Wireless sensor network(WSN) authentication schemes are the basic necessary security tool for protecting the communication authenticity and integrity in the network. In this paper we propose general authentication schemes from linear error-correcting codes and rational functions. There are very many different random code and function based authentication schemes with the same security functionality and cost. When these different and random code and function based authentication schemes are used for various parts of one large scale wireless sensor network, the resulted WSN authentication schemes constructed in this paper can resist the compromising of large number of sensor nodes for this large scale wireless sensor network with moderate communication and storage cost. The verification per node in our novel WSN authentication schemes is very fast. Thus WSN authentication schemes based on codes and rational functions can be used to resist the pollution and DoS attacks from the adversary.

The linear complexity of a periodic sequence over ${\hbox{GF}}(p^m)$ plays an important role in cryptography and communication (see Menezes, van Oorschort, and Vanstone, Handbook of Applied Cryptography. Boca Raton, FL: CRC, 1997 ). In this correspondence, we prove a result which reduces the computation of the linear complexity and minimal connection polynomial of a period un sequence over ${\hbox{GF}}(p^m)$ to the computation of the linear complexities and minimal connection polynomials of $u$ period $n$ sequences. The conditions $u\,\vert\,p^m-1$ and ${\rm gcd}(n,p^m-1)=1$ are required for the result to hold. Some applications of this reduction in fast algorithms to determine the linear complexities and minimal connection polynomials of sequences over ${\hbox{GF}}(p^m)$ are presented.

We prove a result which reduces the computation of the linear complexity of a sequence over GF(pm) (p is an odd prime) with period 2n (n is a positive integer such that there exists an element bisinGF(pm), bn=-1) to the computation of the linear complexities of two sequences with period n. By combining with some known algorithms such as the Berlekamp-Massey algorithm and the Games-Chan algorithm we can determine the linear complexity of any sequence over GF(pm) with period 2tn (such that 2t|pm-1 and gcd(n,pm-1)=1) more efficiently

We apply Steane's enlargement of the Calderbank-Shor-Steane (CSS) codes and additive codes over F/sub 4/ to concatenated algebraic-geometric codes to construct many good quantum codes with fewer restrictions on the parameters compared to some known quantum codes. Some of the quantum codes we have constructed are either optimal or have parameters as good as the best known codes, while some have parameters better than those obtained from other known constructions.