Meicheng Liu

Perfect algebraic immune functions

A perfect algebraic immune function is a Boolean function with perfect immunity against algebraic and fast algebraic attacks. The main results are that for a perfect algebraic immune balanced function the number of input variables is one more than a power of two; for a perfect algebraic immune unbalanced function the number of input variables is a power of two. Also, for n equal to a power of two, the Carlet-Feng functions on n+1 variables and the modified Carlet-Feng functions on n variables are shown to be perfect algebraic immune functions.

Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions

In this correspondence, first we give a decomposition of symmetric Boolean functions, then we show that almost all symmetric Boolean functions, including these functions with good algebraic immunity, behave badly against fast algebraic attacks. Besides, we improve the relations between algebraic degree and algebraic immunity of symmetric Boolean functions.

Identification and construction of Boolean functions with maximum algebraic immunity

AbstractBoolean functions with maximum algebraic immunity have been considered as one class of cryptographically significant functions. It is known that Boolean functions on odd variables have maximum algebraic immunity if and only if a correlative matrix has column full rank, and Boolean functions on even variables have maximum algebraic immunity if and only if two correlative matrices have column full rank. Recently, a smaller matrix was used in the odd case. We find that one or two smaller matrices can be used in the even case and consequently present several sufficient and necessary conditions for Boolean functions with maximum algebraic immunity. This result advances the ability to identify whether Boolean functions on even variables achieve maximum algebraic immunity. We also present a construction algorithm for n-variable Boolean functions with maximum algebraic immunity, specially with the Hamming weights of

Almost perfect algebraic immune functions with good nonlinearity

\sum {_{i = 0}^{\left\lceil {\frac{n} {2}} \right\rceil - 1} } \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)

Degree Evaluation of NFSR-Based Cryptosystems

. It is easily realized for not too large n and helps construct balanced Boolean functions with maximum algebraic immunity on even variables. Furthermore, we present a sufficient and necessary condition for balanced Boolean functions to achieve maximum algebraic immunity and optimum algebraic degree, and modify the construction algorithm to construct Boolean functions on odd variables with maximum algebraic immunity, optimum algebraic degree and high nonlinearity.

New Insights on AES-Like SPN Ciphers

In this paper, we analyze the security of round-reduced versions of the Keccak hash function family. Based on the work pioneered by Aumasson and Meier, and Dinur et al., we formalize and develop a technique named linear structure, which allows linearization of the underlying permutation of Keccak for upi¾?to 3 rounds with large number of variable spaces. As a direct application, it extends the best zero-sum distinguishers by 2 rounds without increasing the complexities. We also apply linear structures to preimage attacks against Keccak. By carefully studying the properties of the underlying Sbox, we show bilinear structures and find ways to convert the information on the output bits to linear functions on input bits. These findings, combined with linear structures, lead us to preimage attacks against upi¾?to 4-round Keccak with reduced complexities. An interesting feature of such preimage attacks is low complexities for small variants. As extreme examples, we can now find preimages of 3-round SHAKE128 with complexity 1, as well as the first practical solutions to two 3-round instances of Keccak challenge. Both zero-sum distinguishers and preimage attacks are verified by implementations. It is noted that the attacks here are still far from threatening the security of the full 24-round Keccak.

Lightweight MDS Generalized Circulant Matrices

In this paper, it is proven that a family of 2k-variable Boolean functions, including the function recently constructed by Tang et al. [IEEE TIT 59(1): 653-664, 2013], are almost perfect algebraic immune for any integer k ≥ 3. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. A lower bound on their nonlinearity is obtained based on the work of Tang et al., which is better than that of Carlet-Feng function. It is also checked for 3 ≤ k ≤ 9 that the exact nonlinearity of such functions is very good, which is slightly smaller than that of Carlet-Feng function, and some functions of this family even have a slightly larger nonlinearity than Tang et al.s function. To sum up, among the known functions with provable good immunity against fast algebraic attacks, the functions of this family make a trade-off between the exact value and the lower bound of nonlinearity.

Searching cubes for testing Boolean functions and its application to Trivium

Boolean functions with large algebraic immunity resist algebraic attacks to a certain degree, but they may not resist fast algebraic attacks (FAAs). It is necessary to study the resistance of Boolean functions against FAAs. In this paper, we localize the optimal resistance of Boolean functions against FAAs and introduce the concept of e-fast algebraic immunity (e-FAI) for n-variable Boolean functions against FAAs, where e is a positive integer and