In this paper, linear block codes over finite fields of the algebraic integer ring of the cyclotomic field Q(eLGL) modulo irreducible elements are presented, where n"5, 8, or 12. These codes can be used for coding over four-dimensional signal space and can correct one error which takes from the group of all roots of unity in the algebraic integer ring of Q(eLGL), where n"5, 8, or 12.1999 Academic Press

Abstract. Wang et al. [Z.Wang, L.Wang, S.Zheng, Y.Yang and Z.Hu, International Journal of Network Security, 14(2012)33-38] proposed an identity-based signature scheme based on cubic residues. But the scheme is insecure because it cannot withstand a conspiracy attack by users. To overcome this security vulnerability, this paper proposes a novel method to compute a cubic root of a cubic residue. Then a modified identity-based signature scheme based on cubic residues is proposed. The security analysis results show that the modified scheme can resist the conspiracy attack.

In order to improve the efficiency of certificate-based signature scheme, a new certificate-based signature scheme based on cubic residue is proposed. The scheme does not need any bilinear pairing computation which is known to be difficult to computation. The scheme is secure against existing forgery on the adaptively chosen message and identity attack under assumption of the hardness of integer factorization.

Abstract.It is well known that to construct the generating polynomials of higher power residue codes over finite fields is difficult. This paper gives explicit expressions of generating idempotents of quintic residue codes over the binary field. Using the result obtained, one can construct the generating polynomials of quintic residue codes over the binary field by computing the greatest common divisors of these generating idempotents and the polynomial 1 n x  with computer software such as Matlab and Maple.

Abstract. Higher power residue codes over finite fields are generated by factors of the polynomial 1 n x − . Unfortunately, to decompose the polynomial 1 n x − over finite fields is difficult. Generating idempotents can also generate higher power residue codes. Thus it is important to get generating idempotents of cyclic codes. We find precise expressions of generating idempotents of some sixth residue codes of length over the binary field, where p is a prime such that ( ) 1 mod 24 p ≡ .