Regarding Classification of n-sharing of Multivariate Mappings over Finite Fields and One NSUCrypto'2019 Olympiad Problem

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A problem of great importance that arises in designing and implementation of a cryptosystem is countering side channel attacks. Often an appropriate mathematical algorithm, implemented on a specific physical device to work in the physical environment, becomes vulnerable to such attacks.The “function sharing” technique is a prospective and efficient way to avoid this problem. In the paper we investigate “non-complete sharing” of Boolean functions and mappings, and functions and mappings over finite fields and provide a complete description of the set of functions with n variables, which have sharing.The main findings are the following: introducing and investigating a new concept of “weak” non-complete n-sharing, establishing its connection with “weak” and “classical” n-sharing, and substantiating its advantages from the algebraic point-of-view as well as establishing and proving a criterion for the existence of weak non-complete n-sharing for an arbitrary function. The results also include an explicit description of a set of functions which have weak sharing in terms of algebraic normal form, obtaining the precise and simple descriptions for the boundary (“border”) cases: n = 2, n=m and binary fields. Applying these results to the AES S-box allows complete solving the problem, i.e. a complete answer to the question of a representability of the S-box of the AES cipher as a sharing is available. We believe that the same way can be successful for other cryptographic algorithms.