Almost All Vectorial Functions Have Trivial Extended-Affine Stabilizers

TL;DR

This paper proves that nearly all vectorial functions over finite fields have trivial extended-affine stabilizers, implying that most functions are unique up to EA-equivalence and supporting random sampling in cryptography.

Contribution

It establishes that asymptotically almost all vectorial functions have trivial EA-stabilizers and provides bounds on collision probabilities for equivalences.

Findings

Most vectorial functions have trivial EA-stabilizers.

Two independently sampled functions are EA-inequivalent with super-exponentially small probability.

Functions with nontrivial EA-stabilizers are exponentially rare.

Abstract

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.