A connection between the boomerang uniformity and the extended differential in odd characteristic and applications
Mohit Pal, Pantelimon Stănică · Advances in Mathematics of Communications · 2025
This paper makes the first bridge between the classical differential/boomerang uniformity and the newly introduced $ c $-differential uniformity. We show that the boomerang uniformity of an odd APN function is given by the maximum of the entries (except for the first row/column) of the function's $ (-1) $-Difference Distribution Table. In fact, the boomerang uniformity of an odd permutation APN function equals its $ (-1) $-differential uniformity. We then use this connection to easily compute the boomerang uniformity of several odd APN functions. In the second part we give two classes of differentially low-uniform functions obtained by modifying the inverse function. The first class of permutations (CCZ-inequivalent to the inverse) over a finite field $ \mathbb F_{p^n} $ ($ p $, an odd prime) is obtained from the composition of the inverse function with an order-$ 3 $ cycle permutation, with differential uniformity $ 3 $ if $ p = 3 $ and $ n $ is odd; $ 5 $ if $ p = 13 $ and $ n $ is even; and $ 4 $ otherwise. The second class is a family of binomials and we show that their differential uniformity equals $ 4 $. Finally, we extend to odd characteristic a result of Charpin and Kyureghyan (2010) providing an upper bound for the differential uniformity of the function and its switched version via a trace function.