In this paper, we compare two generalized cyclotomic binary sequences with length 2p2 in terms of the linear complexity. One classical sequence is defined using the method introduced by Ding and Helleseth, while the other modified sequence is defined in a slightly different manner. We show that the modified sequence has linear complexity of 2p2, which is higher than that of the classical one.

Pairing computations on elliptic curves with odd prime degrees are rarely studied as low efficiency. Recently, Clarisse, Duquesne and Sanders proposed two new curves with odd prime embedding degrees: BW13-P310 and BW19-P286, which are suitable for some special cryptographic schemes. In this paper, we propose efficient methods to compute the optimal ate pairing on this types of curves, instantiated by the BW13-P310 curve. We first extend the technique of lazy reduction into the finite field arithmetic. Then, we present a new method to execute Miller's algorithm. Compared with the standard Miller iteration formulas, the new ones provide a more efficient software implementation of pairing computations. At last, we also give a fast formula to perform the final exponentiation. Our implementation results indicate that it can be computed efficiently, while it is slower than that over the (BLS12-P446) curve at the same security level.

Supersingular isogeny Diffie-Hellman (SIDH) is attractive for its relatively small public key size, but it is still unsatisfactory due to its efficiency, compared to other post-quantum proposals. In this paper, we focus on the performance of SIDH when the starting curve is E6 : y2 = x3 + 6x2 + x, which is fixed in Round-3 SIKE implementation. Inspired by previous works [1], [2], we present several tricks to accelerate key generation of SIDH and each process of SIKE. Our experimental results show that the performance of this work is at least 6.09% faster than that of the SIKE implementation, and we can further improve the performance when large storage is available.

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=9 has 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.

In this paper, we suggest that all pairings are in a group from an abstract angle. Based on the results, some new pairings with the short Miller loop are constructed for great efficiency. It is possible that our observation can be applied into other aspects of pairing-based cryptosystems.