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

##### Sian-Jheng Lin, Wei-Ho Chung, Yunghsiang S. Han

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.

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