The c-differential uniformity of the perturbed inverse function via a trace function $$ {{\,\textrm{Tr}\,}}\big (\frac{x^2}{x+1}\big )$$

Abstract

Differential uniformity is one of the most crucial concepts in cryptography. Recently Ellingsen et al. (IEEE Trans Inf Theory 66:5781–5789, 2020) introduced a new concept, the c-Difference Distribution Table and the c-differential uniformity, by extending the usual differential notion. The motivation behind this new concept is based on having the ability to resist some known differential attacks which is shown by Borisov et. al. (Multiplicative Differentials, 2002). In 2022, Hasan et al. (IEEE Trans Inf Theory 68:679–691, 2022) gave an upper bound of the c-differential uniformity of the perturbed inverse function H via a trace function \( {{\,\textrm{Tr}\,}}\big (\frac{x^2}{x+1}\big )\). In their work, they also presented an open question on the exact c-differential uniformity of H. By using a new method based on algebraic curves over finite fields, we solve the open question in the case \( {{\,\textrm{Tr}\,}}(c)=1= {{\,\textrm{Tr}\,}}(\frac{1}{c})\) for \( c \in {\mathbb {F}}_{2^n}\setminus \{0,1\} \) completely and we show that the exact c-differential uniformity of H is 8. In the remaining case, we almost completely solve the problem and we show that the c-differential uniformity of H is either 8 or 9.