Exposing SIMD Parallelism in SQIsign: An AVX-512 Implementation
Modern isogeny-based cryptosystems spend much of their running time in finite-field, elliptic-curve, and higher-dimensional isogeny arithmetic. Exploiting SIMD parallelism is nontrivial: routines such as Montgomery ladders contain loop-carried dependencies, while point, pairing, and theta-coordinate formulas expose only irregular fine-grained parallelism. We show that substantial SIMD parallelism can be recovered by reorganizing the arithmetic dependency graphs of higher-level primitives rather than vectorizing field multiplication in isolation. We develop an end-to-end AVX-512IFMA implementation of SQIsign in which data remain in a radix-$2^{51}$ vector representation across most of the curve-side computation. Our redesign includes projective xDBLADD schedules, batched point doubling in several coordinate systems, a vectorized biscalar ladder, fused cubical-arithmetic pairing steps, and batched one- and two-dimensional isogeny evaluation. Relative to the reference C implementation, we achieve end-to-end speedups of $1.76\times$, $1.71\times$, and $3.18\times$ for key generation, signing, and verification at NIST level~I; combined with Qlapoti, key-generation and signing speedups rise to $2.90\times$ and $2.69\times$. We further apply the same backend and methodology to CORAL, a recent isogeny group action for post-quantum non-interactive key exchange based on two-dimensional $2$-isogenies. Across five parameter sets, this yields $1.28$--$1.40\times$ speedups for key generation and $1.92$--$2.46\times$ for shared-key computation. These results provide cross-scheme evidence that algorithm-level SIMD scheduling is a reusable optimization dimension for higher-dimensional isogeny cryptography.