Efficient Algorithm for Finding Roots of Error-Locator Polynomials

Abstract

A novel method for finding roots of polynomials over finite fields has been proposed. This method is based on the cyclotomic discrete Fourier transform algorithm. The improvement is achieved by using the normalized cyclic convolutions, which have a small complexity and allow matrix decomposition, as well as methods of adapting the truncated normalized cyclic convolutions calculation. For small values of degree of the error-locator polynomial the novel method has not only the smallest multiplicative complexity, but the full computational complexity of this method is also less than with other methods. Thus, the multiplicative complexity of the novel method in comparison to the method of affine decomposition (the Fedorenko-Trifonov method) is up to ten times less, although the additive complexity is approximately 10-15% more. The novel method has matrix representation convenient for implementation.INDEX TERMS Convolution, decoding, discrete Fourier transforms, error correction codes, fast Fourier transforms, Galois fields, Reed-Solomon codes.

A Spectral Algorithm for Decoding Systematic BCH Codes

“…The new review of the best methods for finding roots of polynomials over finite field GF (2 m ) is published in [16].…”