AbstractThe multiplicative complexity of an Sbox over a finite field is the minimum number of multiplications needed to implement the Sbox as an arithmetic circuit. In this paper we fully characterize bijective Sboxes with multiplicative complexity 1 up to affine equivalence over any finite field. We show that under affine equivalence in odd characteristic there are two classes of bijective functions and in even characteristic there are three classes of bijective functions with multiplicative complexity 1. Moreover, in (Jeon et al., Cryptogr. Commun., 14(4), 849874 (2022)) Aboxes where introduced to lower bound the differential uniformity of an Sbox over $$\mathbb {F}_{2}^{n}$$
F
2
n
via its multiplicative complexity. We generalize this concept to arbitrary finite fields. In particular, we show that the differential uniformity of a (n, m)Sbox over $$\mathbb {F}_{q}$$
F
q
is at least $$q^{n  l}$$
q
n

l
, where $$\lfloor \frac{n  1}{2} \rfloor + l$$
⌊
n

1
2
⌋
+
l
is the multiplicative complexity of the Sbox.