Abstract
Boolean functions should possess high fast algebraic immunity when used in stream ciphers in order to stand up to fast algebraic attacks. However, in previous research, the fast algebraic immunity of Boolean functions was usually calculated by the computer. In 2017, Tang, Carlet, and Tang first mathematically proved that every function belonging to a class of 1-resilient Boolean functions has the fast algebraic immunity no less than n - 6. Inspired by the Tang's method, we also demonstrate that the fast algebraic immunity of another class of the 1-resilient Boolean functions is no less than n - 6. Meanwhile, we also prove some combinator facts originated from the Tu-Deng Conjecture.
Highlights
Boolean functions are the kernel components in some cryptosystems and their cryptographic properties directly determine the security of the cryptosystems
Inspired by Tang’s method, we prove that the fast algebraic immunity (FAI) of the class of 1-resilient Boolean functions proposed by Tang in [27] is no less than n − 6
This paper demonstrates that a class of 1-resilient Boolean functions has FAI greater than or equal to n − 6
Summary
Boolean functions are the kernel components in some cryptosystems and their cryptographic properties directly determine the security of the cryptosystems. Optimal AI and high FAI are attributes that Boolean functions should have in order to resist (fast) algebraic attacks. As everyone knows, it is very difficult to calculate the FAI of given Boolean function with high algebraic degree when the variable is larger than 18 [25]. In 2013, Tang et al proposed two classes of Boolean functions, called T-C-T function, which have excellent cryptographic properties [26] They are not 1-resilient which represents a disadvantage when they are used as a filter in stream ciphers.
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