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# A connection between the boomerang uniformity and the extended differential in odd characteristic and applications

Author:
Mohit Pal, Pantelimon Stanica
Keyword:
Computer Science, Information Theory, Information Theory (cs.IT), Discrete Mathematics (cs.DM), Number Theory (math.NT)
journal:
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date:
2023-12-03 00:00:00
Abstract
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 next apply this result 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\$. We next complete the open case of \$p=3\$ in the investigation started by G\" olo\u glu and McGuire (2014), for \$p\geq 5\$, and continued by K\"olsch (2021), for \$p=2\$, \$n\geq 5\$, on the characterization of \$L_1(X^{p^n-2})+L_2(X)\$ (with linearized \$L_1,L_2\$) being a permutation polynomial. 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.