We prove that asymptotically almost all vectorial functions over finite fields have trivial extended-affine stabilizers. As a consequence, the number of EA-equivalence classes is asymptotically equal to the naive estimate, namely the total number of functions divided by the size of the EA-group, with vanishing relative error. Furthermore, we derive upper bounds on collision probabilities for both extended-affine and CCZ equivalences. For EA-equivalence, we leverage the trivial-stabilizer result to establish a matching lower bound, yielding a tight asymptotic formula that shows two independently sampled functions are EA-equivalent with super-exponentially small probability. The results validate random sampling strategies for cryptographic primitive design and show that functions with nontrivial EA-stabilizers form an exponentially rare subset.

翻译：我们证明了在有限域上，渐近几乎所有向量值函数都具有平凡的扩展仿射稳定子。因此，EA等价类的数量渐近地等于朴素估计值，即函数总数除以EA-群的规模，且相对误差趋近于零。此外，我们推导了扩展仿射等价和CCZ等价中碰撞概率的上界。对于EA等价，我们利用平凡稳定子结果建立了匹配的下界，从而得到一个紧的渐近公式，表明两个独立采样的函数以超指数小的概率是EA等价的。这些结果验证了密码原语设计中的随机采样策略，并表明具有非平凡EA稳定子的函数构成一个指数稀疏的子集。