Cyclic codes have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms. In this paper, for an odd prime p, the weight distributions of two classes of p-ary cyclic codes are completely determined. We show that both codes have at most five nonzero weights.

The Ate pairing has been suggested since it can be computed efficiently on ordinary elliptic curves with small values of the traces of Frobenius t. However, not all pairingfriendly elliptic curves have this property. In this paper, we generalize the Ate pairing and find a series of the variations of the Ate pairing. We show that the shortest Miller loop of the variations of the Ate pairing can possibly be as small as r 1/ϕ(k) on some special pairing-friendly curves with large values of Frobenius trace, and hence speed up the pairing computation significantly.

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Abstract. For AES 128 security level there are several natural choices for pairing-friendly elliptic curves. In particular, as we will explain, one might choose curves with k = 9 or curves with k = 12. The case k = 9has not been studied in the literature, and so it is not clear how efficiently pairings can be computed in that case. In this paper, we present efficient methods for the k = 9 case, including generation of elliptic curves with the shorter Miller loop, the denominator elimination and speed up of the final exponentiation. Then we compare the performance of these choices.From the analysis, we conclude that for pairing-based cryptography at the AES 128 security level, the Barreto-Naehrig curves are the most efficient choice, and the performance of the case k = 9 is comparable to the Barreto-Naehrig curves.

Scalar multiplication is the most important and expensive operation in elliptic curve cryptosystems. In this paper we improve the efficiency of the Elliptic Net algorithm to compute scalar multiplication by using the equivalence of elliptic nets. The proposed method saves f our multiplications in each iteration loop. Experimental results also indicates that our algorithm will be more efficient than the previously known results in this line.