In the last decade, the introduction of advanced cryptographic protocols operating on large finite fields \(\displaystyle \mathbb {F}_{q}\) has raised the need for efficient cryptographic primitives in this setting, commonly referred to as Arithmetization-Oriented (AO). The cryptanalysis of AO hash functions is essentially done through the study of the CICO problem on the underlying permutation. On several AO hash functions relying on S-boxes with low degree inverse, two recent works at Crypto 2024 and Asiacrypt 2024 managed to solve the CICO problem much more efficiently than traditional Gröbner basis methods, using respectively advanced Gröbner basis techniques and resultants. In this paper, we propose an attack framework based on resultants that applies to a wide range of AO permutations and improves significantly upon these two recent works. Our improvements mainly come from an efficient reduction procedure that we propose and rigorously analyze, taking advantage of fast multivariate multiplication. We present the most efficient attacks on Griffin, Arion, Anemoi, and Rescue. We show that most variants of Griffin, Arion and Anemoi fail to reach the claimed security level. For the first time, we successfully break a parameter set of Rescue, namely its 512-bit security variant. The presented theory and complexity estimates are backed up with experimental attacks. Notably, we practically find CICO solutions for 8 out of 10 rounds of Griffin, 11 out of 21 rounds of Anemoi, 6 out of 18 rounds of Rescue, improving by respectively 1, 3 and 1 rounds on the previous best practical attacks.