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2015-05-01
Baofeng Wu, Qingfang Jin, Zhuojun Liu, Dongdai Lin.  2014.  Constructing Boolean functions with potentially optimal algebraic immunity based on additive decompositions of finite fields (extended abstract). Information Theory (ISIT), 2014 IEEE International Symposium on. :1361-1365.

We propose a general approach to construct cryptographic significant Boolean functions of (r + 1)m variables based on the additive decomposition F2rm × F2m of the finite field F2(r+1)m, where r ≥ 1 is odd and m ≥ 3. A class of unbalanced functions is constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case r = 1. Functions belonging to this class have high algebraic degree, but their algebraic immunity does not exceed m, which is impossible to be optimal when r > 1. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks.

2015-04-30
Baofeng Wu, Qingfang Jin, Zhuojun Liu, Dongdai Lin.  2014.  Constructing Boolean functions with potentially optimal algebraic immunity based on additive decompositions of finite fields (extended abstract). Information Theory (ISIT), 2014 IEEE International Symposium on. :1361-1365.

We propose a general approach to construct cryptographic significant Boolean functions of (r + 1)m variables based on the additive decomposition F2rm × F2m of the finite field F2(r+1)m, where r ≥ 1 is odd and m ≥ 3. A class of unbalanced functions is constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case r = 1. Functions belonging to this class have high algebraic degree, but their algebraic immunity does not exceed m, which is impossible to be optimal when r > 1. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks.