This paper presents four optimizations for faster hashing onto elliptic curves. One optimization is for elliptic curves without non-trivial automorphisms when $ q \equiv 2 \mod{3} $. Another is for $ q \equiv 2, 4 \mod{7} $ and an elliptic curve with j-invariant $ -3^3 5^3 $, though this curve is not used in real-world cryptography. The other two optimizations are for subgroups $ G_1 $, $ G_2 $ of pairing-friendly curves. The performance gain comes from fewer exponentiations in $ F_q $ and the absence of the need to hash directly onto $ G_1 $ in certain settings. This allows for faster verification of aggregate BLS signatures in blockchain technologies. The results apply to pairing-friendly curves like BLS12-381 and some NIST curves. The paper also discusses hashing over highly 2-adic fields. The paper explains how to hash onto $ G_2 $ more efficiently and why sometimes hashing directly onto $ G_1 $ is not needed. Clearing the cofactor $ c_2 $ can significantly improve performance. In some cases, clearing the cofactor $ c_1 $ can be avoided. Optimal ate pairings can be extended to $ \mathbb{G}_2 \times E_1(\mathbb{F}_q) $ in certain scenarios. BLS12-381 is a de facto standard in pairing-based cryptography. The paper discusses hash functions to elliptic curves, which are typically compositions of hash functions, encodings, and scalar multiplications. The most complex part is the encoding. Most pairing-based protocols require a hash function to at most one group $ G_1 $ or $ G_2 $. The choice between the two depends on the required compactness of the hash value. Some protocols require both hash functions. Hashing to $ G_2 $ is more cumbersome than hashing to $ G_1 $, so it cannot always be exchanged. The paper also discusses how not to hash onto $ G_1 $, as optimal ate pairings are often used in real-world cryptography. The Miller loop length depends only on the order of $ G_2 $. Multiplying by a cofactor can protect against small-subgroup attacks. The paper also discusses the importance of subgroup security and the use of cofactors in protocols like the Diffie-Hellman key exchange. The paper concludes that the optimizations presented can significantly improve the efficiency of hashing onto elliptic curves in various cryptographic applications.This paper presents four optimizations for faster hashing onto elliptic curves. One optimization is for elliptic curves without non-trivial automorphisms when $ q \equiv 2 \mod{3} $. Another is for $ q \equiv 2, 4 \mod{7} $ and an elliptic curve with j-invariant $ -3^3 5^3 $, though this curve is not used in real-world cryptography. The other two optimizations are for subgroups $ G_1 $, $ G_2 $ of pairing-friendly curves. The performance gain comes from fewer exponentiations in $ F_q $ and the absence of the need to hash directly onto $ G_1 $ in certain settings. This allows for faster verification of aggregate BLS signatures in blockchain technologies. The results apply to pairing-friendly curves like BLS12-381 and some NIST curves. The paper also discusses hashing over highly 2-adic fields. The paper explains how to hash onto $ G_2 $ more efficiently and why sometimes hashing directly onto $ G_1 $ is not needed. Clearing the cofactor $ c_2 $ can significantly improve performance. In some cases, clearing the cofactor $ c_1 $ can be avoided. Optimal ate pairings can be extended to $ \mathbb{G}_2 \times E_1(\mathbb{F}_q) $ in certain scenarios. BLS12-381 is a de facto standard in pairing-based cryptography. The paper discusses hash functions to elliptic curves, which are typically compositions of hash functions, encodings, and scalar multiplications. The most complex part is the encoding. Most pairing-based protocols require a hash function to at most one group $ G_1 $ or $ G_2 $. The choice between the two depends on the required compactness of the hash value. Some protocols require both hash functions. Hashing to $ G_2 $ is more cumbersome than hashing to $ G_1 $, so it cannot always be exchanged. The paper also discusses how not to hash onto $ G_1 $, as optimal ate pairings are often used in real-world cryptography. The Miller loop length depends only on the order of $ G_2 $. Multiplying by a cofactor can protect against small-subgroup attacks. The paper also discusses the importance of subgroup security and the use of cofactors in protocols like the Diffie-Hellman key exchange. The paper concludes that the optimizations presented can significantly improve the efficiency of hashing onto elliptic curves in various cryptographic applications.