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On algebraic immunity of trace inverse functions on finite fields of characteristic two. (English) Zbl 1425.94058
Summary: The trace inverse functions \(\operatorname{Tr}(\lambda x^{-1})\) over the finite field \(\mathbb{F}_{2^n}\) are a class of very important Boolean functions and are used in many stream ciphers such as SFINKS, RAKAPOSHI, the simple counter stream cipher (SCSC) presented by W. Si and C. Ding [Cryptogr. Commun. 4, No. 2, 79–104 (2012; Zbl 1282.94065)], etc. In order to evaluate the security of those ciphers in resistance to (fast) algebraic attacks, the authors need to characterize algebraic properties of \(\operatorname{Tr}(\lambda x^{-1})\). However, currently only some bounds on algebraic immunity of \(\operatorname{Tr}(\lambda x^{-1})\) are given in the public literature, for example, the NGG upper bound and the Bayev lower bound, etc. This paper gives the exact value of the algebraic immunity of \(\operatorname{Tr}(\lambda x^{-1})\) over \({F_{{2^n}}}\), that is, \(\operatorname{AI}(\operatorname{Tr}(\lambda x^{-1}))=\lceil 2\sqrt{n}\rceil - 2\), where \(n\geq 2\), \(\lambda\in\mathbb{F}_{2^n}\) and \(\lambda\neq 0\), which shows that Dalai’s conjecture on the algebraic immunity of \(\operatorname{Tr}(\lambda x^{-1})\) is correct. What is more, the authors demonstrate some weak properties of \(\operatorname{Tr}(\lambda x^{-1})\) against fast algebraic attacks.
MSC:
94A60 Cryptography
06E30 Boolean functions
Software:
JBool; RAKAPOSHI; SNOW
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