Upper bounds of differential branch number of $$n$$-bit permutations

Abstract The differential branch number is a key parameter used to measure the diffusion ability of a permutation. It is of great significance to study the upper bound of the differential branch number of permutations. This paper analyses the existence problem of binary codes with a specified dimension and minimum distance using combinatorial inequality. The result obtained is used to estimate the differential branch number of permutations on $${\mathbb{G}\mathbb{F}(2)}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> associated with those codes, and this paper finally finds two upper bounds, one of which is the tightest upper bound currently known. Furthermore, inspired by the comparison between two specific bounds mentioned above, the asymptotic upper bound of the differential branch number is studied. The conclusion quantitatively illustrates the relation between the upper bound and the range of n . It is shown that when n is sufficiently large, the differential branch number of n -bit permutations has an asymptotic upper bound of 0.44012 n . This is also the most accurate upper bound that can be proved using the scheme presented in this paper.