Decoding Generalized Reed-Solomon Codes and Its Application to RLCE Encryption Scheme

Abstract

This paper compares the efficiency of various algorithms for implementing public key encryption scheme RLCE on 64-bit CPUs. By optimizing various algorithms for polynomial and matrix operations over finite fields, we obtained several interesting (or even surprising) results. For example, it is well known (e.g., Moenck 1976) that Karatsuba\u27s algorithm outperforms classical polynomial multiplication algorithm from the degree 15 and above (practically, Karatsuba\u27s algorithm only outperforms classical polynomial multiplication algorithm from the degree 35 and above ). Our experiments show that 64-bit optimized Karatsuba\u27s algorithm will only outperform 64-bit optimized classical polynomial multiplication algorithm for polynomials of degree 115 and above over finite field $GF(2^{10})$. The second interesting (surprising) result shows that 64-bit optimized Chien\u27s search algorithm ourperforms all other 64-bit optimized polynomial root finding algorithms such as BTA and FFT for polynomials of all degrees over finite field $GF(2^{10})$. The third interesting (surprising) result shows that 64-bit optimized Strassen matrix multiplication algorithm only outperforms 64-bit optimized classical matrix multiplication algorithm for matrices of dimension 750 and above over finite field $GF(2^{10})$. It should be noted that existing literatures and practices recommend Strassen matrix multiplication algorithm for matrices of dimension 40 and above. All experiments are done on a 64-bit MacBook Pro with i7 CPU with a single thread. The reported results should be appliable to 64 or larger bits CPU. For 32 or smaller bits CPUs, these results may not be applicable. The source code and library for the algorithms covered in this paper will be available at http://quantumca.org/