (n = 2 r , k) Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in O(n log 2 (k)) finite field operations, and the erasure decoding in O(n log 2 (n)) finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of O(n log 2 (n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.

Fast Hermite interpolation and evaluation over finite fields of characteristic two

“…Consequently, we omit its proof. The choice of reduction trees provided by the proposition captures the strategy used in recent algorithms [14,21,20] for an arbitrary choice of subspace basis. Accordingly, we use such trees as our baseline for comparison.…”

“…Gao and Mateer [14] also contribute to this special case by providing an algorithm that achieves the same multiplicative complexity while performing only O( (log ) log log ) additions. For the same subspace enumeration used in the additive FFTs, Lin, Chung and Han [21] provide algorithms for converting between the Lagrange and LCH bases that perform O( log ) additions and multiplications. Lin, Chung and Han use their basis and conversion algorithms to provide fast encoding and decoding algorithms for Reed-Solomon codes.…”

“…The bound for conversion with the Newton basis is new. For conversion with the Lagrange basis, we only have the algorithm of Lin, Chung and Han [21] to compare with, and only for the case = 2 n . Their algorithm performs fewer additions in this case, but only after a much larger precomputation, of unanalysed complexity, that stores 2 n − 1 field elements in auxiliary space.…”

Frobenius Additive Fast Fourier Transform

“…Algorithm 1: Basis conversion: monomial to novel polynomial basis constructed from Cantor basis 2.6 Additive FFT Given a polynomial P represented in novel polynomial basis, Lin, Chung and Han [LCH14] proposed a fast method to compute its additive Fourier transform.…”

Coding for Improved Throughput Performance in Network Switches

“…An RS code exists for every choice of k ≤ n ≤ q, where q is the code alphabet size, which is a prime power. RS encoding/decoding can be performed efficiently [10], [12].…”