FFT Algorithm for Binary Extension Finite Fields and its Application to Reed-Solomon Codes Sian-Jheng Lin, Member, IEEE, Tareq Y. Al-Naffouri, Member, IEEE, and Yunghsiang S. Han, Fellow, IEEE Abstract—Recently, a new polynomial basis over binary ex- specifically, since the practical implementations of RS codes tension fields was proposed such that the fast Fourier transform are typically over binary extension finite fields, the complexity (FFT) over such fields can be computed in the complexity of of RS codes over those fields has received more attentions than order O(n lg(n)), where n is the number of points evaluated in FFT. In this work, we reformulate this FFT algorithm such that it that over others [6][7]. can be easier understood and be extended to develop frequency- domain decoding algorithms for (n = 2m; k) systematic Reed- The conventional syndrome-based RS decoding algorithm + Solomon (RS) codes over F2m ; m 2 Z , with n − k a power of has quadratic complexities. Some fast approaches [8][9] are two. First, the basis of syndrome polynomials is reformulated in based on FFTs or fast polynomial arithmetic techniques. How- the decoding procedure so that the new transforms can be applied to the decoding procedure. A fast extended Euclidean algorithm ever, the structures of FFTs over finite fields vary with the sizes is developed to determine the error locator polynomial. The of fields Fq. When q−1 is a smooth number, meaning that q−1 computational complexity of the proposed decoding algorithm can be factorized into many small primes, the Cooley-Tucky is O(n lg(n − k) + (n − k) lg2(n − k)), improving upon the best FFT in O(n lg(n)) field additions and field multiplications 2 currently available decoding complexity O(n lg (n) lg lg(n)), and can be applied. A conventional case involves choosing Fermat reaching the best known complexity bound that was established primes q 2 f2m + 1jm = 1; 2; 4; 8; 16g. Based on such FFTs, by Justesen in 1976. However, Justesen’s approach is only for the 2 codes over some specific fields, which can apply Cooley-Tucky Justesen [8] gave an O(n lg (n)) approach for decoding (n; k) FFTs. As revealed by the computer simulations, the proposed RS code over F2m+1. Another approach to solve the key decoding algorithm is 50 times faster than the conventional one equations of BCH codes was proposed by Pan [10], and it 16 15 for the (2 ; 2 ) RS code over F216 . reduces a factor of lg n when the characteristic of the field is large enough. However, the algorithm [10] does not have I. INTRODUCTION improvement for the codes over binary extension fields. If Reed-Solomon (RS) codes are a class of block error- q − 1 is not smooth, Cooley-Tucky FFTs are inapplicable. correcting codes that were invented by Reed and Solomon [1] In this case, the FFTs over arbitrary fields [11][12] can be in 1960. An (n; k) RS code is constructed over , for applied and it requires O(n lg(n) lg lg(n)) field operations. Fq 2 n = q − 1. Its extended version, called extended Reed- Gao [9] presented an O(n lg (n) lg lg(n)) RS decoding algo- Solomon codes [2], admits a codeword length of up to n = q rithm over arbitrary fields, by utilizing fast polynomial multi- or n = q + 1. The systematic version of (n; k) RS code plications [13]. Further, for the codes over F2m , the additive appends n − k parity symbols to the k message symbols, FFT [14], that requires O(n lg(n) lg lg(n)) operations, can be forming a codeword of length n. RS codes are maximum applied to reduce the leading constant further. To authors’ distance separable (MDS). (n; k) RS codes can correct up to knowledge, the additive FFT [14] is the fastest algorithm over b(n − k)=2c erroneous symbols. Nowadays, RS codes have F2m so far. numerous important applications, including barcodes (such as As RS codes are typically constructed over binary extension arXiv:1503.05761v3 [cs.IT] 14 Aug 2016 QR codes), storage devices (such as Blu-ray Discs), digital television (such as DVB and ATSC), and data transmission fields, we consider this case in this paper. Clearly, if one technologies (such as DSL and WiMAX). RS codes are also wants to remove the extra factor lg lg(n) in the RS algorithms used to design other forward error correction codes, such as over binary extension fields, the FFTs in O(n lg(n)) are regenerating codes [3][4] and local reconstruction codes [5]. required. Recently, Lin et al. [15] showed a new way to solve The wide range of applications of RS codes raises an impor- aforementioned FFT problem. The paper [15] defined a new tant issue concerning their computational complexity. More polynomial basis based on subspace polynomials over F2m . For a polynomial of degree less than h in this new basis, This work was supported in part by CAS Pioneer Hundred Talents Program the h-point multipoint evaluations can be made in O(h lg(h)) and the National Science of Council (NSC) of Taiwan under Grants NSC 102- 2221-E-011-006-MY3, NSC 101-2221-E- 011-069-MY3. S.-J. Lin is with the field operations. Based on the multipoint evaluation algorithm, School of Information Science and Technology, University of Science and encoding/erasure decoding algorithms for (n; k) RS codes [15] Technology of China (USTC), Hefei, China and the Electrical Engineering were proposed to achieve O(n lg(n)). However, the error- Department, King Abdullah University of Science and Technology (KAUST), Kingdom of Saudi Arabia (e-mail: [email protected]), Tareq Y. Al-Naffouri correction RS decoding algorithm based on the new basis was is with the Electrial Engineering Department at King Abdullah University of not yet provided. Science and Technology (KAUST), Thuwal, Makkah Province, Kingdom of Saudi Arabia. (e-mail: [email protected]), and Y. Han is with the Department of Electrical Engineering, National Taiwan University of Science This paper develops an error correction decoding algorithm m and Technology, Taipei, Taiwan. (e-mail: [email protected]). for (n = 2 ; k) RS codes over F2m , for k=n ≥ 0:5 and (n−k) 1 1 a power of two. In practice RS codes usually have rates where vk = (v0; v1; : : : ; vk−1) is a basis of space Vk, and k=n ≥ 0:5. The complexity of the proposed algorithm is given k ≤ m. We can form a strictly ascending chain of subspaces by O(n lg(n − k) + (n − k) lg2(n − k)). Holding constant the given by code rate k=n yields a complexity O(n lg2(n)), which is better than the best existing complexity of O(n lg2(n) lg lg(n)), that f0g = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vm = F2m : was achieved by Gao [9] in 2002. The algorithm is based 2m−1 Let f!igi=0 denote the elements of F2m . Each element is on the non-standard polynomial basis [15]. To embed the defined as new basis into the decoding algorithm, we reformulate the decoding formulas such that all arithmetics are performed on !i = i0 · v0 + i1 · v1 + ··· + im−1 · vm−1; the new basis. The key equation is solved by the Euclidean where i 2 f0; 1g is the binary representation of i. That is, algorithm, and thus the fast polynomial divisions, as well j m−1 as the Euclidean algorithm in the new basis are proposed. i = i0 + i1 · 2 + ··· + im−1 · 2 ; 8ij 2 f0; 1g: Finally, we combine those algorithms, resulting in a fast error- 2k−1 correction RS decoding algorithm. The major contributions of This implies that Vk = f!igi=0 , for k = 0; 1; : : : ; m. Note this paper are summarized as follows. that !0 = 0 is the additive identity in the filed. In this work, !0 1) An alternative description of the algorithms [15] for the and 0 will be used interchangeably when there is no confusion. new polynomial basis is presented. The subspace polynomial [14, 16, 17] of Vk is defined as 2) An O(h lg(h)) fast polynomial division in the new basis Y sk(x) = (x − a); (2) is derived. a2Vk 3) An O(h lg2(h)) fast half-GCD algorithm in the new k basis is presented. and it is clear to see that deg(sk(x)) = 2 . For example, 4) An O(n lg(n − k)) RS encoding algorithm is presented, s0(x) = x, and s2(x) = x(x − v0)(x − v1)(x − v0 − v1). The for n − k a power of two. properties of sk(x) are given in [16, 18]. 5) A syndrome-based RS decoding algorithm that is based Theorem 1 ([16, 18]). (i). sk(x) is an F2-linearlized polyno- on the new basis is demonstrated. 2 mial for which 6) An O(n lg(n − k) + (n − k) lg (n − k)) RS decoding k X 2i algorithm is presented, for n − k a power of two. sk(x) = sk;ix ; (3) Notably, [15] gave the encoding algorithms for RS codes with i=0 O(n lg(k)) k the complexity , for a power of two. The encoding with each sk;i 2 F2m . This implies that algorithm [15] is suitable for coding rate k=n ≤ 0:5; however, the proposed encoding algorithm in this work is suitable for sk(x + y) = sk(x) + sk(y); 8x; y 2 F2m : (4) k=n ≥ 0:5. (ii). The formal derivative of s (x) is a constant The rest of this paper is organized as follows.