Abstract
Efficient encoding and decoding algorithms of the error correcting codes and the cryptosystems have been extensively studied. In general their algorithms are operated on finite fields, polynomial rings and integer rings. Therefore it is very important and interesting to investigate the efficient algorithms for the operations of the fields and the rings.
In this paper we have proposed two efficient algorithms of the modular calculations in polynomial rings over GF(2) and integer rings. Firstly, we shall discuss the fast algorithm, which has been proposed by Kashiwagi and Moriuchi in 1982, for the modular calculations in polynomial rings over GF(2). Unfortunately their algorithm cannot be fast enough unless the dividend is the sparse polynomial. We shall modify their algorithm so that it can be efficient in almost every case. Secondly, we shall show that the modified algorithm can be applied to the modular calculations in interger rings. Our algorithm is useful when enciphering and deciphering the practical RSA public-key cryptosystem using computer.
