Further cryptographic properties of the multiplicative inverse function

Article Type

Research Article

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Discrete Applied Mathematics


Differential analysis is an important cryptanalytic technique on block ciphers. In one form, this measures the probability of occurrence of the differences between certain input vectors and the corresponding output vectors. For this analysis, the constituent S-boxes of a block cipher need to be studied carefully. In this direction, we derive further cryptographic properties of the multiplicative inverse function, especially the ones related to higher order differentials. This improves some theoretical results of Boukerrou et al. [ToSC 2020(1)]. Further, we prove that the multiplicative inverse function defined over F2n has an error (bias) in its second order differential spectrum with probability [Formula presented], and that error occurs in more than one place. Next, we analyze the Gowers uniformity norm of S-boxes, which is also a measure connected to higher order approximations. Finally, the bounds related to the nonlinearity profile of the multiplicative inverse function are derived using both Gowers U3 norm and Walsh–Hadamard spectrum. Some of our findings here provide slightly improved bounds over the work of Carlet [IEEE-IT, 2008]. These theoretical insights might have implications towards non-randomness of a block cipher, where the multiplicative inverse function is used as a primitive.

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