In isogeny‐based cryptography, bilinear pairings are regarded as a powerful tool in various applications, including key compression, public key validation, and torsion basis generation. However, in most isogeny‐based protocols, the performance of pairing computations is unsatisfactory due to the high computational cost of the Miller function. Reducing the computational expense of the Miller function is crucial for enhancing the overall performance of pairing computations in isogeny‐based cryptography. This paper addresses this efficiency bottleneck. To achieve this, we propose several techniques for a better implementation of pairings in isogeny‐based cryptosystems. We use (modified) Jacobian coordinates and present new algorithms for Miller function computations to compute pairings of order 2 ∙ and 3 ∙ . For pairings of arbitrary order, which are crucial for key compression in some SIDH‐based schemes (such as M‐SIDH and binSIDH), we combine Miller doublings with Miller additions/subtractions, leading to a considerable speedup. Moreover, the optimizations for pairing applications in CSIDH‐based protocols are also considered in this paper. In particular, our approach for supersingularity verification in CSIDH is 15.3% faster than Doliskani’s test, which is the state‐of‐the‐art.

Isogeny-based cryptography is famous for its short key size. As one of the most compact digital signatures, SQIsign (Short Quaternion and Isogeny Signature) is attractive among post-quantum cryptography, but it is inefficient compared to other post-quantum competitors because of complicated procedures in the ideal-to-isogeny translation, which is the efficiency bottleneck of the signing phase. In this paper, we recall the current implementation of SQIsign and mainly focus on how to improve the execution of the ideal-to-isogeny translation in SQIsign. Specifically, we demonstrate how to utilize the reduced Tate pairing to save one of the two elliptic curve discrete logarithms. In addition, the efficient implementation of the remainder discrete logarithm computation is explored. We speed up other procedures in the ideal-to-isogeny translation with various techniques as well. It should be noted that our improvements also benefit the performance of key generation and verification in SQIsign. In the instantiation with$ {p_{1973}}$, the improvements lead to a speedup of 5.47%, 8.80% and 25.34% for key generation, signature and verification, respectively.

In recent years, the isogeny-based protocol, namely supersingular isogeny Diffie-Hellman (SIDH) has become highly attractive for its small public key size. In addition, one can utilize several techniques to further compress the public key. However, compared to other post-quantum protocols, the computational cost of SIDH is relatively high, and so is that of its public-key compression. On the other hand, the storage for pairing computation and discrete logarithms to speed up the current implementation of the key compression is somewhat large. In this paper, we mainly improve the performance of public-key compression of SIDH, especially the efficiency and the storage of pairing computation involved. Our experimental results show that the memory requirement for pairing computation is reduced by a factor of about 1.5. Meanwhile, the instantiation of public-key compression of SIDH is$6.95\%--10.44\%$faster than the current state-of-the-art. Although SIKE is broken now, the techniques in this paper may benefit other isogeny-based cryptosystems which are still secure.