Ubiquitous Weak-Key Classes of BRW-Polynomial Function

Abstract

BRW-polynomial function is suggested as a preferred alternative of polynomial function, owing to its high efficiency and seemingly non-existent weak keys. In this paper we investigate the weak-key issue of BRW-polynomial function as well as BRW-instantiated cryptographic schemes. Though, in BRW-polynomial evaluation, the relationship between coefficients and input blocks is indistinct, we give out a recursive algorithm to compute another \((2^{v+1}-1)\)-block message, for any given \((2^{v+1}-1)\)-block message, such that their output-differential through BRW-polynomial evaluation, equals any given s-degree polynomial, where \(v\ge \lfloor \log _2(s+1)\rfloor \). With such algorithm, we illustrate that any non-empty key subset is a weak-key class in BRW-polynomial function. Moreover any key subset of BRW-polynomial function, consisting of at least 2 keys, is a weak-key class in BRW-instantiated cryptographic schemes like the Wegman-Carter scheme, the UHF-then-PRF scheme, DCT, etc. Especially in the AE scheme DCT, its confidentiality, as well as its integrity, collapses totally, when using weak keys of BRW-polynomial function, which are ubiquitous.